Amal Attouchi

Philippe Souplet

Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with p-Laplacian diffusion

We study the initial-boundary value problem for the Hamilton-Jacobi equation with nonlinear diffusion u t = ∆ p u + |∇u| q in a two-dimensional domain for q > p > 2. It is known that the spatial derivative of solutions may become unbounded in finite time while the solutions themselves remain bounded. We show that, for suitably localized and monotone initial data, the gradient blow-up occurs at a single point of the boundary. Such a result was known up to now only in the case of linear diffusion (p = 2). The analysis in the case p > 2 is considerably more delicate.

Introduction 1.Problem and main result

This article is a contribution to the study of the influence of nonlinear diffusion on the qualitative properties of equations of Hamilton-Jacobi type and, in particular, on the formation of finite-time singularity. More specifically, we consider the following problem

     u t = ∆ p u + |∇u| q ,
(x, y) ∈ Ω, t > 0, u(x, y, t) = µy, (x, y) ∈ ∂Ω, t > 0, u(x, y, 0) = u 0 (x, y), (x, y) ∈ Ω, (1.1) where ∆ p denotes the p-Laplace operator, ∆ p = ∇ • (|∇u| p-2 ∇u). Throughout this paper, we assume that µ ≥ 0 is a constant and that q > p > 2.

(1.2)

For reasons that will appear later, we restrict ourselves to a class of planar domains Ω which satisfy certain geometric properties. We assume that, for some L 1 , L 2 > 0, Ω ⊂ R 2 is a smooth bounded domain of class C 2+ǫ for some ǫ ∈ (0, 1); (1.3) Ω is symmetric with respect to the axis x = 0;

(1.4) Ω ⊂ {y > 0} and Ω contains the rectangle (-L 1 , L 1 ) × (0, 2L 2 );

(1.5) Ω is convex in the x-direction.

(

In particular, by (1.5), ∂Ω has a flat part, centered at the origin (0, 0). Note that assumption (1.6) is equivalent to the fact that Ω ∩ {y = y 0 } is a line segment for each y 0 . The initial data u 0 is taken in V µ , where

V µ := u 0 ∈ C 1 (Ω), u 0 ≥ µy in Ω, u 0 = µy on ∂Ω .

We shall use the following notation throughout:

Ω + := {(x, y) ∈ Ω; x > 0} .
For T > 0, set Q T = Ω × (0, T ), S T = ∂Ω × (0, T ) and ∂ P Q T = S T ∪ (Ω × {0}) its parabolic boundary. Problem (1.1) is well posed locally in time (see Section 2 for details), with blow-up alternative in W 1,∞ norm. For brevity, when no confusion arises, the existence time of its maximal solution u will be denoted by

T := T max (u 0 ) ≤ ∞.
It is known (see [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]) that global nonexistence, i.e. T < ∞, occurs for suitably large initial data (more generally, for problem (1.1) in an n-dimensional bounded domain with Dirichlet boundary conditions). Note that the condition q > p is sharp, since the solutions are global and bounded in W 1,∞ if 1 < q ≤ p (see [START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF]). Since it follows easily from the maximum principle that u itself remains uniformly bounded on Q T , global nonexistence can only occur through gradient blow-up, namely sup

Q T |u| < ∞ and lim t→T ∇u(•, t) ∞ = ∞.
This is different from the usual blow-up, in which the L ∞ norm of the solution tends to infinity as t → T max , which occurs for equations with zero-order nonlinearities, such as u t = ∆ p u + u q (see [START_REF] Fujii | Asymptotic behavior of blowup solutions of a parabolic equation with the p-Laplacian[END_REF][START_REF] Galaktionov | Single point blow-up for N-dimensional quasilinear equations with gradient diffusion and source[END_REF]). The study of -L ∞ or gradient-blow-up singularities, in particular their location, time and spatial structure is very much of interest for the understanding of the physical problems modelled by such equation, as well as for the mathematical richness that they involve. The L ∞ blow-up for the equation u t = ∆ p u + u q has been extensively studied, both in the case of linear (p = 2) and nonlinear (p > 2) diffusion; see respectively the monographs [START_REF] Quittner | Superlinear parabolic problems: Blow-up, global existence and steady states[END_REF] and [START_REF] Samarskii | Blow-up in quasilinear parabolic equations[END_REF] and the numerous references therein.

As for equation (1.1) with p = q = 2, it is known as the (deterministic version of the) Kardar-Parisi-Zhang (KPZ) equation, describing the profile of a growing interface in certain physical models (see [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF]), where u then represents the height of the interface profile. The case of p = 2 and q ≥ 1 is a more general model which was developed by Krug and Spohn, aiming at studying the effect of the nonlinear gradient term on the properties of solutions (see [START_REF] Krug | Universality classes for deterministic surface growth[END_REF]). Our main interest in this paper is to study the effect of a quasilinear gradient diffusivity on the localization of the singularities.

For the case of linear diffusion p = 2, various sufficient conditions for gradient blowup and global existence were provided and qualitative properties were investigated, such as: nature of the blow-up set, rate and profile of blow-up, maximum existence time and continuation after blow-up, boundedness of global solutions and convergence to a stationary state. We refer for these to the works [START_REF] Arrieta | Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena[END_REF][START_REF] Guo | Blow-up rate estimates for the heat equation with a nonlinear gradient source term[END_REF][START_REF] Ph | Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem[END_REF][START_REF] Ph | Global solutions of inhomogeneous Hamilton-Jacobi equations[END_REF][START_REF] Quittner | Superlinear parabolic problems: Blow-up, global existence and steady states[END_REF][START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] and the references therein.

The case p > 2 is far from being completely understood and fewer works deal with the nonlinear diffusion. The large time behavior of global solutions in bounded or unbounded domains has been studied in [START_REF] Ph | Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption[END_REF][START_REF] Stinner | Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF][START_REF] Ph | Convergence to separate variables solutions for a degenerate parabolic equation with gradient source[END_REF][START_REF] Bartier | Gradient estimates for a degenerate parabolic equation with gradient absorption and applications[END_REF][START_REF] Barles | Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation[END_REF][START_REF] Ph | Non-diffusive large time behavior for a degenerate viscous Hamilton-Jacobi equation[END_REF]. Concerning the asymptotic description of singularities, results on the gradient blow-up rate in one space dimension can be found in [START_REF] Attouchi | Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term[END_REF][START_REF] Zhang | Boundedness of Global Solutions for a Heat Equation with Exponential Gradient Source[END_REF][START_REF] Zhang | Gradient blowup rate for a viscous Hamilton-Jacobi equation with degenerate diffusion[END_REF]. On the other hand, in any space dimension, it is known [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF] that gradient blow-up can take place only on the boundary, i.e.

GBUS(u 0 ) ⊂ ∂Ω,

where the gradient blow-up set is defined by GBUS(u 0 ) = x 0 ∈ Ω; for any ρ > 0, sup

(Ω∩Bρ(x 0 ))×(T -ρ,T ) |∇u| = ∞ .
Moreover, the following upper bounds for the space profile of the singularity were obtained in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]:

|∇u| ≤ Cδ -1 q-p+1
and u ≤ Cδ q-p q-p+1 in Q T , where δ(x) = dist(x, ∂Ω), (1.7) and they are sharp in one space dimension [START_REF] Attouchi | Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term[END_REF]. (Our nondegeneracy lemma 7.1 below indicates that they are also sharp in higher dimensions.)

It is easy to see, by considering radially symmetric solutions with Ω being a ball, that GBUS(u 0 ) can be the whole of ∂Ω. A natural question is then: Can one produce examples (in more than one space dimension) when GBUS(u 0 ) is a proper subset of ∂Ω, especially a single point ?

The goal of this article is to provide an affirmative answer to this question. Our main result is the following. (i) For any ρ ∈ (0, L 1 ), there exists µ 0 = µ 0 (p, q, Ω, ρ) > 0 such that, for any µ ∈ (0, µ 0 ], there exist initial data u 0 in V µ ∩ C 2 (Ω) such that the corresponding solution u of (1.1) enjoys the following properties:

T := T max (u 0 ) < ∞ and GBUS(u 0 ) ⊂ [-ρ, ρ] × {0}, (1.8)
u(•, t) is symmetric with respect to the line x = 0, for all t ∈ (0, T ), (1.9)

u x ≤ 0 in Ω + × (0, T ), (1.10)
u y ≥ µ/2 in Q T . (1.11)
(ii) For any such µ and u 0 , we have

GBUS(u 0 ) = {(0, 0)}.
A class of initial data satisfying the requirements of Theorem 1.1 is provided in Lemma 6.1 below. We note that, in the semilinear case p = 2, a single-point boundary gradient blow-up result was obtained in [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF]. Although we follow the same basic strategy, the proof here is considerably more complicated. We point out right away that, in view of property (1.11), the equation is not degenerate for the solutions under consideration. However, since the essential goal of this article is to study the effect of nonlinear diffusion on gradient blow-up, what is relevant here are the large values of the gradient in the diffusion operator (rather than the issues of loss of higher regularity that would arise from the degenerate nature of the equation near the level ∇u = 0). It is an open question whether or not single-point gradient blow-up can still be proved in the case µ = 0. Actually, the lower bound (1.11) on |∇u| is crucially used at various points of the proof, which is already very long and involved, due to the presence of the nonlinear -even though nondegenerate -diffusion term (see the next subsection for more details).

Section 2 is devoted to local in time well-posedness and regularity results. The proof of Theorem 1.1 will be split into several sections, namely sections 3-7 for assertion (i) and sections 8-10 for assertion (ii) (the latter uses also section 5). Finally, in two appendices, we provide the proofs of some regularity properties and a suitable parabolic version of Serrin's corner lemma. Since the proof is quite long and involved, for the convenience of readers, we now give an outline of the main steps of the proof.

Outline of proof

For sake of clarity, we have divided the proof into a number of intermediate steps, each of which being relatively short (one or two pages, say), except for step (f), which involves long and hard computations. For the convenience of readers, we outline the structure of the proof.

(a) Preliminary estimates (Lemmas 3.1-3.4): The symmetry in the variable x and the decreasing property for x > 0 are basic features in order to expect single-point gradient blow-up. Besides u being bounded, we also have boundedness of u t . Moreover, for sufficiently small µ and under a suitable assumption on u 0 , we show that u y ≥ µ/2. These bounds on u t and u y seem necessary in the very long calculations of the key step (f) below. In turn, the positivity of u y guarantees that solutions are actually classical and that D 2 u, D 3 u satisfy some bounds which seem also necessary to the argument, especially in view of the application of the Hopf lemma and the Serrin corner lemma in step (g).

(b) Finite time gradient blow-up for suitably concentrated initial data (Lemma 4.1): by using a rescaling argument and known blow-up criteria, we show that the solution blows up in finite time provided the initial data is suitably concentrated in a small ball near the origin.

(c) Local boundary gradient control (Lemma 5.1): if the gradient remains bounded on the boundary near a given boundary point, then the gradient remains also bounded near that point inside the domain, hence it is not a blow-up point. This is proved by a local Bernstein type argument.

(d) Localization of the gradient blow-up set (Lemma 5.3): if an initial data is suitably concentrated near the origin, then the gradient blow-up set is contained in a small neighborhood of the origin. This is proved by constructing comparison functions which provide a control of the gradient on the boundary outside a small neighborhood of the origin, and then applying step (c). One then constructs (Lemma 6.1) initial data which also fulfill the assumptions in (a) and (b). This ensures the existence of "well-prepared" initial data and thereby completes the proof of assertion (i) of Theorem 1.1.

(e) Nondegeneracy of gradient blow-up (Lemma 7.1): if the solution is only "weakly singular" in a neighborhood of a boundary point, then the singularity is removable. 1More precisely, we show the existence of m = m(p, q) ∈ (0, 1) such that, for a given point (x 0 , 0) on the flat part of ∂Ω, if u(x, y, t) ≤ c(x)y m near (x 0 , 0) for t close to T , then (x 0 , 0) is not a gradient blow-up point. In view of step (c), it suffices to control the gradient on the boundary near the point (x 0 , 0). This is achieved by constructing special comparison functions, taking the form of "regularizing (in time) barriers".

(f) Verification of a suitable parabolic inequality for an auxiliary function J, of the form J(x, y, t) = u x + kxy -γ u α .

(Proposition 8.1). This is the most technical step and gives rise to very long computations. Those computations make use, among many other things, of the singular, Bernstein-type boundary gradient estimate (1.7), obtained in [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF]. They use the bound on u t and the lower bound on u y , obtained at step (a) (it is not clear if the latter could be relaxed2 ). We note that a similar function J was introduced in [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] to treat the semilinear case p = 2. The function in [START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF] was a 2D-modification of a one-dimensional device from [START_REF] Friedman | Blow-up of positive solutions of semilinear heat equations[END_REF], used there to show single point L ∞ blow-up for radial solutions of equations of the form u t -∆ p u = u q (for p = 2, see also [START_REF] Galaktionov | Single point blow-up for N-dimensional quasilinear equations with gradient diffusion and source[END_REF] for p > 2). Although the ideas are related, the calculations here are considerably harder than in [START_REF] Friedman | Blow-up of positive solutions of semilinear heat equations[END_REF][START_REF] Galaktionov | Single point blow-up for N-dimensional quasilinear equations with gradient diffusion and source[END_REF][START_REF] Li | Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains[END_REF].

(g) Verification of initial-boundary conditions for the auxiliary function J in a small subrectangle near the origin. This requires a delicate parabolic version of Serrin's corner lemma, which we prove in Appendix 2 (see Proposition 11.1).

(h) Derivation of a weakly singular gradient estimate near the origin and conclusion. Steps (d) and (e) imply J ≤ 0 by the maximum principle. By integrating this inequality we obtain an inequality of the form u(x, y, t) ≤ c(x)y k as y → 0, for each small x = 0 and some k > m. In view of step (e), this shows that (0, 0) is the only gradient blow-up point.

Local well-posedness and regularity

In this section we consider the question of local existence and regularity for problem (1.1). Actually, we consider the slightly more general problem

u t -∆ p u = |∇v| q in Q T , (2.1)
u = g on S T , (2.2) u(•, 0) = u 0 in Ω, (2.3)
where the boundary data g and initial data u 0 satisfy: g ≥ 0 is the trace on ∂Ω of a regular function in C 2+γ (Ω) for some γ ∈ (0, 1) (2.4) and u 0 ∈ W 1,∞ (Ω), u 0 ≥ 0, u 0 = g on ∂Ω.

(2.5)

A function u is called a weak super-(sub-) solution of problem (2.1)-(2.3) on Q T if u(•, 0) ≥ (≤) u 0 in Ω, u ≥ (≤) g on S T , u ∈ C(Ω × [0, T )) ∩ L q (0, T ; W 1,q (Ω)), u t ∈ L 2 (0, T ; L 2 (Ω))
and the integral inequality

Q T u t ψ + |∇u| p-2 ∇u • ∇ψ dx dt ≥ (≤) Q T |∇u| q ψ dx dt holds for all ψ ∈ C 0 (Q T ) ∩ L p (0, T ; W 1,p (Ω)
) such that ψ ≥ 0 and ψ = 0 on S T . A function u is a weak solution of (2.1)-(2.3) if it is a super-solution and a sub-solution.

The following result was established in [2, Theorem 1.1] (actually in any space dimension).

Theorem 2.1. Assume (1.3) and q > p -1 > 1. Let M 1 > 0, let u 0 , g satisfy (2.4)-(2.5) and ∇u 0 ∞ ≤ M 1 . Then: (i) There exists a time T 0 = T 0 (p, q, Ω, M 1 , g C 2 ) > 0 and a weak solution u of (2.1)-(2.3) on [0, T 0 ), which moreover satisfies u ∈ L ∞ (0, T 0 ; W 1,∞ (Ω)). Furthermore, ∇u is locally Hölder continuous in Q T 0 .

(ii) For any τ > 0, the problem (2.1)-(2.3) has at most one weak solution u such that u ∈ L ∞ (0, τ ; W 1,∞ (Ω)).

(iii) There exists a (unique) maximal, weak solution of (2.1)-(2.3) in L ∞ loc ([0, T ); W 1,∞ (Ω)), still denoted by u, with existence time denoted by T = T max (u 0 ). Then

min Ω u 0 ≤ u ≤ max Ω u 0 in Q T , (2.6)
∇u is locally Hölder continuous in Q T and if T < ∞, then lim t→T ∇u(t) ∞ = ∞ (gradient blow up, GBU).

Remark 2.1. We also have a comparison principle for problem

(2.1)-(2.3), cf. [2, Propo- sition 2.1]. More precisely, if v 1 , v 2 ∈ C(Q T ) are weak sub-/super-solutions of (2.1)-(2.3) in Q T , then sup Q T (v 1 -v 2 ) ≤ sup ∂Q T (v 1 -v 2 ).
As one expects, the solution will possess additional regularity if we know that |∇u| remains bounded away from 0. This is made precise by the following result, which is a consequence of regularity theory [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF][START_REF] Lieberman | Second order parabolic differential equations[END_REF] for quasilinear uniformly parabolic equations. However, for completeness, we provide a proof in Appendix 1.

Theorem 2.2. Under the assumptions of Theorem 2.1, suppose also that inf Q T |∇u| > 0.

(i) Then u is a classical solution in Q T and

u ∈ C 2+α,1+α/2 loc (Ω × (0, T )). (2.7)
for some α ∈ (0, 1).

(ii) Moreover, ∇u ∈ C 2+β,1+β/2 loc (Ω × (0, T )). (2.8)
for some β ∈ (0, 1).

(iii) If the boundary conditions in (2.2) depend only on y, then

u x ∈ C 2+β,1+β/2 loc
(Ω × (0, T )).

(2.9)

for some β ∈ (0, 1).

3 Preliminary estimates: x-symmetry, lower bound on u y and bound on u t Notation. Throughout the paper, we shall use the summation convention on repeated indices, in expressions of the form a ij u ij or a ij u i u j . Also, the letter C will denote positive constants which may vary from line to line, and whose dependence will be indicated if necessary.

Lemma 3.1. Let µ > 0 and u 0 ∈ V µ . Assume u 0 is symmetric with respect to the line x = 0, (3.1)

∂ x u 0 ≤ 0 in Ω + .
(3.2)

Then we have u(x, y, t) ≥ µy in Q T . (3.3)
and properties (1.9)-(1.10) are satisfied.

Proof. Property (1.9) is a direct consequence of (3.1) and the local-in-time uniqueness. Due to the assumption u 0 ∈ V µ , v = µy is a subsolution of (1.1). This implies (3.3).

To prove (1.10), fix h > 0 and let

u ± = u(x ± h, y, t) for (x, y) ∈ Ω h := {(x, y) ∈ Ω + ; (x + h, y) ∈ Ω} and t ∈ (0, T ).
Owing to (1.4) and (1.6), we see that (x -h, y) ∈ Ω for all (x, y) ∈ Ω h , so that u -is well defined. The functions u ± are weak solutions of (1.1)

1 in Ω h × (0, T ). Also u + ≤ u -at t = 0, due to (3.1)-(3.
2) and (1.6). Let (x, y) ∈ ∂Ω h . If x = 0, then u + (x, y, t) = u(h, y, t) = u(-h, y, t) = u -(x, y, t) by (1.9). If x > 0, then (x + h, y) ∈ ∂Ω as a consequence of (x, y) ∈ ∂Ω h and (1.6). So, by (3.3), we have

u + (x, y, t) = u(x + h, y, t) = µy ≤ u -(x -h, y, t).
We deduce from the comparison principle that u + ≤ u -in Ω h × (0, T ), which implies (1.10).

Our next lemma provides a useful supersolution of problem (1.1).

Lemma 3.2. For 0 < ρ < L 1 , denote Σ ρ = [-ρ/2, ρ/2] × {0} and Σ ′ ρ = ∂Ω \ ((-ρ, ρ) × {0}).
There exist µ 0 = µ 0 (p, q, Ω, ρ) > 0 and a function U ∈ C 2 (Ω), depending on p, q, ρ, with the following properties:

U > 0 on Ω ∪ Σ ρ , (3.4) U = 0 on Σ ′ ρ , (3.5) |U y | ≤ 1/2 in Ω (3.6)
and, for all 0 < µ ≤ µ 0 , the function Ū = µ(y + U) satisfies

-∆ p Ū ≥ |∇ Ū| q in Ω. (3.7)
Proof. Fix a nonnegative function φ ∈ C 3 (R 2 ) such that φ = 1 on Σ ρ and φ = 0 on Σ ′ ρ . We shall look for U under the form U = εV , ε > 0, where V ∈ C 2 (Ω) is the classical solution of the linear elliptic problem

-V xx + (p -1)V yy = 1 in Ω, V = φ on ∂Ω. (3.8)
Note that V > 0 in Ω by the strong maximum principle which, along with the boundary conditions in (3.8), will guarantee (3.4)- (3.5). Let Ū = µ(y

+ U). Assume 0 < ε < 1/( V x ∞ + 2 V y ∞ ),
∆ p Ū = |∇ Ū| p-2 ∆ Ū + (p -2) Ūi Ūj Ūij |∇ Ū| 2 = µε|∇ Ū| p-2 V xx + V yy + (p -2) (εV x ) 2 V xx + 2(εV x )(1 + εV y )V xy + (1 + εV y ) 2 V yy (1 + εV y ) 2 + (εV x ) 2 = µε|∇ Ū| p-2 V xx + (p -1)V yy + (p -2) (εV x ) 2 (V xx -V yy ) + 2(εV x )(1 + εV y )V xy (1 + εV y ) 2 + (εV x ) 2 ≤ µε|∇ Ū| p-2 -1 + Cε 2 + Cε
in Ω, with C > 0 depending only on Ω, ρ, p (through V ). We may then choose ε depending only on Ω, ρ, p, such that -∆ p Ū ≥ µε 2 |∇ Ū| p-2 . Next, since q > p, for any µ ≤ µ 0 with µ 0 = µ 0 (p, q, Ω, ρ) > 0 sufficiently small, we have

-∆ p Ū ≥ µε 2 |∇ Ū | p-2 ≥ µε 2(2µ) q+2-p |∇ Ū| q ≥ |∇ Ū| q in Ω.
Based on Lemma 3.2, we construct a class of solutions such that u y satisfies a positive lower bound. Lemma 3.3. Let 0 < ρ < L 1 and let µ 0 , Ū be given by Lemma 3.2. Assume that 0 < µ ≤ µ 0 and u 0 ∈ V µ satisfy

u 0 (x, y) ≤ µ y + cχ (-ρ/2,ρ/2)×(0,L 2 )
in Ω, (3.9)

with c = c(p, q, Ω, ρ) > 0 sufficiently small. (i) Then u ≤ Ū in Q T . (ii) Assume in addition that ∂ y u 0 ≥ µ/2 in Ω. (3.10) Then ∂ y u ≥ µ/2 in Q T . (3.11)
Proof. (i) Let Ū = µ(y + U) be given by Lemma 3.2. From (3.4), we know that

c := min [-ρ/2,ρ/2]×[0,L 2 ] U > 0.
Under assumption (3.9), we thus have u 0 ≤ Ū in Ω. Since u = µy ≤ Ū on S T , we infer from the comparison principle that u ≤ Ū in Q T .

(ii) Set δ 0 = µ/2, fix h > 0 and let Ωh = {(x, y) ∈ Ω; (x, y + h) ∈ Ω}. We observe that u 1 := u(x, y, t) + δ 0 h and u 2 := u(x, y + h, t)

are weak solutions of (1.1)

1 in Ωh × (0, T ). Let (x, y) ∈ ∂ Ωh . If (x, y) ∈ ∂Ω, then, by (3.
3), we have

u 2 (x, y, t) = u(x, y + h, t) ≥ µ(y + h) = u(x, y, t) + µh ≥ u 1 (x, y, t).
Otherwise, we have (x, y) ∈ Ω and (x, y + h) ∈ ∂Ω. So there is a minimal h ∈ (0, h] such that (x, y + h) ∈ ∂Ω. By the mean-value inequality, it follows that, for some θ ∈ (0, 1),

U(x, y) = U(x, y + h) -hU y (x, y + θ h) ≤ |U y (x, y + θ h)|h ≤ h/2,
where we used (3.6). Therefore, Ū (x, y) ≤ µ(y + h/2), hence

u(x, y + h, t) -u(x, y, t) ≥ µ(y + h) -Ū (x, y) ≥ µh 2 .
We have thus proved that

u 2 ≥ u 1 on ∂ Ωh . (3.12)
On the other hand, using (3.10) and the fact that u 0 = µy on ∂Ω, it is not difficult to show that y → u 0 (x, y) -δ 0 y is nondecreasing in Ω. (Note that in case Ω is nonconvex, this is not a mere consequence of (3.10) alone). It follows that u(x, y + h, 0) ≥ u(x, y, 0) + δ 0 h in Ωh .

(3.13)

Owing to (3.12)-(3.13), we may then apply the comparison principle to deduce that u 2 ≥ u 1 in Ωh × (0, T ). Since h is arbitrary, the desired conclusion (3.11) follows immediately.

Assuming that u 0 is sufficiently regular, we also get an estimate on the time derivative.

Lemma 3.4. Let µ ≥ 0 and assume that u 0 ∈ V µ ∩ C 2 (Ω). Then

|u t | ≤ C1 := ∆ p u 0 + |∇u 0 | q ∞ in Q T . (3.14)
Proof. It is easy to see that v ± (x, y, t) := u 0 (x, y) ± C1 t are respectively super-and sub-solution of (1.1) in Q T . The comparison principle implies that

u 0 (x, y) -C1 t ≤ u(x, y, t) ≤ u 0 (x, y) + C1 t in Q T . (3.15)
Now fix h ∈ (0, T ) and set w ± (x, y, t) := u(x, y, t + h) ± C1 h. By (3.15), we have w -(x, y, 0) ≤ u 0 (x, y) ≤ w + (x, y, 0) and it follows that w ± are respectively super-and sub-solution of (1.1) in Q T -h . By a further application of the comparison principle, we deduce that |u(x, y, t

+ h) -u(x, y, t)| ≤ C1 h in Q T -h .
Since h is arbitrary, we conclude by dividing by h and sending h → 0.

Finite-time gradient blow-up for concentrated initial data

In this section, by a rescaling argument, we show that the solution of (1.1) blows up in finite time provided the initial data is suitably concentrated in a small ball near the origin. For such concentrated initial data, under some additional assumptions, we will show in section 6 that the gradient blow-up set is contained in a small neighborhood of the origin.

The following lemma shows in particular that gradient blow-up may occur for initial data of arbitrarily small L ∞ -norm (but the W 1,∞ norm has to be sufficiently large). We note that we do not assume (3.10) here, so that in the proof, we work only with weak solutions (in particular, we cannot use the continuity of ∇u up to the boundary).

Lemma 4.1. Let κ = (q -p)/(q -p + 1). There exists

C 1 = C 1 (p, q) > 0 such that, if ε ∈ (0, min(L 1 , L 2 )), µ ≥ 0 and u 0 ∈ V µ satisfies u 0 (x, y) ≥ C 1 ε κ in B ε/3 (0, ε) ⊂ Ω, (4.1) then T max (u 0 ) < ∞. Proof. We denote B r = B r (0, 0) ⊂ R 2 for r > 0. Fix a radially symmetric function h ∈ C ∞ 0 (B 1 ), h ≥ 0, such that supp(h) ⊂ B 1/3 and h ∞ = 1. We consider the following problem    v t -∆ p v = |∇v| q , x ∈ B 1 , t > 0 v(x, y, t) = 0, x ∈ ∂B 1 , t > 0 v(x, y, 0) = v 0 (x, y) := C 1 h(x, y), x ∈ B 1 . (4.2)
We know from [2, Theorem 1.4] that there exists C 0 = C 0 (p, q) > 0 such that if v 0 L 1 ≥ C 0 , then T max (v 0 ) < ∞. Therefore, if we take C 1 = C 1 (p, q) > 0 large enough then ∇v blows up in finite time in L ∞ norm.

Next we use the scale invariance of the equation, considering the rescaled functions

v ε (x, y, t) := ε κ v x ε , y -ε ε , t ε (2q-p)/(q-p+1) .
Pick ε ∈ (0, min(L 1 , L 2 )) and denote Bε = B ε (0, ε), which is included in Ω and tangent to ∂Ω at the origin. Set T ε = ε (2q-p)/(q-p+1) T max (v 0 ) and Tε = min T max (u 0 ), T ε . We shall show that, for each τ ∈ (0, Tε ),

∇v ε L ∞ (Qτ ) ≤ max ∇v ε (•, 0) ∞ , ∇u L ∞ (Qτ ) . (4.3)
Since gradient blow-up occurs in finite time T ε for v ε , this will guarantee

T max (u 0 ) ≤ T ε < ∞.
First observe that v ε solves (4.2) in Bε ×(0, T ε ), with initial data v ε (x, y, 0) ≤ u 0 (x, y), due to (4.1). It follows from the comparison principle that v ε ≤ u in Bε × (0, Tε ). In particular, for each h ∈ (0, ε) and t ∈ (0, Tε ), since u(0, 0, t) = 0, we get that

v ε (0, h, t) h ≤ u(0, h, t) h ≤ ∇u(•, t) ∞ . (4.4)
We shall next show that the first quantity in (4.4) can be suitably bounded from below in terms of the sup norm of ∇v ε . Fix h ∈ (0, ε) and let Bh

ε := B ε (0, ε -h) Since v h ε (x, y, t) := v ε (x, y + h, t
) is a solution of (4.2) in Bh ε , it follows from the comparison principle (see Remark 2.1) that, for any 0 < τ < Tε , sup

( Bε∩ Bh ε )×(0,τ ) |v ε (x, y + h, t) -v ε (x, y, t)| ≤ max sup Bε∩ Bh ε |v h ε (x, y, 0) -v ε (x, y, 0)|, sup ∂( Bε∩ Bh ε )×(0,τ ) |v h ε (x, y, t) -v ε (x, y, t)| . (4.5)
We claim that, for any 0 < t < T ε , sup

∂( Bε∩ Bh ε ) |v h ε (x, y, t) -v ε (x, y, t)| ≤ v ε (0, h, t). (4.6)
First consider the case (x, y) ∈ ∂ Bε . Then (x, y + h) -(0, ε) ≥ (0, h) -(0, ε) , due to

x 2 + (y + h -ε) 2 -(h -ε) 2 = ε 2 -(y -ε) 2 + (y + h -ε) 2 -(h -ε) 2 = 2hy ≥ 0.
Since v ε is radially symmetric and non-increasing with respect to the point (0, ε), we deduce that

|v ε (x, y + h, t) -v ε (x, y, t)| = v ε (x, y + h, t) ≤ v ε (0, h, t). Next consider the case (x, y) ∈ ∂ Bh ε that is, (x, y + h) ∈ ∂ Bε . Then (x, y) -(0, ε) ≥ (0, h) -(0, ε) , due to x 2 + (y -ε) 2 -(h -ε) 2 = ε 2 -(y + h -ε) 2 + (y -ε) 2 -(h -ε) 2 = 2h(2ε -h -y) ≥ 0. Therefore, |v ε (x, y + h, t) -v ε (x, y, t)| = v ε (x, y, t) ≤ v ε (0, h, t)
, and the claim (4.6) is proved. Now fix y ∈ [0, ε). It follows from (4.4)-(4.6) that, for each 0 < h < ε -y and 0 < t < τ < Tε ,

|v ε (0, y + h, t) -v ε (0, y, t)| h ≤ max ∇v ε (•, 0) ∞ , ∇u L ∞ (Qτ )
hence, letting h → 0,

|∂ y v ε (0, y, t)| ≤ max ∇v ε (•, 0) ∞ , ∇u L ∞ (Qτ ) .
For each 0 < τ < Tε , taking supremum over y ∈ [0, ε) and t ∈ (0, τ ) and using the fact that v ε is radially symmetric, we obtain (4.3). This concludes the proof of the lemma.

Local boundary control for the gradient and localization of the gradient blow-up set

For simplicity we shall here assume (3.9) and (3.10), so as to have the continuity of ∇u up to the boundary (although one might possibly relax this assumption at the expense of additional work).

Lemma 5.1. Let ρ, µ, u 0 be as in Lemma 3.3(ii) and let (x 0 , y 0 ) ∈ ∂Ω. If there exist M 0 , R > 0 such that

|∇u| ≤ M 0 in (B R (x 0 , y 0 ) ∩ ∂Ω) × [0, T max (u 0 )), (5.1)
then (x 0 , y 0 ) is not a gradient blow-up point.

The proof is based on a local Bernstein technique. For (x 0 , y 0 ) ∈ ∂Ω, R > 0 and given α ∈ (0, 1), we may select a cut-off function η ∈ C 2 (B R (x 0 , y 0 )), with 0 < η ≤ 1, such that

η = 1 on B R/2 (x 0 , y 0 ), η = 0 on ∂B R (x 0 , y 0 )
and

|∇η| ≤ CR -1 η α |D 2 η| + η -1 |∇η| 2 ≤ CR -2 η α on B R (x 0 , y 0 ), (5.2)
where C = C(α) > 0 (see e.g. [START_REF] Ph | Global solutions of inhomogeneous Hamilton-Jacobi equations[END_REF] for an example of such function). Also, for 0 < t 0 < τ < T = T max (u 0 ), we denote

Q t 0 τ,R = (B R (x 0 , y 0 ) ∩ Ω) × (t 0 , τ ).
For the proof of Lemma 5.1, we then rely on the following lemma from [START_REF] Attouchi | Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion[END_REF] (cf. [2, Lemma 3.1]; it was used there to derive upper estimates on |∇u| away from the boundary).

Lemma 5.2. Let µ ≥ 0 and u 0 ∈ V µ . Let (x 0 , y 0 ) ∈ ∂Ω, R > 0, 0 < t 0 < τ < T and choose α = (q + 1)/(2q -p + 2). Denote w = |∇u| 2 and z = ηw. Then z ∈ C 2,1 (Q t 0 τ,R ) and satisfies the following differential inequality

Lz + C 2 z 2q-p+2 2 ≤ C 3 u 0 ∞ t 0 2q-p+2 q + C 3 R -2q-p+2 q-p+1 , (5.3)
where

C i = C i (p, q) > 0, Lz = z t -Āz -H • ∇z, (5.4) Āz = |∇u| p-2 ∆z + (p -2)|∇u| p-4 (∇u) t D 2 z∇u, (5.5)
and H is defined by

H := (p -2)w p-4 2 ∆u + (p -2)(p -4) 2 w p-6 2 ∇u • ∇w + qw q-2 2 ∇u + p -2 2 w p-4
2 ∇w.

(5.6)

Proof of Lemma 5.1. Let t 0 = T /2 < τ < T and set

M 1 := sup 0≤t≤t 0 ∇u L ∞ < ∞.
By Lemma 3.3(ii) and Theorem 2.2, we know that ∇u is a continuous function on Ω × (0, T ), hence z ∈ C(Q t 0 τ,R ). Therefore, unless z ≡ 0 in Q t 0 τ,R , z must reach a positive maximum at some point (x 1 , y 1 , t 1 ) ∈ Q t 0 τ,R . Since

z = 0 on ∂B R (x 0 , y 0 ) ∩ Ω × [t 0 , τ ], (5.7)
we deduce that either (x 1 , y 1 )

∈ B R (x 0 , y 0 ) ∩ Ω or (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ ∂Ω. • If t 1 = t 0 , then z(x 1 , y 1 , t 1 ) ≤ ∇u(t 0 ) 2 L ∞ ≤ M 2 1 .
(5.8)

• If t 0 < t 1 ≤ τ and (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ ∂Ω, then, by (5.1), z(x 1 , y 1 , t 1 ) ≤ M 2 0 .
(5.9)

• Next consider the case t 0 < t 1 ≤ τ and (x 1 , y 1 ) ∈ B R (x 0 , y 0 ) ∩ Ω. Then we have ∇z(x 1 , y 1 , t 1 ) = 0, z t (x 1 , y 1 , t 1 ) ≥ 0 and D 2 z(x 1 , y 1 , t 1 ) ≤ 0, and therefore Lz ≥ 0. Using (5.3) we arrive at

C 2 z(x 1 , y 1 , t 1 ) 2q-p+2 2 ≤ C 3 u 0 ∞ t 0 2q-p+2 q + C 3 R -2q-p+2 q-p+1
that is,

z(x 1 , y 1 , t 1 ) ≤ C u 0 ∞ t 0 1 q + CR -1 q-p+1 =: M 2 > 0.
(5.10)

It follows from (5.7)-(5.10) that max

Q t 0 τ,R z ≤ M 2 3 , with M 3 = max {M 0 , M 1 , M 2 }. Since z = |∇u| 2 in B R/2 (x 0 , y 0 ) ∩ Ω × (t 0 , τ ) and τ ∈ (t 0 , T ) is arbitrary, we get |∇u| ≤ M 3 in B R/2 (x 0 , y 0 ) ∩ Ω × (t 0 , T ),
and we conclude that (x 0 , y 0 ) is not a gradient blow-up point.

By combining Lemmas 3.3 and 5.1, we can now easily obtain a class of initial data whose possible gradient blow-up set is contained in a small neighborhood of the origin. But (5.11) easily follows from a comparison with the function Ū provided in Lemma 3.2. Indeed, under the assumptions of Lemma 3.3(i), we already know that u ≤ Ū in Q T . Also, u = µy = Ū on Σ ′ ρ × (0, T ). From this, along with (3.3), it follows that

∂ ν Ū ≤ ∂ ν u ≤ µ ∂ ν y on Σ ′ ρ × (0, T ).
(5.12)

From (5.12) and (1.1) 2 , we get

|∇u| 2 ≤ µ 2 + |∂ ν u| 2 ≤ C on Σ ′ ρ × (0, T ),
hence (5.11), and the lemma is proved.

6 Existence of well-prepared initial data: proof of Theorem 1.1(i)

We need to construct initial data meeting the requirements from sections 3-5. This will be achieved in the following lemma. Let us fix an even function ϕ ∈ C ∞ (R) such that sϕ ′ (s) ≤ 0, with

ϕ(s) = 1 for |s| ≤ 1/3 0 for |s| ≥ 2/3. (6.1)
Lemma 6.1. Let κ = (q -p)/(q -p+1) and let C 1 = C 1 (p, q) > 0 be given by Lemma 4.1.

For ε ∈ (0, min(L 1 , L 2 /2)), define

ψ ε (y) =      ϕ y -ε ε for 0 ≤ y ≤ ε ϕ y -ε L 2 for y ≥ ε (6.2)
and let u 0 be defined by

u 0 (x, y) = µy + C 1 ε κ ϕ x ε ψ ε (y).
Next fix 0 < ρ < L 1 and let µ 0 = µ 0 (p, q, Ω, ρ) > 0 and c = c(p, q, Ω, ρ) > 0 be given by Lemmas 3.2 and 3.3. For any µ ∈ (0, µ 0 ], there exists ε 0 = ε 0 (p, q, Ω, µ, ρ) > 0 such that, for all ε ∈ (0, ε 0 ], the function u 0 ∈ V µ and satisfies u 0 is symmetric with respect to the line x = 0, (6.3)

∂ x u 0 ≤ 0 in Ω + , (6.4) ∂ y u 0 ≥ µ/2 in Ω, (6.5) u 0 (x, y) ≤ µ y + cχ (-ρ/2,ρ/2)×(0,L 2 )
in Ω, (6.6)

u 0 (x, y) ≥ C 1 ε κ in B ε/3 (0, ε) ⊂ Ω. (6.7) Proof. Assume ε ≤ min(L 1 , L 2 /12). Then ψ ε (y) = 0 for y ∈ [ ε 3 , 3L 2 4 ] (6.8) (indeed, y ≥ 3L 2 4 implies y-ε L 2 ≥ 3 4 -1 12 = 2
3 ) and therefore u 0 ∈ V µ . Properties (6.3)-(6.4) are clear by the choice of ϕ.

To check (6.5), we note that

∂ y u 0 = µ + C 1 ε κ ϕ x ε ψ ′ ε (y).
For 0 ≤ y ≤ ε, we have ψ ′ ε (y) ≥ 0, hence ∂ y u 0 ≥ µ. Whereas, for y ≥ ε, we have

ψ ′ ε (y) = L -1 2 ϕ ′ (y -ε)/L 2 ≥ -L -1 2 ϕ ′ ∞ , hence ∂ y u 0 ≥ µ -C 1 ε κ L -1 2 ϕ ′ ∞ ≥ µ/2 whenever ε κ ≤ µL 2 /(2C 1 ϕ ′ ∞
). As for (6.6), if C 1 ε κ ≤ µc and ε ≤ ρ/2, it immediately follows from ϕ, ψ ε ≤ 1 and supp(ϕ) ⊂ (-1, 1). Finally, since ϕ(x/ε) = 1 for |x| ≤ ε/3 and ψ ε (y) = 1 for |y -ε| ≤ ε/3, we have (6.7). The lemma is proved.

Proof of Theorem 1.1(i). Let µ and u 0 be as in Lemma 6.1.

• The fact that T max (u 0 ) < ∞ follows from Lemma 4.1.

• Next, we have GBUS(u 0 ) ⊂ [-ρ, ρ] × {0} as a consequence of Lemma 5.3.

• Properties (1.9)-(1.10) follow from Lemma 3.1.

• Finally, property (1.11) is a consequence of Lemma 3.3(ii). This proves the assertion.

Nondegeneracy of gradient blow-up points

In this section, we show that if u is only "weakly singular" in a neighborhood of a boundary point (x 0 , 0), then the singularity is removable.

Lemma 7.1. Let ρ, µ, u 0 be as in Lemma 3.3(ii) and let x 0 ∈ (-L 1 , L 1 ). There exist c 0 = c 0 (p, q) > 0 such that, if u 0 ∈ V µ with T := T max (u 0 ) < ∞ and u(x, y) ≤ c 0 y (q-p)/(q-p+1) in (B R (x 0 , 0) ∩ Ω) × [t 0 , T ), (

for some R > 0 and t 0 ∈ (0, T ), then (x 0 , 0) is not a gradient blow-up point.

Proof. Let x 0 ∈ (-L 1 , L 1 ). Then for some constants r ∈ (0, R) and d ∈ (0, L 2 ), we have that

ω 1 := (x, y) ∈ R 2 ; |x -x 0 | < r, 0 < y < d ⊂ B R (x 0 , 0) ∩ Ω.
Setting β = 1/(q -p + 1), we define the comparison function

v = v(x, y, t) = εyV -β in Q := ω 1 × (t 0 , T ) with V = y + η r 2 -(x -x 0 ) 2 (t -t 0 ),
where η, ε > 0 are to be determined later. We compute, in Q,

v t = -εβηy(r 2 -(x -x 0 ) 2 )V -β-1 , v x = 2εβηy(x -x 0 )(t -t 0 )V -β-1 , v y = εV -β -εβyV -β-1 = εV -β 1 -β y V , v xx = 2εβηy(t -t 0 )V -β-1 -4εβη 2 y(x -x 0 ) 2 (t -t 0 ) 2 (-β -1)V -β-2 = 2εβη(t -t 0 )V -β-1 y + 2(β + 1)η(x -x 0 ) 2 (t -t 0 ) y V , v yy = -2εβV -β-1 + εβ(β + 1)yV -β-2 = εβV -β-1 -2 + (β + 1) y V , v xy = 2εβη(x -x 0 )(t -t 0 )V -β-1 -2εβ(β + 1)ηy(x -x 0 )(t -t 0 )V -β-2 = 2εβη(x -x 0 )(t -t 0 )V -β-1 1 -(β + 1) y V .
Noting that β < 1 and y V ≤ 1, we see that, in Q,

0 ≤ v xx ≤ 2εβηT d + 2(β + 1)ηr 2 T V -β-1 , |v xy | ≤ 2εβηrT V -β-1 and v yy ≤ εβ(β -1)V -β-1 < 0. It follows that ∆ p v = |∇v| p-2 ∆v + (p -2) v i v j v ij |∇v| 2 ≤ |∇v| p-2 (p -1)v xx + v yy + (p -2)|v xy | ≤ εβ|∇v| p-2 V -β-1 2(p -1)ηT d + 2(β + 1)ηr 2 T + (β -1) + 2(p -2)ηrT .
On the other hand, we have

|∇v| ≥ |v y | ≥ ε(1 -β)V -β ≥ ε(1 -β)(d + ηT r 2 ) -β , hence v t ≥ -εβηdr 2 V -β-1 ≥ -εβ|∇v| p-2 V -β-1 ηdr 2 ((1 -β)ε) 2-p (d + ηT r 2 ) (p-2)β .
Therefore,

v t -∆ p v ≥ εβ|∇v| p-2 V -β-1 × -ηdr 2 ((1 -β)ε) 2-p (d + ηT r 2 ) (p-2)β -2(p -1)ηT d + 2(β + 1)ηr 2 T -2(p -2)ηrT + (1 -β) .
Since also

|v x | ≤ 2εβηrT V -β , |v y | ≤ εV -β ,
if we choose η = η(p, q, d, r, T, ε) > 0 small enough, we get that, in Q,

|∇v| ≤ 2εV -β and v t -∆ p v ≥ εβ(1 -β) 2 |∇v| p-2 V -β-1 , hence v t -∆ p v ≥ εβ(1 -β) 2 |∇v| p-2 (2ε) -β+1 β |∇v| β+1 β = β(1 -β) 4 (2ε) -1 β |∇v| q ,
due to β = 1/(q -p + 2). If ε = ε 0 (p, q) > 0 is small enough, we thus obtain

v t -∆ p v ≥ |∇v| q . (7.2)
Now we shall check the comparison on the parabolic boundary of ω 1 × (t 0 , T ). On

ω 1 × {t 0 }, choosing c 0 = 2 -β ε 0 , we have u ≤ c 0 y 1-β = 2 -β ε 0 y 1-β ≤ v. (7.3)
On the lateral boundary part {(x, y) ∈ R 2 ; |x -x 0 | = r, 0 ≤ y ≤ d} × (t 0 , T ), inequality (7.3) holds also. On the surface {(x, y) ∈ R 2 ; |x -x 0 | ≤ r, y = 0} ⊂ ∂Ω, we have for t 0 < t < T , u(., ., t) = v(., ., t) = 0 Finally, on {(x, y) ∈ R 2 ; |x -x 0 | ≤ r, y = d}×(t 0 , T ), assuming in addition that η satisfies η ≤ dT -1 r -2 , we get

u ≤ c 0 d 1-β ≤ ε 0 d(d + ηr 2 T ) -β ≤ v.
Using the comparison principle, we get that u ≤ v in ω 1 × (t 0 , T ). (7.4) This implies that

|u y | ≤ ε η(r 2 -|x -x 0 | 2 )(t -t 0 ) -β ≤ M 0
on (B r/2 ((x 0 , 0)) ∩ ∂Ω) × ((t 0 + T )/2, T ) for some constant M 0 > 0. Lemma 7.1 is then a direct consequence of Lemma 5.1.

The auxiliary function J and the proof of singlepoint gradient blow-up

In all this section, we fix ρ, x 1 with

0 < ρ < x 1 < L 1 (8.1)
and we assume that µ and u 0 ∈ V µ ∩ C 2 (Ω) satisfy the assumption of Theorem 1.1(ii) i.e., the corresponding solution of (1.1) fulfills properties (1.8)- (1.11). We denote as before T = T max (u 0 ). We consider the auxiliary function

J(x, y, t) := u x + c(x)d(y)F (u), with    F (u) = u α , c(x) = kx, k > 0, d(y) = y -γ and 1 < α < 1 + q -p, γ = (1 -2σ)(α -1), (8.2)
where 0 < σ < 1 2(q -p + 1) (8.3) is fixed. Letting D = (0, x 1 ) × (0, y 1 ), our goal is to use a comparison principle to prove that

J ≤ 0 in D × (T /2, T ),
provided α > 1 is chosen close enough to 1 (hence making γ > 0 small) and y 1 ∈ (0, L 2 ) and k > 0 are chosen sufficiently small.

Parabolic inequality for the auxiliary function J

By the regularity of u (see Theorem 2.2), we have

J ∈ C 2,1 (Q T ).
A key step is to derive a parabolic inequality for J. To this end, we define the operator

PJ := J t -|∇u| p-2 ∆J -(p -2)|∇u| p-4 D 2 J ∇u, ∇u + H • ∇J + AJ, (8.4)
where the functions H = H(x, y, t) and A = A(x, y, t) are given by formulae (9.9)-(9.12) below.

Proposition 8.1. Assume (8.1), (8.3) and let µ, u 0 satisfy the assumption of Theorem 1.1(ii). There exist α, γ satisfying (8.2), y 1 ∈ (0, L 2 ) and k 0 > 0, all depending only on p, q, Ω, µ, u 0 C 2 , σ, such that, for any k ∈ (0, k 0 ], the function J satisfies

PJ ≤ 0 in D × (T /2, T ). (8.5)
Moreover,

H, A ∈ C(D × (0, T )) and A ∈ L ∞ (D × (T /2, τ )) for each τ ∈ (T /2, T ). (8.6)
The proof of Proposition 8.1 is very long and technical. In order not to disrupt the main line of argument, we postpone it to section 9 and now present the rest of the proof of Theorem 1.1(ii).

Boundary conditions for the auxiliary function J

The verification of the appropriate boundary and initial conditions for the function J depends on an essential way on the applicability to u x of the Hopf boundary lemma at the points (x 1 , 0) and (0, y 1 ), up to t = T . To this end, besides the nondegeneracy of problem (1.1), guaranteed by (1.11), we also need the following local regularity lemma, which ensures that D 2 u remains bounded up to t = T away from the gradient blow-up set.

Lemma 8.2. Let ρ ∈ (0, L 1 ) and let µ, u 0 satisfy the assumption of Theorem 1.1(ii). Let ω ′ ⊂ ω ⊂ Ω be such that dist(ω ′ , Ω \ ω) > 0 and t 0 ∈ (0, T ). If

sup ω×(0,T ) |∇u| < ∞, then sup ω ′ ×(t 0 ,T ) |D 2 u| < ∞. Proof. Introduce an intermediate domain ω ′′ with ω ′ ⊂ ω ′′ ⊂ ω, such that dist(ω ′′ , Ω\ω) > 0 and dist(ω ′ , Ω \ ω ′′ ) > 0. Write the PDE in (1.1) as -∇ • (|∇| p-2 ∇u) = |∇u| q -u t .
Using |∇u| ≥ ∂ y u ≥ µ/2 (cf. (1.11)) and the boundedness of u t in Q T (cf. Lemma 3.4), it follows from the elliptic estimate in [13, Theorem V.5.2] that there exists θ ∈ (0, 1) such that ∇u(

•, t) C θ (ω ′′ ) ≤ C, t 0 /2 ≤ t < T.
The boundedness of u t in Q T and the interpolation result in [13, Lemma II.

u t -a ij u ij = f, where a ij = |∇u| p-2 δ ij + (p -2) u i u j |∇u| 2 , f = |∇u| q .
Indeed, the matrix (a ij ) = (a ij (x, y, t)) is uniformly elliptic due to (1.11) and

a ij ξ i ξ j = |∇u| p-2 |ξ| 2 + (p -2)|∇u| p-4 |∇u • ξ| 2 ≥ |∇u| p-2 |ξ| 2 ≥ (µ/2) p-2 |ξ| 2 , (8.7)
and there exists ν and there exists k 1 > 0 (depending in particular on y 1 ) such that, for any k ∈ (0, k 1 ], the function J satisfies J ≤ 0 on ∂D × (T /2, T ). (8.9)

∈ (0, 1) such that a ij , f ∈ C ν (ω ′ × [t 0 /2, T ′ ]), for each T ′ < T , with norm independent of T ′ .
Proof. Since u = 0 for y = 0 and |∇u| ≤ C(τ ) in Ω × [0, τ ] for each τ < T , we have

u ≤ C(τ )y in D × [0, τ ].
Due to γ < α, we may therefore extend the function c(x)d(y)F (u) continuously to be 0 for y = 0. Property (8.8) then follows from the regularity of u (see Theorem 2.2) and we have J = 0 on (0, x 1 ) × {0} × (T /2, T ). (8.10)

By (1.10), we have u x = 0 on {0} × (0, y 1 ) × (0, T ), hence J = 0 on {0} × (0, y 1 ) × (T /2, T ). (8.11) Next, the function w = u x is ≤ 0 in Ω + × (0, T ) (cf. (1.10)) and satisfies there:

w t = a ij (x, y, t)w ij + B(x, y, t) • ∇w, (8.12)
with

a ij (x, y, t) = |∇u| p-2 δ ij + (p -2) u i u j |∇u| 2 , B(x, y, t) = (p -2)|∇u| p-4 ∇u∆u + (p -2)(p -4)|∇u| p-6 D 2 u ∇u, ∇u ∇u + q|∇u| q-2 ∇u + 2(p -2)|∇u| p-4 (D 2 u ∇u). Fix ρ < x 3 < x 2 < x 1 . Since GBUS ⊂ [-ρ, ρ] × {0}, we have |∇u| ≤ C in Ω\ {(-x 3 , x 3 ) × (0, y 1 /3)} × (0, T ).
It follows from Lemma 8.2 that

|D 2 u| ≤ C in Ω\ {(-x 2 , x 2 ) × (0, y 1 /2)} × (T /4, T ), hence |B| ≤ C in Ω\ {(-x 2 , x 2 ) × (0, y 1 /2)} × (T /4, T ).
Moreover, the matrix A(x, y, t) is uniformly elliptic (cf. (8.7)). We may thus apply the strong maximum principle and the Hopf boundary point Lemma [19, Theorem 6 p. 174], to get

u x ≤ -c 1 y on {x 1 } × (0, y 1 ) × (T /2, T ), u x ≤ -c 1 x on (0, x 1 ) × {y 1 } × (T /2, T ).
Also, since x 1 > ρ and u(x, 0, t) = 0, we get that, for some c 2 > 0,

u ≤ c 2 y on {x 1 } × (0, y 1 ) × (T /2, T ).
Consequently, using α > γ + 1 and (2.6), we have for 0 < k ≤ k 1 (y 1 ) sufficiently small

J(x, y 1 , t) ≤ -c 1 x + kxy -γ 1 u 0 α ∞ ≤ 0 on (0, x 1 ) × {y 1 } × (T /2, T ), J(x 1 , y, t) ≤ -c 1 y + kx 1 y α-γ c α 2 ≤ 0 on {x 1 } × (0, y 1 ) × (T /2, T
). This, along with (8.10)- (8.11), proves (8.9).

Initial conditions for J

Lemma 8.4. Assume (8.1)-(8.3) and let µ, u 0 satisfy the assumption of Theorem 1.1(ii). There exists k 2 > 0 such that, for any k ∈ (0, k 2 ], the function J satisfies

J(x, y, T /2) ≤ 0 in [0, x 1 ] × [0, L 2 ].
The proof relies on a parabolic version of the Serrin corner lemma applied to u x . This is provided by Proposition 11.1, which we state and prove in Appendix 2.

Proof. The function z = u x satisfies equation (8.12). We shall apply Proposition 11.1 to this equation, with

τ 1 = T /4, τ 2 = 3T /4, X 1 = x 1 , Y 1 = L 2 , X1 = L 1 , Ŷ1 = 2L 2 .
We thus need to check the assumption (11.2). Let us denote DT = (0, X1 ) × (0, Ŷ 1 ) × (T /4, 3T /4).

For x = 0 or y = 0, we have u x = 0, hence a 12 = a 21 = (p -2)|∇u| p-2 u x u y = 0. Due to the regularity of u (cf. (2.7)), we deduce that

a 12 + a 21 ≥ -C(x ∧ y) in DT . (8.13)
On the other hand, for x = 0 and 0 < y < Ŷ1 , we have u xy = u x = 0. Also, by (2.9),

we have |(u xy ) x | = |(u x ) yx | ≤ C in DT . Consequently |u x | + |u xy | ≤ Cx in DT . Using (2.7
) and (1.11), we deduce

B 1 = (p -2)|∇u| p-4 (∆u)u x + (p -2)(p -4)|∇u| p-6 D 2 u ∇u, ∇u u x + q|∇u| q-2 u x + 2(p -2)|∇u| p-4 u xx u x + u xy u y ≥ -Cx in DT . (8.14)
Next, for y = 0 and 0 < x < X1 , we have u t = 0 and u x = u xx = 0. Recalling (2.7), we thus have

(u y ) q = |∇u| q = u t -∆ p u = -|∇u| p-2 ∆u + (p -2) D 2 u ∇u, ∇u |∇u| 2 = -|∇u| p-2 u xx + u yy + (p -2) u xx u 2 x + 2u xy u x u y + u yy u 2 y u 2 x + u 2 y = -(p -1)(u y ) p-2 u yy .
It follows that, for 0 < x < X1 and t ∈ [T /4, 3T /4],

B 2 (x, 0, t) = (p -2)(p -4)|∇u| p-6 u xx u 2 x + 2u xy u x u y + u yy u 2 y u y + (p -2)|∇u| p-4 (u xx + u yy )u y + q|∇u| q-2 u y + 2(p -2)|∇u| p-4 u yx u x + u yy u y = (p -2)(p -4)(u y ) p-3 u yy + (p -2)(u y ) p-3 u yy + q(u y ) q-1 + 2(p -2)(u y ) p-3 u yy = (p -2)(p -1)(u y ) p-3 u yy + q(u y ) q-1 = (q + 2 -p)(u y ) q-1 ≥ (q + 2 -p)(µ/2) q-1 > 0.
Therefore, owing to (2.7), there exists η > 0 such that

B 2 (x, y, t) ≥ 0 on (0, X1 ) × [0, η] × [T /4, 3T /4],
which implies

B 2 ≥ -Cy in DT . (8.15)
In view of (8.13)-(8.15), we may thus apply Proposition 11.1 to deduce

u x (x, y, T /2) ≤ -c 3 xy in [0, x 1 ] × [0, L 2 ]. Let C := ∇u(•, T /2) ∞ . Since α > γ + 1, we get that, for k ∈ (0, k 2 ] with k 2 > 0 small enough, J(x, y, T /2) ≤ -c 3 xy + kxC α y α-γ ≤ [kC α L α-γ-1 2 -c 3 ]xy ≤ 0 in [0, x 1 ] × [0, L 2 ].

Proof of Theorem 1.1(ii)

Let α, γ, y 1 , k 0 be given by Proposition 8.1 and let k 1 , k 2 be given by Lemmas 8.3-8.4. We take k = min(k 0 , k 1 , k 2 ). By these results and the maximum principle, we have

J ≤ 0 in D × (T /2, T ). (8.16)
Integrating inequality (8.16) over (0, x) for 0 < x < x 1 , with fixed y, we get that

u ≤ Cx -2/(α-1) y 1-2σ in D × (T /2, T ),
where

C = C(α, k, σ) > 0. Using that 1 -2σ > q -p q -p + 1
, it follows from the nondegeneracy property in Lemma 7.1 that no point (x 0 , 0) with 0 < |x 0 | ≤ ρ can be a gradient blow-up point. In view of (1.8), we conclude that GBUS(u 0 ) = {(0, 0)}.

9 Proof of the main parabolic inequality (Proposition 8.1)

The proof is quite technical. For sake of clarity, some of the intermediate calculations will be summarized in Lemma 9.1 and 9.2 below. We first compute

J t = u xt + cdF ′ (u)u t = (∆ p u) x + (|∇u| q ) x +cdF ′ ∆ p u (0p) + cdF ′ |∇u| q (0q)
. and

(∆ p u) x = |∇u| p-2 ∆(u x ) + (p -2)∆u|∇u| p-4 ∇u • ∇u x + (p -2)|∇u| p-4 D 2 u x ∇u, ∇u + (p -2)(p -4)|∇u| p-6 ∇u • ∇u x D 2 u ∇u, ∇u + 2(p -2)|∇u| p-4 D 2 u ∇u, ∇u x .
Using that u x = J -cdF (u), we write

∇u x = ∇J -cdF ′ ∇u -F c ′ d d ′ c , D 2 u x = D 2 J -cdF ′ D 2 u -cdF ′′ u 2 x u x u y u x u y u 2 y -F (u) c ′′ d c ′ d ′ c ′ d ′ d ′′ c -F ′ (u) 2c ′ du x cd ′ u x + c ′ du y cd ′ u x + c ′ du y 2cd ′ u y , ∆u x = T race(D 2 u x ) = ∆J -cdF ′ ∆u -cdF ′′ |∇u| 2 -F [c ′′ d + d ′′ c] -2F ′ c ′ dJ + 2F ′ F c ′ cd 2 -2F ′ (u)d ′ cu y , D 2 u x , ∇u, ∇u = D 2 J, ∇u, ∇u -cdF ′′ |∇u| 4 -cdF ′ D 2 u, ∇u, ∇u -F c ′′ du 2 x -2F c ′ d ′ u x u y -F cd ′′ u 2 y -2F ′ |∇u| 2 (cd ′ u y + dc ′ u x )
and

D 2 u, ∇u, ∇u x = D 2 u, ∇u, ∇J -cdF ′ D 2 u, ∇u, ∇u -F c ′ d∇u • ∇J + cc ′ d 2 F ′ F |∇u| 2 + d 2 F 2 (c ′ ) 2 J -d 2 F 2 (c ′ ) 2 cdF + c ′ cd ′ dF 2 u y -F d ′ c(u x u xy + u y u yy ).
Therefore,

(∆ p u) x = |∇u| p-2 ∆J -cdF ′ ∆u -cdF ′′ |∇u| 2 -2c ′ dF ′ J +2cc ′ d 2 F F ′ -2d ′ cF ′ u y -F (c ′′ d + d ′′ c) + (p -2)∆u|∇u| p-4 ∇u • ∇J -cdF ′ |∇u| 2 -u y cd ′ F -c ′ dF J + c ′ cd 2 F 2 + (p -2)|∇u| p-4 D 2 J∇u, ∇u -cdF ′ D 2 u∇u, ∇u -cdF ′′ |∇u| 4 -2c ′ dF ′ |∇u| 2 J + 2c ′ cd 2 F ′ F |∇u| 2 -2cd ′ F ′ u y |∇u| 2 -c ′′ dF (u x ) 2 -2c ′ d ′ F Ju y + 2c ′ cd ′ dF 2 u y -d ′′ cF (u y ) 2 + (p -2)(p -4)|∇u| p-6 D 2 u∇u, ∇u [∇u • ∇J -cdF ′ |∇u| 2 -u y cd ′ F -c ′ dF J + c ′ cd 2 F 2 + 2(p -2)|∇u| p-4 D 2 u ∇u, ∇J -cdF ′ D 2 u ∇u, ∇u -c ′ dF ∇u • ∇J + c ′ cd 2 F ′ F |∇u| 2 + (c ′ d) 2 F 2 J -(c ′ d) 2 F 3 cd + c ′ d ′ cdF 2 u y -d ′ cF u yy u y -d ′ cF ∇J • L + d ′ dc 2 F ′ F u y J -d ′ d 2 c 3 F ′ F 2 u y + (d ′ cF ) 2 J -(d ′ cF ) 2 cdF ,
where L = 0 u x . This can be rewritten as

(∆ p u) x = |∇u| p-2 ∆J + (p -2)|∇u| p-4 D 2 J ∇u, ∇u + H 1 • ∇J + A 1 (x, y, t)J -F |∇u| p-2 [c ′′ d + d ′′ c] -(p -2)F |∇u| p-4 c ′′ du 2 x + d ′′ cu 2 y -2(p -2)cdF |∇u| p-4 [(c ′ dF ) 2 + (d ′ cF ) 2 ] -(p -1)cdF ′′ |∇u| p +4(p -2)F 2 c ′ cd ′ d|∇u| p-4 u y    (1) ≤ 0 +(4p -6)c ′ cd 2 F ′ F |∇u| p-2 -2(p -1)cd ′ F ′ |∇u| p-2 u y -2(p -2)d ′ d 2 F ′ F 2 c 3 |∇u| p-4 u y (2) ≥ 0 -(p -1)cdF ′ ∆ p u (3) -2(p -2)d ′ cF |∇u| p-4 u y u yy (4)
-(p -2)cd ′ F u y |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u

+(p -2)c ′ cd 2 F 2 |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u (5)
,

where

H 1 := (p -2) |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u ∇u -2(p -2)cd ′ F |∇u| p-4 L -2(p -2)c ′ dF |∇u| p-4 ∇u + 2(p -2)|∇u| p-4 D 2 u, ∇u (9.1)
and

A 1 := -2(p -1)F ′ c ′ d|∇u| p-2 -(p -2)F c ′ d |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u +2(p -2)|∇u| p-4 F 2 (c ′ d) 2 + (d ′ c) 2 + 2(p -2)d ′ dF ′ F c 2 |∇u| p-4 u y -2(p -2)c ′ d ′ F |∇u| p-4 u y . (9.2)
On the other hand, we have

(|∇u| q ) x = q|∇u| q-2 ∇u • ∇u x = q|∇u| q-2 ∇u • ∇J -qc ′ dF |∇u| q-2 J + (7), where (7)
:= -qcdF ′ |∇u| q (7 -)≤0 + qcc ′ d 2 F 2 |∇u| q-2 -qcd ′ F |∇u| q-2 u y (7 + )≥0
.

Setting A 2 := A 1 -qc ′ dF |∇u| q-2 , H 2 := H 1 + q|∇u| q-2 ∇u, (9.3)
we have thus proved the following lemma.

Lemma 9.1. Define the parabolic operator:

LJ := J t -|∇u| p-2 ∆J -(p -2)|∇u| p-4 D 2 J ∇u, ∇u -H 2 • ∇J -A 2 J.
Then LJ = (0 p ) + (0 q ) + ( 1) + ( 2) + ( 3) + ( 4) + ( 5) + ( 6) + [START_REF] Fujii | Asymptotic behavior of blowup solutions of a parabolic equation with the p-Laplacian[END_REF]. (9.4) As a significant difficulty as compared with the semilinear case p = 2, many additional terms appear in the contributions (1), ( 2), ( 4)-( 6), and especially nonlinear, second order terms in ( 4)- [START_REF] Friedman | Blow-up of positive solutions of semilinear heat equations[END_REF]. To proceed further, we need to observe that, among the second derivatives of u, u yy needs a special treatment, since it is not immediately expressed in terms of ∇J unlike u xx and u xy . Namely we shall eliminate u yy by expressing it in terms of u t , ∇u, u xx and u xy by using the equation. Although this will make the computation even more involved, by producing a lot of additional terms, this seems to be the only way to control the effects of u yy . The bound on u t given by Lemma 3.4 will be helpful in this process.

First we have

(3) = -(p -1)cdF ′ ∆ p u = -cdF ′ ∆ p u -(0p) -(p -2)cdF ′ u t (3t) +(p -2)cdF ′ |∇u| q .
(3q) (9.5) To deal with (4), we set Since u x = J -cdF , we get

u yy = u t -|∇u| q -∇u x • M w ,
∇u x • M = -cdF ′ J |(p -1)∇u| p-2 + (p -2)|∇u| p-4 u 2 y -2(p -2)cd ′ F u y |∇u| p-4 J +2(p -2)c 2 d ′ dF 2 u y |∇u| p-4 + c 2 d 2 F ′ F |(p -1)∇u| p-2 + (p -2)|∇u| p-4 u 2 y +∇J • M -c ′ dF |∇u| p-2 + (p -2)|∇u| p-4 u 2 x .
It follows that

u yy = u t -|∇u| q w - ∇J • M w + cdF ′ J (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w + 2(p -2)cd ′ F u y |∇u| p-4 J w + c ′ dF [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w - c 2 d 2 F F ′ (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w - 2(p -2)c 2 d ′ dF 2 u y |∇u| p-4 w .
Now, to treat the contribution of u yy in ( 5) and ( 6), we set N = u 2

x 2u x u y and rewrite

|∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u ∇u, ∇u = ∆ p u |∇u| 2 -2|∇u| p-6 D 2 u ∇u, ∇u = u t -|∇u| q |∇u| 2 -2|∇u| p-6 ∇u x • N + u 2 y u yy .
We have

∇u x • N = ∇J • N -cdF ′ J u 2 x + 2u 2 y + c 2 d 2 F F ′ u 2 x + 2u 2 y -c ′ dF u 2 x -2cd ′ F Ju y + 2d ′ dc 2 F 2 u y .
The expression in (4) then becomes

(4) = 2(p -2)d ′ cF |∇u| p-4 u y M • ∇J w (4 ∇ ) - 2(p -2)c 2 d ′ dF ′ F |∇u| p-4 u y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y J w - 4(p -2) 2 (d ′ cF ) 2 |∇u| p-4 u 2 y |∇u| p-4 J w      (4 J ) + 2d ′ cF (p -2)|∇u| p-4 |∇u| q u y w + 2(p -2)c 3 d ′ d 2 F ′ F 2 |∇u| p-4 u y (p -1)|∇u| p-2 + (p -2)|∇u| p-4 u 2 y w      (4 -) ≤ 0 + 4(p -2) 2 (d ′ cF ) 2 cdF |∇u| p-4 u 2 y |∇u| p-4 w - 2(p -2)d ′ dc ′ cF 2 u y |∇u| p-4 [|∇u| p-2 + (p -2)|∇u| p-4 u 2 x ] w      (4 + ) ≥ 0 - 2d ′ cF (p -2)|∇u| p-4 u t u y w (4 t ).
where we recall that

H 1 := (p -2) |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u ∇u -2(p -2)cd ′ F |∇u| p-4 L -2(p -2)c ′ dF |∇u| p-4 ∇u + 2(p -2)|∇u| p-4 D 2 u, ∇u , (9.11)
with L = 0 u x , and and using ( f ), ( ĩ), ( j), ( l) ≤ 0, we obtain the following lemma.

A 1 := -2(p -1)F ′ c ′ d|∇u| p-2 -(p -2)F c ′ d |∇u| p-4 ∆u + (p -4)|∇u| p-6 D 2 u∇u, ∇u +2(p -2)|∇u| p-4 F 2 (c ′ d) 2 + (d ′ c) 2 + 2(p -2)d ′ dF ′ F c 2 |∇u| p-4 u y -2(p -2)c ′ d ′ F |∇u| p-4 u y . ( 9
Lemma 9.2. Recalling the definition (8.4) of the parabolic operator P: |∇u| ≤ C 0 y -1/(q-p+1) and u ≤ C 0 y (q-p)/(q-p+1) in [0, x 1 ] × (0, L 2 ] × (0, T ), (9.13) where C 0 = C 0 (p, q, Ω, µ, ∇u 0 ∞ ) > 0, and we will use also the lower bound from (1.11):

PJ := J t -|∇u| p-2 ∆J -(p -2)|∇u| p-4 D 2 J ∇u, ∇u -H • ∇J -A(x,
|∇u| ≥ u y ≥ δ 0 = µ/2 > 0. (9.14) First, using w ≥ |∇u| p-2 , we get (f ) ≤ -2(p -2)pc ′ cd ′ dF 2 u y |∇u| p-4 .
Assume y 1 ≤ 1. Due to (9.13), we have dF ≤ C α 0 y -γ+α(q-p)/(q-p+1) ≤ C α 0 . (9.15)

Combining (9.17)-(9.26), we get that

PJ cdF ≤ |∇u| q u q 2 γC q-p+1 0 2σ + 25(p -2) 2 γ K 2 u 0 ∞ 2σδ q+p 0 -α(q -p + 1) + |∇u| p-4 u 2 y y 2 k (q + 2(p -2))C α+q-p+2 0 δ -2 0 y (α-1)(2σ-1 q-p+1 )+ 2(q-p) q-p+1 +k α(4p -6)δ -2 0 C α+1 0 y (α-1)(2σ-1 q-p+1 )+ 2(q-p) q-p+1 +4(p -2)(p -1)k 2 γ 2 x 2 C 2α 0 y -2γ+ 2α(q-p) q-p+1 δ -2 0 +(p -2) αKy µδ p-2 0 + 3(p -2) k u 0 α ∞ Ky 2-γ δ p-2 0 -(2p -3)σγ . (9.27)
Now, we may choose α = α(p, q, Ω, µ, u 0 C 2 , σ) > 1 close enough to 1 in such a way that γ = (α -1)(1 -2σ) is small enough to satisfy

γ q 2 2σ C q-p+1 0 + 25(p -2) 2 K 2 u 0 ∞ 2σδ q-p 0 ≤ (q -p + 1)α (9.28) and (α -1) 2σ - 1 q -p + 1 + 2(q -p) q -p + 1 ≥ 0, (9.29) α q -p q -p + 1 -(α -1)(1 -2σ) ≥ 0. (9.30)
Finally, once α is fixed (hence γ is also fixed small), recalling that y ≤ y 1 ≤ 1, x ≤ L 1 and γ ≤ 2, we take k 0 = k 0 (p, q, Ω, µ, u 0 C 2 , σ) > 0 possibly smaller, in such a way that

k 0 (q + 2(p -2))C α+q-p+2 0 δ -2 0 + α(4p -6)δ -2 0 C α+1 0 +k 2 0 4(p -2)(p -1)γ 2 L 2 1 C 2α 0 δ -2 0 + k 0 3(p -2) u 0 α ∞ K δ p-2 0 ≤ 2p -3 2 σγ, (9.31)
and next we take y 1 = y 1 (p, q, Ω, µ, u 0 C 2 , σ) > 0 small enough such that (p -2)αKy Finally, we need to check (8.6). The continuity statement is clear from the definition of A, H. Let us show that A is bounded in D × (T /2, τ ) for each τ < T . For this purpose, let us observe that due to |∇u| ≤ C(τ ), u ≤ C(τ )y and α -1 ≥ γ, we have for y ≤ 1 and τ ∈ (T /2, T )

|F ′ d| = αu α-1 y -γ ≤ C α-1 (τ )y α-1-γ ≤ αC α-1 (τ ) (9.34) |F d ′ | = γu α y -γ-1 ≤ γC α (τ ) (9.35) |F d| = u α y -γ ≤ C α (τ ). (9.36) Proof. (i) Since, by Theorem 2.1(iii), ∇u is continuous in Q T , there exist λ, ρ, M 2 > 0 such that λ ≤ |∇u| ≤ M 2 in Q ρ := B ρ (x 0 , y 0 ) × [t 0 -ρ, t 0 + ρ] ⊂ Q T . ( 10.1)
For any unit vector e and 0 < h < ρ/2, let us introduce the differential quotients

D h u = h -1 (τ h u -u)
, where τ h u = u (x, y) + h e, t .

We have

|∇τ h u| q -|∇u| q = d h (x, y, t) • ∇(τ h u -u) in Q ρ/2 , where |d h (x, y, t)| ≤ C independent of h. Next denote b(s) = s (p-
a i (∇τ h u) -a i (∇u) = 1 0 d ds a i (s∇τ h u + (1 -s)∇u)) ds = ãh ij ∂ j (τ h u -u),
where

ãh ij (x, y, t) = 1 0 ∂a i ∂p j (s∇τ h u + (1 -s)∇u)) ds.
Subtracting the PDE in (1.1) for u and for τ h u and dividing by h, we see that D h u is a local weak solution of

∂ t (D h u) -∂ i ãh ij ∂ j (D h u) = d h (x, y, t) • ∇(D h u) in Q ρ/2 . (10.2) Moreover, since ∂a i ∂p j ξ i ξ j = b(|p| 2 )|ξ| 2 + 2b ′ (|p| 2 )p i p j ξ i ξ j ≥ b(|p| 2 )|ξ| 2 ≥ λ p-2 |ξ| 2 in Q ρ/2 by (10.1), we have ãh ij ξ i ξ j ≥ λ p-2 |ξ| 2 in Q ρ/2 . We then test (10.2) with ϕ 2 D h u, where ϕ ∈ C ∞ 0 (Q ρ/2
) is a cut-off function such that ϕ = 1 on Q ρ/3 . By integration by parts and some simple manipulations, it is easy to see that λ p-2

Q ρ/3 |∇D h u| 2 dxdydt ≤ Q ρ/2 ãh ij ∂ i (D h u)∂ j (D h u)ϕ 2 dxdydt ≤ C. It follows that D 2 u ∈ L 2 (Q ρ/3
). Consequently, we obtain that u ∈ W 2,1 2 (Q ρ/3 ) and is a local strong solution of equation (1.1) written in nondivergence form, i.e.:

u t -a ij u ij = f in Q ρ/3 , where a ij = |∇u| p-2 δ ij + (p -2) u i u j |∇u| 2 , f = |∇u| q . (10.3)
Since, by Theorem 2.1(iii), a ij , f are Hölder continuous in Q ρ/3 , it follows from interior Schauder parabolic regularity [13, Theorem III.12.2] that, for some α ∈ (0, 1), u ∈ C 2+α,1+α/2 (Q ρ/4 ). (10.4) (ii) Thanks to (10.4), we know that u is a classical solution of (10.3) in Q ρ/4 . Keeping the above notation, for 0 < h < ρ/8, we then have

(D h u) t -a ij (D h u) ij = F h := D h f + (D h a ij )(τ h u ij ) in Q ρ/8 .
Moreover, as a consequence of (10.4), we have, for 1 < r < ∞,

F h L r (Q ρ/8 ) ≤ C ∇f L r (Q ρ/4 ) + ∇A L r (Q ρ/4 ) D 2 u L ∞ (Q ρ/4 ) ≤ C, 0 < h < ρ/8.
It thus follows from interior parabolic L r estimates (see [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF]Theorem III.12.2]) that, for 0 < h < ρ/8,

D 2 D h u L r (Q ρ/16 ) + ∂ t D h L r (Q ρ/16 ) ≤ C(ρ) F h L r (Q ρ/8 ) + D h u L r (Q ρ/8 ) ≤ C.
We deduce that Du t , D 3 u ∈ L r loc (Q T ). Then differentiating (10.3) in space, we see that the function z = ∂ ℓ u x (ℓ = 1, 2) is a local strong solution of

z t -a ij z ij = f in Q ρ/16 , (10.5)
where f = ∂ ℓ f -u ij ∂ ℓ a ij . Since a ij , f are Hölder continuous in Q ρ/16 due to (10.4), it follows from interior Schauder parabolic regularity [13, Theorem III.12.2] that z ∈ C 2+α,1+α/2 (Q ρ/20 ) for some α ∈ (0, 1).

Proof of Theorem 2.2 (continued). (ii) This is a direct consequence of Lemma 10.1.

(iii) It follows from (i)(ii) that v = u x ∈ C 2,1 (Q T ) ∩C(Ω×(0, T )) is a classical solution of (10.5) in Q T , where a ij are defined in (10.3). Moreover, v = g x = 0 on S T . Taking θ(t) a cut-off in time and setting w = θv, we see that w solves

w t -a ij w ij = f := θ f + θ t v in Q T , (10.6)
with 0 initial-boundary conditions. By [START_REF] Lieberman | Second order parabolic differential equations[END_REF]Theorem 4.28], since f is locally Hölder continuous in Ω×[0, T ) due to (i), there exists a solution to this problem in C 2+β,1+β/2 (Ω× [0, T )) for some β ∈ (0, 1). Since we have uniqueness in the class C 2,1 (Q T ) ∩ C(Ω × [0, T )) by the maximum principle, the conclusion (2.9) follows.

APPENDIX 2. A parabolic version of Serrin's corner lemma

In [17, p. 512], a Serrin corner property in a rectangle was shown for a parabolic equation involving the Laplacian. This was proved by comparison with a suitable product of functions of x, t and y, t. This result and method are no longer sufficient here and we shall establish a result for general nondivergence operators by modifying the original proof of [START_REF] Serrin | A symmetry problem in potential theory Arch[END_REF] for the elliptic case. On the one hand, on K, we have y < a/2, hence x 2 + (y -a) 2 > a 2 /4. On the other hand, using part of assumptions (11.2) along with 0 ≤ a -y ≤ a and 0 ≤ 1 -e α(x 2 +(y-a) 2 +(t-t 1 ) 2 -a 2 ) ≤ 1 on K, it follows that for α > 1 large enough, Lh ≤ αxv -αλa 2 + 2(t 1 -t) + 6a 11 + 2a 22 + 2Ca + 2xB 1 + 2(y -a)B 2 + C ≤ -λα 2 a 2 xv 2 < 0 in K. (11.5) We now set w = z + εh where ε is a positive constant to be chosen. By (11.3) and (11.5), we have Lw < 0 in K. (11.6) Denote M = max K w ≥ 0. Since L is (uniformly) parabolic, by the usual proof of the maximum principle, it follows from (11.6) that w cannot attain the value M in K (observe that for each s ∈ [τ 3 , t 1 ], the section K ∩ {t = s} is an open, possibly empty, subset of R 2 ). To show M = 0 (for sufficiently small ε > 0), it thus suffices to verify that w ≤ 0 on ∂ P K = ∂K \ (K ∩ {t = t 1 }). We have

∂ P K = Γ 1 ∪ Γ 2 , where Γ 1 = ∂K 1 ∩ K 2 and Γ 2 = ∂K 2 ∩ K 1 .
On Γ 1 we have either

x 2 + (y -a) 2 + (t 1 -t) 2 = a 2 or x = 0, so that h = 0 and z ≤ 0, hence w ≤ 0. Next observe that on Γ 2 we have

x 2 + (y -a) 2 + (t 1 -t) 2 < a 2 and x 2 + y 2 = a 2 /4, hence τ 1 < τ 3 ≤ t ≤ t 1 and a/8 < y < a/2 (in other words, (x, y) is "far" from the corners of D). Therefore, by the Hopf boundary point lemma [START_REF] Protter | Maximum principles in differential equations[END_REF]Theorem 6 p. 174] and the strong maximum principle, there exists c > 0 (independent of t 1 ), such that z ≤ -cx on Γ 2 . Choosing ε ∈ (0, c), we then have w ≤ -cx + εx < 0 on Γ 2 .

We have thus proved that M = 0 that is, w ≤ 0 in K. Letting ã := a/(2 √ 2) and noting that {0 < x ≤ y < ã} × {t 1 } ⊂ K, we get z(x, y, t 1 ) ≤ -εh(x, y, t 1 ) = -εxe -αa 2 e α(a 2 -x 2 -(y-a) 2 ) -1 ≤ -εαe -αa 2 x a 2 -x 2 -(y -a) 2 = -εαe -αa 2 x 2ay -x 2 -y 2 ≤ -aεαe -αa 2 xy for 0 < x ≤ y < ã. Now exchanging the roles of x, y and noticing that the assumptions (11.2) are symmetric in x, y, the conclusion already obtained guarantees that also z(x, y, t 1 ) ≤ -aεαe -αa 2 xy for 0 < y ≤ x < ã, hence (11.4) in (0, ã) 2 × [t 0 , τ 2 ). The extension to the remaining part of the rectangle (0, X 1 ) × (0, Y 1 ) (away from the corner (0, 0)) follows from the Hopf boundary lemma and the strong maximum principle.

Theorem 1 . 1 .

 11 Assume (1.2)-(1.6).

Lemma 5 . 3 .

 53 Let ρ, µ, u 0 be as inLemma 3.3(ii). Then GBUS(u 0 ) ⊂ [-ρ, ρ] × {0}.Proof. Denote again Σρ = [-ρ/2, ρ/2] × {0} and Σ ′ ρ = ∂Ω \ ([-ρ, ρ] × {0}). In view of Lemma 5.1, it suffices to show that sup (x,y)∈Σ ′ ρ , t∈(0,T ) |∇u(x, y, t)| < ∞.(5.11)

  3.1] then guarantee the estimate ∇u(x, y, •) C β ([t 0 /2,T )) ≤ C, (x, y) ∈ ω ′ , where β = θ/(1 + θ). Therefore ∇u C β (ω ′ ×[t 0 /2,T )) ≤ C. The conclusion now follows by applying standard Schauder parabolic estimates to the PDE in (1.1), rewritten under the form

Lemma 8 . 3 .

 83 Assume (8.1)-(8.3), let y 1 ∈ (0, L 2 ) and let µ, u 0 satisfy the assumption of Theorem 1.1(ii). Then J ∈ C(D × (0, T ))(8.8)

where M := |∇u| p- 2 + (p - 2 )|∇u| p-4 u 2 x 2 (p - 2 )

 22222 |∇u| p-4 u x u y and w = |∇u| p-2 + (p -2)|∇u| p-4 u 2 y .

. 12 )

 12 Finally observing that(a) + (ã) ≤ 0, (b) + ( b) ≤ 0, (c) + (c) ≤ 0, (d) + ( d) ≤ 0, (e) + (ẽ) ≤ 0

Proposition 11 . 1 .

 111 Let 0 < X 1 < X1 , 0 < Y 1 < Ŷ1 , D = (0, X1 ) × (0, Ŷ1 ) ⊂ R 2 , 0 < τ 1 < τ 2 , Dτ = D × (τ 1 , τ 2 ). Let the coefficients a ij = a ij (x, y, t) satisfy a ij ξ i ξ j ≥ λ|ξ| 2 in Dτ (11.1)for some λ > 0 and assume thata ij , B i ∈ C(Dτ ), a 12 + a 21 ≥ -C(x ∧ y), B 1 ≥ -Cx, B 2 ≥ -Cy in Dτ . (11.2)

  2)/2 and a i (p) = b(|p| 2 )p i where p = (p 1 , p 2 ), so that ∆ p u = ∂ i (a i (∇u)). Following[13, p.445], we write

In this context, the term "nondegeneracy" describes a property of finite-time singularities (like in, e.g., Giga and Kohn[START_REF] Giga | Nondegeneracy of blowup for semilinear heat equations[END_REF]) and should not be confused with the notion of nondegenerate diffusion mentioned after Theorem 1.1.

It seems that the constants in some of the key estimates there are nonuniform as µ → 0 + , which prevents us to argue by a limiting procedure from the case µ > 0.

The other two terms can be rewritten as

and (noticing that (6) can be obtained from ( 5) by formally multiplying with

We shall now collect and relabel the numerous positive and negative terms that we just obtained, when expanding (1)- [START_REF] Fujii | Asymptotic behavior of blowup solutions of a parabolic equation with the p-Laplacian[END_REF] in the process of eliminating u yy . A number of positive and negative terms will then be paired together according to certain cancellations. Then, the remaining positive terms, as well as the terms involving u t , will be eventually controlled by using the negative terms.

Using that d ′ ≤ 0 and F ′ , F ′′ , u y ≥ 0, we first have positive terms:

They give rise to the following decompositions:

We next have terms with a negative sign:

With these terms, we have the following decompositions (using c ′′ = 0):

It follows from (9.4) in Lemma 9.1 and (9.5)-(9.7) that

Reordering the terms, we obtain

Collecting the terms with J (reps., ∇J) in (9.8), together with those in A 2 (resp., H 2 ) and using (9.3), we define

Here we used that γ ≤ α -1 and hence α(q -p)/(q -p + 1) -γ ≥ 1 -α/(q -p + 1) ≥ 0. Assume k 0 = k 0 (p, q, Ω, µ, ∇u 0 ∞ ) > 0 sufficiently small so that

which is possible due to (9.14), x ≤ L 1 and (9.15)). We then have

Next, we have

and, owing to Lemma 3.4,

Here and in the rest of the proof, K denotes a constant depending on u 0 C 2 , p and q. Consequently PJ ≤ (h) + (i) + (j) + (l) + (m) + (3 t ) + (4 t ) + (5 t ) + (6

By (9.13), we have also

Using again Young's inequality, and (9.19), we have

Next, using (9.13) and (9.14), it follows that

and

Finally, using that u ≥ µy, we have

Using the bounds u ≤ u 0 ∞ and (9.14) we have

We also have by (9.13) and (9.14):

The assertion then follows easily from (9.10), (9.12) and (2.7).

10 APPENDIX 1. Proof of regularity results (Theorem 2.2)

Proof of Theorem 2.2(i). We assume

By the results in [13, Chapter V] (see Remark 10.1 below for details), there exists a (unique

, for some α ∈ (0, 1), of the problem

Since v is also a weak solution of (2.1)-(2.3) in Q τ , by uniqueness of weak solutions (cf. Theorem 2.1(ii)), it follows that u = v τ on Q τ , hence (2.7).

Remark 10.1. More precisely, in the special case when u 0 ∈ C 2+α (Ω) and u 0 satisfies the second order compatibility conditions, the existence of v claimed in the above proof follows from [13, Theorem V.6.1]. In the general case u 0 ∈ W 1,∞ (Ω), with u 0 = g on ∂Ω, this follows by a standard approximation procedure of u 0 by such smooth u 0,n . Namely, if v n denotes the solution originating from u 0,n , then, by [13, Theorems V.4.1, V.1.1 and V.5.4] respectively, we get uniform a priori estimates for the sequence v n in the spaces

(Ω × (0, τ ]) for some α ∈ (0, 1). We may then pass to the limit along a subsequence and obtain a solution with the announced properties.

In the proof of Theorem 2.2(ii)(iii), we shall use the following local regularity lemma. We note that only statement (ii) will be used here. The global version of statement (i) was already proved in Theorem 2.2(i). However, we give and prove its local version for completeness, since it was mentioned without proof in [2, p. 2487].

Lemma 10.1. Under the assumptions of Theorem 2.1, let u be the (maximal) weak solution of (1.1) and let P 0 = (x 0 , y 0 , t 0 ) ∈ Q T . Assume |∇u(P 0 )| > 0. Then:

(i) for some α ∈ (0, 1), u is a classical C 2+α,1+α/2 -solution on a space-time neighborhood of P 0 ;

(ii) for some β ∈ (0, 1), ∇u is C 2+β,1+β/2 on a space-time neighborhood of P 0 .

Let z ∈ C 2,1 ( Dτ ) ∩ C( Dτ ) satisfy

Then, for each t 0 ∈ (τ 1 , τ 2 ), there exists c 0 > 0 such that

2 ) and τ 3 = τ 1 +t 0 2 , so that τ 1 < τ 3 < t 0 < τ 2 . Fix t 1 ∈ [t 0 , τ 2 ) and let K 1 := (x, y, t); x 2 + (a -y) 2 + (t 1 -t) 2 < a 2 , x > 0, t ≤ t 1 .

Observe that K 1 ⊂ D × [τ 3 , t 1 ] and set K 2 = B (0, 0), a/2 × [τ 3 , t 1 ] and K = K 1 ∩ K 2 . Now set v(x, y, t) := e -α(x 2 +(y-a) 2 +(t-t 1 ) 2 ) -e -αa 2 , v(x, y, t) = e -α(x 2 +(y-a) 2 +(t-t 1 ) 2 ) , with α > 0 to be chosen later on, and define the auxiliary function h = xv. It is clear that h > 0 in K. We compute 1 -e α(x 2 +(y-a) 2 +(t-t 1 ) 2 -a 2 ) .


R Benouaret

E Goujon

Pascale Goupil
email: pascale.goupil@univ-bpclermont.fr

Grape marc extract causes early perception events, defence reactions and hypersensitive response in cultured tobacco cells

Keywords: Cell death, Grape marc, Hypersensitive reaction, Plant defence, Protease inhibitors, Tobacco BY-2

A polyphenol-rich extract from grapes induced chemoperception in tobacco suspension cells

Introduction

In plant cells, pathogen-derived substances elicit a cascade of reactions conferring plant disease resistance.

Resistant plants respond rapidly to elicitors at the infection site by the hypersensitive response (HR), forming a localized cell collapse in order to restrict the systemic spread of a virulent pathogen. Hypersensitive cell death, which is distinct from necrosis caused by metabolic toxins or severe trauma, is genetically programmed (programmed cell death, PCD), and requires active host cell metabolism [START_REF] Van Doorn | Morphological classification of plant cell deaths[END_REF]. In plants, cell death during HR is similar in some features to apoptosis (a specialized form of PCD) in animals [START_REF] Coll | Programmed cell death in the plant immune system[END_REF].

During plant-pathogen interactions, the perception of elicitor substances by plant cells, and before the HR reaction, leads to specific physiological perturbations such as ion fluxes across the plasma membrane (Ca 2+ influx and K + , Cl -efflux), pH changes, plasma membrane depolarization, oxidative burst and induction of rapid cell death. The HR is subsequently accompanied by defence gene activation, leading to synthesis of phytoalexins and accumulation of pathogenesis-related (PR) proteins [START_REF] Garcia-Brugger | Early signaling events induced by elicitors of plant defenses[END_REF][START_REF] Yang | Signal perception and transduction in plant defence responses[END_REF].

Elicitors include oligosaccharides, glycoproteins, peptides, lipopolysaccharides, and sterols derived from microorganisms or challenged plants [START_REF] Kasparovsky | Ergosterol elicits oxidative burst in tobacco cells via phospholipase A2 and protein kinase C signal pathway[END_REF][START_REF] Montesano | Pathogen derived elicitors: searching for receptors in plants[END_REF]. Highly sophisticated and complex biological processes underlie the interaction between these compounds and the host plants. Early events occurring in the host have largely been deciphered using tobacco cell suspensions [START_REF] Garcia-Brugger | Early signaling events induced by elicitors of plant defenses[END_REF]. Modification of membrane properties is not systematically correlated with the activation of downstream defence events. Also, resistance to pathogen invasion is not necessarily associated with cell death [START_REF] Gilchrist | Programmed cell death in plant disease: the purpose and promise of cellular suicide[END_REF][START_REF] Klarzynski | Linear -1,3 glucans are elicitors of defense responses in tobacco[END_REF]. The structural diversity of elicitors can be expected to have a significant influence on plant cell perception, cell death induction and/or defence gene activation.

The prospect of disease control using a plant's own resistance mechanisms has prompted increasing interest in the development of agents that can mimic natural inducers of plant defence systems. These are named plant defence inducers (PDIs), and act at various points in the signalling pathway leading to disease resistance. PDIs can be organic, inorganic, botanical or synthetic [START_REF] Lyon | Agents that can elicit induced resistance[END_REF][START_REF] Walters | Induced resistance for plant disease control: maximizing the efficacy of resistance elicitors[END_REF]. Although many of these compounds are non-specific and induce resistance in a wide range of crop species against a diverse range of plant pathogens, differences in efficiency are reported [START_REF] Lyon | Agents that can elicit induced resistance[END_REF]. Grape marc extract (GME) is a plant extract Author manuscript, published in Plant Physiology and Biochemistry (2014) 77: 84-89 DOI:10.1016/j.plaphy.2014.01.021 3 that acts as an efficient PDI. On application to tobacco leaves, this wine by-product elicits a variety of defence reactions such as local injury, biochemical changes and systemic molecular response with upregulation of PR proteins [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. As a potent defence elicitor in tobacco, Arabidopsis or tomato [START_REF] Benouaret | Plant defence inducer properties of grape marc extract[END_REF], GME might thus prove to be a useful alternative tool for environmentally oriented phytoprotection.

The present report investigates the ability of GME to initiate chemoperception and induce defence mechanisms in tobacco cultured cells. We show that GME triggered rapid extracellular alkalinization and activated defencerelated genes, ultimately leading to cell death. Using pharmacological inhibition of proteases, we evidenced a hypersensitive status of tobacco BY-2 cells that led to cell death on GME treatment.

Materials and methods

Biological compounds and chemicals

The grape marc extract (GME), a Vitis vinifera L. hydroalcoholic extract, was supplied as a red powder by Grap'Sud (Cruviers-Lascours, France) as described by [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. Batch #11332 used in our experiments was produced from grapes harvested from several red wine varieties in 2011. According to the company, this grape end-product is 95% dry weight (DW) and contains 66.1% polyphenols including 25.4% anthocyanins. The flavonoid-based compounds are responsible for the red colour of the dry extract, and the acidity (pH 4.2) of the aqueous solution when it is dissolved in water. Control experiments were conducted with ultrapure water acidified with HCl to pH 4.2.

Plant material

Tobacco cell suspensions of Nicotiana tabacum L. cv. Bright Yellow-2 (BY-2), were grown at pH 5.8 in MS medium (Duchefa, the Netherlands) supplemented with sucrose (30 g.L -1 ), thiamine (1 mg.L -1 ), myo-inositol (102 mg.L -1 ) and 2,4-dichlorophenoxyacetic acid (0.2 mg.L -1 ). Cells were maintained in the dark on a rotary shaker (140 rpm, 25°C) in 250 mL conical flasks. Subcultures were made weekly by dilution at a 1:15 ratio in fresh medium. All the experiments were performed using cells in exponential growth phase at day 5 after subculture. For alkalinization analysis, cells were collected by filtration, washed with 175 mM mannitol, 0.5 mM CaCl 2 , 0.25 mM MgCl 2 , 1 mM KCl and 1 mM sucrose, and resuspended at 0.1 g fresh weight (FW)/mL in the same medium. The initial pH was about 4.8. For transcript accumulation analysis, cells were collected by filtration, and resuspended at 0.3 g FW/L in MS medium. Cells were equilibrated for 2 h in open 6well microplates (Greiner Bio-one, Germany) with continuous stirring, and then elicited with GME.

Appendix S1 Sequences of gene-specific primers used for quantitative real-time RT-PCR PR: pathogenesis-related protein

Extracellular alkalinization

The pH variations of the culture medium were recorded by introducing a glass microelectrode (Hanna Instrument, HI1330B, France) into 6 ml of the equilibrated cell suspension culture.

The pH measurement, performed every 10 min, started immediately after treatment with elicitors or acidified water. The pH was measured at 10 min intervals, relative to the pH measured at t 0 immediately after adding GME or acidified water (control). Each experiment was repeated three to six times to check for reproducibility.

Figures describe the results of typical experiments. according to the manufacturer's instructions.

Real-time RT-PCR

PCR reactions were prepared using the qPCR kit Mastermix for SYBR green (Eurogentec) according to the manufacturer's protocol. The quantitative assessment of mRNA levels was performed using an iCycler iQ5 (Bio-Rad). The cDNA concentration used produced a threshold cycle value (C T ) of between 15 and 30 cycles.

Amplification specificity was checked by melting-curve analysis. The EF-3 was used as an internal control [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. Quantification of expression ratios was performed according to the mathematical model developed by [START_REF] Pfaffl | A new mathematical model for relative quantification in real-time RT-PCR[END_REF]. Primers and amplicon sizes are given in Appendix S1. DOI:10.1016/j.plaphy.2014.01.021 5

Cell death assay

The Evans Blue assay was performed according to [START_REF] Amano | A versatile assay for the accurate, time-resolved determination of cellular viability[END_REF]. Tobacco cells (500 µL) were withdrawn from the culture medium immediately after GME elicitation (t 0 ) and then every 24 h for 3 days. The cells were spread onto a cotton swab placed at the bottom of a 10 mL plastic syringe. The cells were washed with 10 mL of distilled water and stained with 500 µL of Evans Blue solution (0.25% w/v in distilled water) for 5 min at room temperature. Excess unbound dye was removed by thorough washing with 50 mL of distilled water. The cotton swab containing cells was soaked with 500 µL SDS solution (0.5%, w/v), and the syringe was immediately heated for 3 min in a boiling water bath. The dye bound to dead cells was eluted with 2.5 mL of distilled water, and remaining drops of dye were squeezed out using the syringe plunger.

Absorbance at 600 nm was measured to estimate cell death. Each experiment was repeated three times.

Pharmacological treatments

We used EGTA (2 mM and 5 mM), LaCl 3 (2 mM) as chelator and Ca 2+ channel blocker respectively, pepstatin (1 µM), E-64 (1.5 µM), aprotinin (0.2 µM) as protease inhibitors, and cycloheximide (50 µM) as protein synthesis inhibitor. Pepstatin was dissolved in ethanol and the other chemicals in water. All the chemicals were purchased from Sigma-Aldrich (France).

Statistics

All the experiments were performed in biological and technical triplicates. The values were expressed as mean ± standard error of the mean (SE). Letters above bars indicate statistically significant differences between samples, according to one-way ANOVA followed by LSD Significant test (p ≤ 0.05) using Statistix 9.® software.

Results

Effect of GME on proton flux and Ca 2+ mobilization GME elicits tobacco BY-2 cells. Suspension

cells responded to GME by a rapid, sustained alkalinization of the incubation medium (Fig. 1). Medium alkalinization was detected immediately after addition of GME (0.25%), and the pH shift reached a plateau with a pH of 1.6 units at 80 min. No pH shift was detected with the control cells treated with water. GME added to the cell-free medium did not cause any pH modifications. The intensity of the GME effect was dose-dependent.

The pH shift reached 0.9 units with 0.125% GME, and the 0.0625% concentration did not induce any extracellular alkalinization.

Author manuscript, published in Plant Physiology and Biochemistry (2014) 77: 84-89 DOI:10.1016/j.plaphy.2014.01.021 6

The involvement of calcium in GME-induced extracellular alkalinization was investigated by adding EGTA (2 mM or 5 mM) or LaCl 3 (2 mM) to the culture medium concomitantly with GME. The pH kinetics were measured for 120 min (Fig. 2). Both Ca 2+ -chelator and the Ca 2+ surrogate La 3+ radically suppressed the alkalinization. The extracellular pH rise was reduced to the same extent with both EGTA concentrations, suggesting that the saturating chelation threshold was reached at 2 mM. These compounds dramatically suppressed the pH shift when added subsequently to GME elicitation. They had no effect on the pH medium in control cells. water control with 2 mM EGTA ( * ), GME (□), GME with 2 mM EGTA (o), GME with 5 mM EGTA (∆), GME followed by addition of 2 mM EGTA (-x---x-) at 30 min (arrow). B:

water control (◊), water control with 2mM EGTA ( * ), GME (□), GME with 2 mM LaCl 3 (o), GME followed by addition of 2 mM LaCl 3 (-x---x-) at 30 min (arrow). The data are from representative experiments out of a total of five. GME resulted in an increased CDR, which was 3 to 4 times higher than in the control. The CDR reached 22%, 32% and 39% by 24 h, 48 h and 72 h respectively. The 0.125% GME concentration reduced the magnitude of CDR, which reached 19%, 29% and 30% by 24 h, 48 h and 72 h respectively. A lower concentration of GME (0.0625%) resulted in an even lower CDR, with an average of 21% stabilized over the treatment.

Induction of defence-related gene expression following GME treatment

Pharmacological analysis of GME-induced cell death

To determine whether GME-induced cell death was related to hypersensitive reaction (HR), the action of plant cell death inhibitors was investigated on GME-treated cells. Experimentally, BY-2 cells were treated with or without 0.25% GME for 24 h and 48 h, and cell death was quantified in the presence or absence of chemicals added 30 min before CDR determination.

We first examined the effect of cycloheximide, a widely-used inhibitor of protein synthesis. Cycloheximide reduced CDR to 18% and 25% at 24 h and 48 h respectively in GME-treated tobacco cells (Fig. 5), suggesting that the GME-induced cell death process depends on de novo protein synthesis. Thereafter, we investigated the action of protease inhibitors on GME-induced cell death. As shown in Fig. 5, aprotinin (0.2 µM), a serine protease inhibitor, suppressed the GME-induced cell death at 24 h or 48 h by 43% and 38% respectively.

Pepstatin and E-64, inhibitors of cystein and aspartic proteases, also affected GME-induced cell death. Pepstatin reduced CDR by 36% and 42%, and E-64 inhibitor less efficiently, by 16% and 27% at 24 h and 48 h respectively. Neither inhibitor significantly affected cell viability by itself in control tobacco cells. Taken together, these data evidence the involvement of proteolytic events in GME-induced cell death.

Discussion

Medium alkalinization is a common parameter employed to monitor the primary action of elicitors. It has been described as a highly sensitive assay of elicitor perception [START_REF] Boller | Chemoperception of microbial signals in plant cells[END_REF]. GME induced extracellular alkalinization of BY-2 cells characterized by long-sustained pH rise. Kinetics of changes in extracellular pH vary among elicitors, and do not predict their type or nature. Tobacco cells reacted to GME with an alkalinization response in a manner resembling the perception of several elicitors such as cyclodextrins, methyljasmonate, oligogalacturonides or cryptogein [START_REF] Almagro | Early signaling network in tobacco cells elicited with methyl jasmonate and cyclodextrins[END_REF][START_REF] Binet | Comparison of the effects of cryptogein and oligogalacturonides on tobacco cells and evidence of different forms of desensitization induced by these elicitors[END_REF], whereas the reaction with laminarin or ergosterol was transient, with a return to the resting value [START_REF] Klarzynski | Linear -1,3 glucans are elicitors of defense responses in tobacco[END_REF][START_REF] Vatsa | Cytosolic calcium rises and related events in ergosterol-treated Nicotiana cells[END_REF]. GME is a polyphenol-enriched extract without measurable jasmonic acid [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF] or methyljasmonate (data not shown). The GME early elicitor responses involve rapid modification in proton influx and calcium mobilization, as evidenced by the decreased pH shift in the presence of the Ca 2+ influx inhibitors EGTA or LaCl 3 . These data show that tobacco cells perceived GME by modifying plasma membrane biophysical properties. Later on, GME could activate late elicitor responses with PR3, PAL and CCoAOMT gene overexpression in tobacco cells. Since GME induced upregulation of defence-related genes, the GME-bioactive Our group had previously demonstrated that GME was an inducer of dose-dependent cell death in tobacco plants [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. Here, using a calibrated Evans Blue assay, we evidenced GME-induced cell death in cultured tobacco cells. GME reduced BY-2 cell viability by 24 h. The three GME doses (0.25%, 0.125% and 0.0625%) tested were effective. The lowest GME concentration was ineffective on tobacco leaves [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF], but sufficient to induce cell death in tobacco cells. Such a difference could be related to tobacco suspension cells and leaf tissues varying in their sensitivities to GME-bioactive molecules.

Localized cell death on plant tissue resulting from attack by pathogens might occur either in hypersensitive resistant response or in susceptible reactions such as necrotic symptom potentially caused by toxins [START_REF] Iakimova | Hypersensitive cell death in plants-Its mechanisms and role in plant defence against pathogens[END_REF]. Plant defence inducers (PDIs) used as stimulus-triggering factors could mobilize plant defence and provoke local necrotic tissue when infiltrated in plant leaves [START_REF] Kulye | Hrip1, a novel protein elicitor from necrotrophic fungus, Alternaria tenuissima, elicits cell death, expression of defence-related genes and systemic acquired resistance in tobacco[END_REF][START_REF] Mercier | The algal polysaccharide carrageenans can act as an elicitor of plant defence[END_REF]. Whether the PDI mediates hypersensitive lesions or phytotoxicity should be investigated to provide an exploitable perspective of these biomolecules for phytoprotection. Our previous investigation of GME elicitor activity on tobacco leaves demonstrated local defence reaction with tissue injury on tobacco, Arabidopsis and tomato leaves [START_REF] Benouaret | Plant defence inducer properties of grape marc extract[END_REF][START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. The growth stage of the plant organ could markedly influence the GME-induced resistance reactions. Young growing leaves were much less reactive to GME elicitor molecules than mature ones. The importance of phenology in achieving the GME-defence reaction supports the idea that GME elicits cells that enter a highly regulated cell death process rather than a phytotoxicity process.

In the present study, we investigated the GME-induced HR on BY-2 cells using pharmacological approaches.

The inhibitory effect of cycloheximide on GME-induced cell death indicates that GME triggered an active process requiring protein synthesis. Also, proteolytic events may take part in the mechanisms leading to cell death. Serine-, aspartic-and cysteine-protease inhibitors all partially suppressed GME-induced cell death. All these classes of proteases have been shown to be involved in programmed cell death (PCD) in mammalian systems, with the cysteinyl-aspartic-proteases (caspases) as key cell death executioners [START_REF] Grütter | Caspases: key players in programmed cell death[END_REF]. Caspaselike activity also executes PCD in plant cells [START_REF] Coll | Programmed cell death in the plant immune system[END_REF][START_REF] Woltering | Do plant caspases exist?[END_REF], and the active process operates through constitutively expressed machinery [START_REF] Elbaz | Constitutive caspase-like machinery executes programmed cell death in plant cells[END_REF]. Their potential involvement in HR has

Fig. 1 :

 1 Fig. 1: Extracellular alkalinization in tobacco BY-2 suspension cells on addition of different concentrations of GME. (◊) water control, (o) 0.0625% GME, (∆) 0.125% GME, (□) 0.25% GME. (x) 0.25% GME without BY-2 cells. The data are from representative experiments out of a total of five.

Fig. 2 :

 2 Fig. 2: Extracellular alkalinization in tobacco BY-2 suspension cells of 0.25% GME and Ca 2+ inhibitors. A: water control (◊),

  We verified the ability of GME to induce some defence reactions in suspension tobacco cells. Transcript accumulation was measured for two PR genes, the antimicrobial PR1 and the endochitinase PR3 involved in defence responses. The induction of mRNA accumulation was also assessed for genes involved in the synthesis of phenylpropanoids and lignin encoding phenyl-ammonia-lyase (PAL) and caffeoyl CoA 3-O-methyltransferase (CCoAOMT). Transcript levels were quantified in tobacco BY-2 cells treated with 0.25% GME at different time intervals (from 0 to 24 h). The ratios of transcript levels in treated cells to those in control cells are shown in Fig.3.Interestingly, PAL and CCoAOMT were induced rapidly in GME-treated tobacco cells. The maximum accumulation for both genes was detected at 12 h treatment, and thereafter there was a noticeable reduction. The induction was unexpectedly high, 36-fold for PAL, 110-fold for CCoAOMT and noticeable for PR3. Surprisingly, GME had no effect on the transcriptional induction of the defence marker PR1, even though upregulation had previously been demonstrated on tobacco leaves sprayed or infiltrated with the same GME extracts[START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. Since the real-time RT-PCR provided PR1 C T values similar in control and GME elicited cells (Appendix S2), BY-2 cultured cells should produce high PR1 transcript levels. This could finally reduce the ratio of transcript levels in treated cells to those in control cells making PR1 overaccumulation undetectable in GME-treated cells. GME that might induce different defence responses in tobacco cell cultures and plant cannot be rule out.

Fig. 3 :

 3 Fig.3: Effects of GME on transcript accumulation of defence-related genes in tobacco BY-2 cells. Relative transcript levels of genes of PR1, PR3, PAL and CCoAOMT in cells challenged for 0, 6, 12 and 24 h with 0.25 % GME. Transcript levels assessed by real-time RT-PCR are expressed as mean ± SE (n = 3) of transcript ratios relative to control cells (water treatment).

Fig. 4 :

 4 Fig. 4: Time course of cell death rate assayed by Evans Blue staining of tobacco BY-2 suspension cells treated with different concentrations of GME. Bars represent the mean values ± SE, p ≤ 0.05.

Fig. 5 :

 5 Fig. 5: Effects of cycloheximide and protease inhibitors on GME-induced cell death. Chemicals were added 30 min before cell death measurement in BY-2 cells treated with water (control) or 0.25% GME for 24 h and 48 h. The final concentration of each chemical was 50 µM cycloheximide (+CHX), 0.2 µM aprotinin (+APR), 1 µM pepstatin (+PEPS), and 1.5 µM E-64 (+E-64). Bars represent the mean values ± SE, p ≤ 0.05.

  Tobacco BY-2 cell samples (100 mg) were ground in liquid nitrogen, and total RNAs

were isolated using CTAB extraction buffer according to

[START_REF] Chang | A simple and efficient method for isolating RNA from pine trees[END_REF]

. Total RNAs were cleaned up with 1 U DNase I solution (Euromedex, France) containing 40 U RNase inhibitor (Euromedex, France). RNA integrity was verified on a 1% agarose gel by detecting ribosomal RNAs. First-strand cDNA was synthesized from 1 µg total RNA using oligo d(T) 15 primers and Euroscript Reverse Transcriptase

(Eurogentec, France)

Author manuscript, published in Plant Physiology and Biochemistry (2014) 77: 84-89 DOI:10.1016/j.plaphy.2014.01.021 Acknowledgements: We thank Céline Sac for help with the tobacco cell cultures and laboratory assistance. This work was supported by grants from the FUI: Fonds Unique Interministériel.

Contributions: RB performed the research and statistical analysis, EG set up the cell death procedure, RB and PG drew the figures and analysed the data. PG designed the research and drafted the manuscript. RB and PG revised the manuscript.

Author manuscript, published in Plant Physiology and Biochemistry (2014) 8

Taken together, modified proton flux, the Ca 2+ commitment and the defence-related transcript accumulation in GME-treated tobacco cells are evidence that the suspension cellular biological system reacts to GME treatment by inducing characteristic elicitation and defence responses.

Determination of GME-treated tobacco cell death GME triggers a hypersensitive-like response

corresponding to cell death in tobacco plants [START_REF] Goupil | Grape marc extract acts as elicitor of plant defence responses[END_REF]. The ability of GME to cause cell death in tobacco suspension culture was investigated using the Evans Blue assay. The pigment enters cells with damaged plasmalemma [START_REF] Baker | An improved method for monitoring cell death in cell suspension and leaf disc assays using evans blue[END_REF]. The assay was first calibrated with a set of viability standards using mixtures of living and dead tobacco BY-2 cells (Appendix S3). A near-perfect correlation coefficient (0.989) attests the accuracy of the Evans Blue method for theoretical dead cell content and absorbance at 600 nm.

The time course of cell death rate (CDR) was investigated in tobacco cells exposed to 0.25%, 0.125% and 0.0625% GME up to 72 h. When BY-2 cells were cultivated in the absence of GME (control cells), the CDR was below 10%, and this level remained constant for up to 72 h (Fig. 4). The exposure of tobacco cells to 0.25% 12 been reported following the infection of TMV in tobacco [START_REF] Del Pozo | Expression of the baculovirus p35 protein in tobacco affects cell death progression and compromises N gene mediated disease resistance response to TMV[END_REF]. Our data demonstrate that proteolytic events with caspase-like proteases take part in the mechanisms leading to cell death in GME-elicited BY-2 cells, and the regulated process was arrested by chemicals within minutes (experimentally 30 minutes).

Since plant cells constitutively express caspase-like machinery to run cell death programs [START_REF] Elbaz | Constitutive caspase-like machinery executes programmed cell death in plant cells[END_REF], our data suggest that chemicals could rapidly reverse the induction of PCD and block the morphological changes that tobacco cells undergo under the influence of GME.

GME is typically regarded as a waste by-product in the lucrative winemaking industry [START_REF] Bustamante | Agrochemical characterisation of the solid by-products and residues from the winery and distillery industry[END_REF].

This biomass is a rich source of antioxidative phenolic compounds, typically including anthocyanins, catechins, resveratrol, phenolic acids and procyanidins [START_REF] Lu | The polyphenol constituents of grape pomace[END_REF], with a broad spectrum of pharmacological, medicinal, and therapeutic properties [START_REF] Vislocky | Biomedical effects of grape products[END_REF]. The anticancer effects of grape antioxidants have been shown to induce cell cycle arrest and apoptosis in cancer cells with proposed mechanisms involving antioxidant, anti-inflammatory, and antiproliferative activities [START_REF] Seeram | In vitro antiproliferative, apoptotic and antioxidant activities of punicalagin, ellagic acid and a total pomegranate tannin extract are enhanced in combination with the other polyphenols as found in pomegranate juice[END_REF][START_REF] Zhou | Potential anticancer properties of grape antioxidants[END_REF]. Grape antioxidants acting as free radical scavengers and chelating agents help to reduce physiological reactive oxygen species (ROS), which are known as important mediators of apoptosis (Matès and Sánchez-Jiménez, 2000). In plants, research on redox-dependent cell death has demonstrated that ROS levels may orchestrate necrosis vs. PCD. ROS overaccumulation building up to phytotoxic levels leads to a necrotic phenotype. By contrast, when the accumulation of ROS is insufficient to kill the cell directly, changes in cellular redox homeostasis appear to switch on a signalling cascade, leading to PCD encountered during HR [START_REF] Van Breusegem | Reactive oxygen species in plant cell death[END_REF]. In our plant biological system, we demonstrate that the grape antioxidant-rich extract induced cell death closely associated with proteases and dynamic changes in protein synthesis. The GMEinduced cell death may be part of a defensive process related to the HR. An investigation of ROS as a potential player in the activation of the HR, along with the identification of GME-containing bioactive ingredients, should help characterize the pathways leading to hypersensitive GME-induced cell death in tobacco.

Author manuscript, published in Plant Physiology and Biochemistry (2014) 77: 84-89 DOI:10.1016/j.plaphy.2014.01.021