P Cannarsa
email: cannarsa@mat.uniroma2.it

G Da Prato
email: daprato@sns.it

H Frankowska
email: helene.frankowska@imj-prg.fr

DOMAIN INVARIANCE FOR LOCAL SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS IN HILBERT SPACES

Keywords: . 2010 Mathematics Subject Classification. 58D25, 47H06, 37L25 Evolution equations, invariance, viability, dissipative operators, distance function

come

Introduction

It is well known that the maximum principle is a very powerful and flexible tool in the study of partial differential equations of elliptic and parabolic type. For instance, given a domain O ⊂ R n , such a principle guarantees that the solution to the semilinear initial-boundary value problem

     ∂X ∂t = ∆X + F (t, ξ, X) in ]0, ∞[×O X = 0 on ]0, ∞[×∂O X(0, ξ) = x 0 (ξ)
ξ ∈ O a.e.

(1.1) is nonnegative whenever x 0 0 and F : R + × O × R → R satisfies F (t, x, 0) 0 on ]0, ∞[×∂O together with some growth condition ensuring the well-posedness of (1.1).

On the other hand, it is natural to ask whether the above condition is necessary for X to be nonnegative. More generally, in the spirit of inverse problems, it is interesting to detect the structure of all possible nonlinearities F which are compatible with the nonnegativity of solutions. In particular, as we show below for the above system under suitable growth assumptions, solutions stay nonnegative if and only if for a.e. (t, ξ) ∈ R + × O F (t, ξ, v) Cv ∀v 0 for some constant C 0.

In this paper, we address the above problem in a rather general context giving various applications to parabolic initial-boundary value problems like (1.1) as well as equations with nonlocal terms.

We now introduce the set-up of this paper. In a real separable Hilbert space H, with scalar product •, • and norm | • |, consider the equation X (t) = AX(t) + f t, X(t) , t > t 0 X(t 0 ) = x 0 , (1.2) where A : D(A) ⊂ H → H is a densely defined negative self-adjoint linear operator on H and f : [0, ∞[×H → H is assumed to be locally square summable in t and locally Lipschitz in x. Then, it is well-known that, for any (t 0 , x 0 ) ∈ [0, ∞[×H, problem (1.2) has a unique maximal solution X(•; t 0 , x 0 ), defined on some interval [t 0 , T (t 0 , x 0 )[ which can be proved to equal [t 0 , ∞[ under additional assumptions. In the latter case, we say that the maximal solution is global.

We say that a nonempty closed set K ⊂ H is invariant under (1.2) if, for all t 0 0 and x 0 ∈ K, we have that X(t; t 0 , x 0 ) ∈ K for all t ∈ [t 0 , T (t 0 , x 0 )[. More generally, given a complete separable metric space U and a Lebesgue measurable map u : [t 0 , ∞[→ U , called control, one can also consider the control system X (t) = AX(t) + f (t, X(t), u(t)), u(t) ∈ U X(t 0 ) = x 0 .

(1.3)

Given a control u(•) we denote by X(•; t 0 , x 0 , u) the maximal solution of (1.3) and by T (t 0 , x 0 , u) the right end-point of the interval on which it is defined. Then, K is called viable under (1.3) if for every initial condition (t 0 , x 0 ) ∈ R + × K there exists a control u : [t 0 , ∞) → U such that X(t; t 0 , x 0 , u) ∈ K for all t ∈ [t 0 , T (t 0 , x 0 , u)[. Notice that the notion of invariance can be introduced even for control systems by requiring that, for every (t 0 , x 0 ) ∈ R + × K and every control u : [t 0 , ∞) → U , X(t; t 0 , x 0 , u) ∈ K for all t ∈ [t 0 , T (t 0 , x, u)[. Such a property is clearly more restrictive than viability.

There is an extensive literature addressing domain invariance issues in infinite dimensional spaces, although only part of it can be applied to partial differential equations. For instance, Martin [START_REF] Martin | Differential equations on closed subsets of a Banach space[END_REF] studied the invariance of K under equation (1.2) in Banach spaces, in the special case of A = 0, extending the classical condition introduced by Nagumo [START_REF] Nagumo | Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen[END_REF]. Then, Pavel [START_REF] Pavel | Invariant sets for a class of semi-linear equations of evolution[END_REF] established necessary and sufficient conditions for the invariance of K under (1.2) in Banach spaces, assuming the semigroup generated by A, e tA , to be compact and f to be continuous in both variables. In [START_REF] Pavel | Semilinear equations with dissipative time-dependent domain perturbations[END_REF], the same author removed the compactness hypothesis on e tA replacing it with the dissipativity of f . In both papers [START_REF] Pavel | Invariant sets for a class of semi-linear equations of evolution[END_REF] and [START_REF] Pavel | Semilinear equations with dissipative time-dependent domain perturbations[END_REF], the condition for invariance takes the form

lim λ↓0 d K e λA x + λf (t, x) λ = 0 ∀t ∈ [0, ∞[ , ∀x ∈ K. (1.4)
Later on, in [START_REF] Shi | Viability theorems for a class of differential-operator inclusions[END_REF], Shi studied an analogous viability problem for a differential inclusion, that is, for a set-valued map f . Assuming both K and e tA to be compact, he derived a necessary and sufficient condition, analogous to (1.4), with lim λ↓0 replaced by lim inf λ↓0 (lower Dini derivative). A detailed exposition of this theory, and more, can be found in the monograph [START_REF] Cârjȃ | Viability, invariance and applications[END_REF] by O. Carja, M. Necula and K. Vrabie. We observe that a common feature of the above results is that conditions for invariance are:

• imposed at (boundary) points of K, and • expressed in terms of the semigroup e tA .

In [START_REF] Cannarsa | Invariance for quasi-dissipative systems in Banach spaces[END_REF], assuming f = f (x) in (1.2) to be continuous and quasi-dissipative, we showed that a necessary and sufficient condition for the invariance of K is that all points x ∈ D(A) \ K, sufficiently close to K, satisfy the inequality

D -d K (x) (Ax + f (x)) C d K (x) (1.5)
for some constant C 0, where D -d K (x) (Ax + f (x)) denotes the lower Dini directional derivative of d K at x in the direction Ax + f (x). Notice that (1.5) is formulated just in terms of the generator A, which is simpler to use in applications than (1.4).

Next, consider the controlled heat equation

     ∂X ∂t (t, ξ) = ∆X(t, ξ) + g(t)X(t, ξ) (t > 0, ξ ∈ O) X = 0, on (0, ∞) × ∂O X(0, ξ) = x 0 (ξ), ξ ∈ O, (1.6)
where O ⊂ R n is a bounded domain with C 2 boundary and g ∈ L ∞ (0, T ) is called a bilinear control. For a given smooth convex function φ : R → R, an interesting viability problem is to find conditions to ensure the existence of a control g such that the corresponding maximal solution X g of (1.6) remains in the φ-energy level of x 0 , that is

O φ X g (t, ξ) dξ = O φ x 0 (ξ) dξ ∀t ∈ [0, T (0, x 0 )[. (1.7)
We note that a special case of the above problem, for φ(s) = s 2 , is considered by Caffarelli and Lin in [START_REF] Caffarelli | Nonlocal heat flows preserving the L 2 energy[END_REF]. As we show in section 4, the problem of determining g to satisfy (1.7) can be reduced to the invariance, under (1.2), of suitable level sets of the integral functional associated with φ.

More generally, in section 3, we study the invariance of a closed set K under (1.2) assuming that f is defined just on the domain of some fractional power of A. First, we give sufficient conditions for invariance that are formulated-like (1.5)-in terms of the lower Dini derivatives of the distance function d K at points of D(A) which are exterior to K (Theorem 3.2 and Corollary 3.3). Then, under further assumptions, we show that the above conditions are also necessary (Theorem 3.7 and Corollary 3.9). These results become particularly simple when K is proximally smooth since, in this case, Dini derivatives are replaced by scalar products with proximal normals (Theorem 3.4 and Theorem 3.10). Notably, in the latter case, one only needs to impose conditions on ∂K ∩ D(A).

As for control system (1.3), when K is invariant under the action of e tA , we provide a sufficient condition for viability under an additional compactness assumption for e tA .

Finally, we would like to stress that this paper is restricted to Hilbert space settings for two main reasons: the use of maximal L 2 -regularity for solutions of linear evolution equations and the convenience of formulating our condition for invariance in terms of proximal normals. Part of our results could certainly be extended to suitable classes of Banach spaces, and also to set-valued operators associated with maximal dissipative graphs. Nevertheless we prefer to avoid giving here such partial extensions for the sake of generality of final results obtained in the setting of Hilbert spaces.

This paper is organized as follows. In section 2, we discuss assumptions and well-posedness for problems (1.2) and (1.3). In section 3, we derive our conditions for invariance. In section 4, we study a bilinear control problem for system (1.6), transforming it into an invariance problem, and use invariance to extract nontrivial information on the maximal time of existence. In section 5, we analyse viability under the control system (1.3). In section 6, we prove proximal smoothness for level sets of certain integral functionals.

Preliminaries

Let H be a real separable Hilbert space with scalar product •, • and norm | • |. For any x ∈ H and r > 0 we set

B r (x) = y ∈ H : |y -x| < r , B r = B r (0) .
We denote by L 1 the Lebesgue measure on R and set

R + = [0, ∞[ , R * + =]0, ∞[ , and r = min n ∈ N : r n ∀r ∈ R + . We say that a measurable set F ⊂ R + is of full measure if L 1 R + \ F = 0.
For any (t 0 , x 0 ) ∈ R + × H, consider the Cauchy problem

X (t) = AX(t) + f t, X(t) , t > t 0 X(t 0 ) = x 0 (2.1)
where A is a linear operator on H and f is a nonlinear term. In this paper, both A and f are allowed to be unbounded.

2.1. Assumptions. In this section, we introduce the assumptions that will be enforced in the rest of the paper. We begin with operator A which is hereafter supposed to satisfy the following.

(H A ): A : D(A) ⊂ H → H is a densely defined self-adjoint linear operator such that A -ωI for some ω > 0. Assumption (H A ) ensures that A is the infinitesimal generator of a strongly continuous semigroup of contractions on H, which we denote by e tA (t 0). Moreover, e tA is analytic and e tA x ∈ D(A) for all t > 0 and x ∈ H.

For any θ ∈ R we denote by (-A) θ the fractional powers of -A (see, for instance, [17, Section 2.6]) and we set

H θ = D (-A) θ/2 with |x| θ = |(-A) θ/2 x|, ∀x ∈ H θ (θ 0)
We recall that, for all θ 0,

ω θ/2 |(-A) -θ/2 x| |x| ∀x ∈ H, (2.2
)

ω θ/2 |x| |x| θ ∀x ∈ H θ , (2.3
) and, for all 0 θ 1,

ω (1-θ)/2 |x| θ |x| 1 ∀x ∈ H 1 , (2.4)
where ω > 0 is the constant given by (H A ).

We now state the assumptions on the nonlinear term f . (H f ):

f : R + × H θ → H for some θ ∈ [ 0, 1 ] and (a) for all x ∈ H θ , t → f (t, x) is Lebesgue measurable on [ 0, ∞ [; (b) there exists a function L : R + × R + → R + with r → L(t,
f (t, x) -f (t, y), x -y f (t, x) -f (t, y) |x -y| L(t, r) |x -y| θ |x -y| (2.8)
for all x, y ∈ H θ with |x| θ , |y| θ r. Consequently, for every > 0, every r 0, and all x, y as above we have that

f (t, x) -f (t, y), x -y 2 |x -y| 2 θ + L(t, r) 2 2 |x -y| 2 . (2.9)
Typical examples of partial differential equations which can be recast in the form (2.1), with A and f satisfying assumptions (H A ) and (H f ) respectively, are semilinear parabolic equations.

Example 2.2. Let O ⊂ R n (n 3) (1) be a bounded domain with boundary of class C 2 . Consider the semilinear initial-boundary value problem

     ∂X ∂t = ∆X + F (t, ξ, X) in ]0, ∞[×O X = 0 on ]0, ∞[×∂O X(0, ξ) = x 0 (ξ) ξ ∈ O a.e.
(2.10)

where

F : R + × O × R → R is such that v → F (t, ξ, v) is of class C 1 (R) for a.e. (t, ξ) ∈ R + × O, (t, ξ) → F (t, ξ, v) is Lebesgue measurable for all v ∈ R, (2.11)
and satisfies, for some given (2.12) the growth conditions

p ∈ 1, n n -2 , ϕ 0 ∈ L 2 loc (R + ; L 2 (O)), and ϕ 1 ∈ L 2 loc (R + ; L n (O)),
|F (t, ξ, v)| C 0 ϕ 0 (t, ξ) + |v| p (2.13) ∂F ∂v (t, ξ, v) C 1 ϕ 1 (t, ξ) + |v| p-1 (2.14)
for a.e. (t, ξ) ∈ R + × O, every v ∈ R, and some constants C 0 , C 1 0.

We will now show that the above problem can be recast as an evolution equation like (2.1), for some operator A and nonlinear map f satisfying assumptions (H A ) and (H f ), respectively. In the Hilbert space

H = L 2 (O), with norm |x| = O |x(ξ)| 2 dξ 1 2 ∀x ∈ L 2 (O),
define the linear operator A by

D(A) = H 2 (O) ∩ H 1 0 (O) Ax = ∆x ∀x ∈ D(A).
(2.15) (1) We assume n 3 for simplicity. The analysis of this example is even simpler for n = 1, 2.

It is well known that A satisfies (H A ) and

H 1 = H 1 0 (O). Moreover, in view of Poincaré's inequality O |x(ξ)| 2 dξ C 2 O O |∇x(ξ)| 2 dξ ∀x ∈ H 1 0 (O), (2.16)
an equivalent norm in H 1 is the following

|x| 1 = O |∇x(ξ)| 2 dξ 1 2 ∀x ∈ H 1 0 (O),
with the associated scalar product

x, y 1 = O ∇x(ξ) • ∇y(ξ) dξ ∀x, y ∈ H 1 0 (O).
Moreover, appealing to the Sobolev embedding theorem, we have that

H 1 0 (O) ⊂ L p (O) ∀p ∈ 1, 2n n -2 and O |x(ξ)| p dξ 1 p C p (O) O |∇x(ξ)| 2 dξ 1 2 ∀x ∈ H 1 0 (O). (2.17)
Therefore, (2.11), (2.13), and (2.17) ensure that the map f : R + × H 1 → H, defined by

f (t, x)(ξ) = F t, ξ, x(ξ) ∀(t, x) ∈ R + × H 1 , ξ ∈ O a.e., (2.18)
satisfies (H f )-(a), (2.5), and (2.6). Moreover, by (2.14) we have that

|f (t, x) -f (t, y)| 2 = O F t, ξ, x(ξ) -F t, ξ, y(ξ) 2 dξ C O ϕ 1 (t, ξ) 2 + |x(ξ)| + |y(ξ)| 2(p-1) x(ξ) -y(ξ) 2 dξ C O ϕ 1 (t, ξ) n + |x| + |y| n(p-1) dξ 2 n O x -y 2n n-2 dξ n-2 n
which gives (2.7). So, f satisfies (H f ) with θ = 1 (see also [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF]Example 3.6]).

As we show next, the abstract model (2.1) also allows to treat equations with nonlocal terms.

Example 2.3. In O ⊂ R n as in the above example, consider the initialboundary value problem for the heat operator with a nonlocal source term

       ∂X ∂t = ∆X + O |∇X| 2 dξ X in ]0, ∞[×O X = 0 on ]0, ∞[×∂O X(0, ξ) = x 0 (ξ) ξ ∈ O a.e.
(2. [START_REF] Shi | Viability theorems for a class of differential-operator inclusions[END_REF] In the Hilbert space H = L 2 (O), define A as in (2.15) and take

f (t, x)(ξ) = |x| 2 1 x(ξ) ∀(t, x) ∈ R + × H 1 0 (O) , ξ ∈ O a.e.
Then, one immediately has that f satisfies (H f )-(a), (2.5), and (2.6) with θ = 1. We proceed to check that (2.7) also holds true. Fix R > 0 and let x, y ∈ H 1 0 (O) be such that |x| 1 , |y| 1 R. Then, by (2.16),

f (t, x) -f (t, y) = |x| 2 1 x -|y| 2 1 y |x| 2 1 |x -y| + |y| |x| 2 1 -|y| 2 1 R 2 |x -y| + 2C O R 2 |x -y| 1 which yields (2.7) with L(t, R) = 3C O R 2 .
2.2. Well-posedness. In this section, we discuss well-posedness for problem (2.1). These results will be needed in the next sections. Let us consider the linear problem

X (t) = AX(t) + g(t), t ∈]0, T [ X(0) = x 0 , (2.20)
where g ∈ L 1 (0, T ; H) and x 0 ∈ H. We recall that X ∈ C([0, T ]; H) given by

X(t) = e tA x 0 + t 0 e (t-s)A g(s)ds ∀t ∈ [0, T ] (2.
|X(t) -Y (t)| e t t 0 (s,R)ds |x 0 -y 0 | ∀ t ∈ [t 0 , t 0 + τ ]. (2.27)
and

|X(t) -Y (t)| θ e t t 0 (s,R)ds |x 0 -y 0 | θ ∀ t ∈ [t 0 , t 0 + τ ].
(2.28)

Furthermore, if θ = 1, then X ∈ H 1 (t 0 , t 0 + τ ; H) ∩ L 2 (t 0 , t 0 + τ ; D(A)).
(2.29)

Proof. Let T and R be fixed positive numbers. The existence and uniqueness of the mild solution to (2.1)-as well as estimate (2.26)-can be obtained by a standard fixed-point argument for the map Φ :

X τ → X τ defined by Φ(X)(t) = e (t-t 0 )A x 0 + t t 0 e (t-s)A f s, X(s) ds X ∈ X τ , t ∈ [t 0 , t 0 + τ ] ,
where 0 < τ T -t 0 has to be properly chosen and

X τ = X ∈ C [t 0 , t 0 + τ ]; H θ : sup t 0 t t 0 +τ |X(t)| θ 2R .
Once the mild solution X of (2.1), satisfying (2.26), has been constructed, one can appeal to Proposition 2.4 with

g(t) = f t, X(t) (t ∈ [t 0 , t 0 + τ ]),
which belongs to L 2 (t 0 , t 0 + τ ; D(A)), to deduce the fact that

X (t) = AX(t) + f t, X(t)
for a.e. t ∈ [t 0 , , t 0 + τ ]. For θ = 1, (2.29) follows from the same proposition.

We now proceed to justify (2.27) and (2.28). Let Y be the mild solution of (2.1) with any initial condition

y 0 ∈ H θ such that |y 0 | θ ≤ R. Then, (X -Y ) (t) = A(X -Y )(t) + f t, X(t) -f (t, Y (t) (t ∈ [t 0 , t 0 + τ ] a.e.)
So, taking the scalar product of both sides of the above identity by X -Y , in view of (2.9) with = 2ω 1-θ and (2.4) we obtain 1 2

d dt |X -Y | 2 (2.30) -|X -Y | 2 1 + ω 1-θ |X -Y | 2 θ + L(t, 2R) 2 4ω 1-θ |X -Y | 2 L(t, 2R) 2 4ω 1-θ |X -Y | 2 .
Thus, (2.27) follows by Gronwall's lemma with (t, R) = L(t,2R) 2 4ω 1-θ . Estimate (2.28) can be deduced in a similar way. This time, taking the scalar product with (-A) θ (X -Y ) and recalling (2.7), we have that

1 2 d dt |X -Y | 2 θ = -|X -Y | 2 1+θ + f (t, X) -f (t, Y ), (-A) θ (X -Y ) -|X -Y | 2 1+θ + L(t, 2R) |X -Y | θ |X -Y | 2θ . Thus, instead of (2.30) we obtain 1 2 d dt |X -Y | 2 θ (2.31) -|X -Y | 2 1+θ + ω 1-θ |X -Y | 2 2θ + L(t, 2R) 2 4ω 1-θ |X -Y | 2 θ .
Since (2.4) yields

ω 1-θ |X -Y | 2 2θ = ω 1-θ |(-A) θ (X -Y )| 2 |(-A) θ/2 (X -Y )| 2 1 = |X -Y | 2 1+θ
, we derive (2.28) by Gronwall's lemma, with the same function as above.

Once the existence and uniqueness of the local mild solution has been proved, one constructs the maximal solution of problem (2.1), X(•; t 0 , x 0 ), which exists on the maximal interval [t 0 , T (t 0 , x 0 )[ defined as follows:

T (t 0 , x 0 ) = sup T t 0 : ∃ X(•) mild solution of (2.1) on [t 0 , T ] . (2.32)
Then, for any fixed T ∈ [t 0 , T (t 0 , x 0 )[, thanks to the uniqueness property of Proposition 2.6 one has that X(t; t 0 , x 0 ) = X(t) for all t ∈ [t 0 , T ] where X is the mild solution of (2.1) on [t 0 , T ]. Moreover, X(•; t 0 , x 0 ) satisfies (2.29), (2.27), and (2.28) on all compact subintervals of [t 0 , T (t 0 , x 0 )[.

Control systems.

We now want to extend the above well-posedness result to the semilinear control system

X (t) = AX(t) + f (t, X(t), u(t)), u(t) ∈ U X(t 0 ) = x 0 , (2.33)
where u : [t 0 , ∞[→ U is Lebesgue measurable and U is a complete separable metric space. This goal is obtained at essentially no cost, because in the previous section f has been assumed to be measurable in time. We just need to adapt assumption (H f ) to control systems, as we do next.

(H f ): f : R + × H θ × U → H for some θ ∈ [ 0, 1 ]
and satisfies the following:

(a) for all (x, u) ∈ H θ × U , the map t → f (t, x, u) is Lebesgue measurable on R + ; (b) for a.e. t 0, the map (x, u) → f (t, x, u) is continuous; (c) f u (t, x) := f (t, x, u) satisfies (H f ) uniformly in u ∈ U .
Directly from Proposition 2.6 we deduce the well-posedness of (2.33) for any (t 0 , x 0 ) ∈ R + × H θ and any measurable control u : [t 0 , ∞[→ U . We denote by X(•; t 0 , x 0 , u) the maximal solution of such a problem.

As is well-known, X(•; t 0 , x 0 , u) is global under an additional assumption.

Proposition 2.7. Assume (H A ), (H f ), and suppose there exists a nonneg-

ative function c ∈ L 2 loc (R + ) such that f (t, x, u) c(t)(1 + |x| θ ) ∀ x ∈ H θ , (2.34)
for a.e. t 0 and every u ∈ U . Then, for any (t 0 , x 0 ) ∈ R + × H θ , the maximal solution of (2.33) is global, that is, T (t 0 , x 0 ) = ∞ and for every T > t 0 there exists a constant C T > 0 depending only on the magnitude of

T 0 c(t) 2 dt such that |X(t; t 0 , x 0 , u)| θ C T (1 + |x 0 | θ ) ∀t ∈ [t 0 , T ]. (2.35) Proof. Fix any (t 0 , x 0 ) ∈ R + × H θ and let T ∈]t 0 , T (t 0 , x 0 )[. Since X (t) = AX(t) + f t, X(t), u(t) (t ∈ [t 0 , T ] a.e.),
where X(•) = X(•; t 0 , x 0 , u), taking the scalar product of both sides of the above identity by (-A) θ X and arguing as in the proof of (2.31) we obtain 1 2

d dt |X(t)| 2 θ -|X(t)| 2 1+θ + ω 1-θ |X(t)| 2 2θ + c(t) 2 4ω 1-θ 1 + |X(t)| θ 2 c(t) 2 4ω 1-θ 1 + |X(t)| θ 2 (t ∈ [t 0 , T ] a.e.)
Then, by Gronwall's lemma we conclude that

|X(t)| θ C T (1 + |x 0 | θ ) ∀t ∈ [t 0 , T ]
for some constant C T > 0 depending only on the magnitude of T 0 c(t) 2 dt. This implies that the maximal solution of (2.33) is global and satisfies (2.35).

Next, we state a Scorza-Dragoni type theorem that can be easily deduced from [4, Theorem 1].

Theorem 2.8. Let T > 0, Y and Z be separable metric spaces and consider a Carathéodory map F : [0, T ] × Y → Z. Then, for any ε > 0 there exists a compact set

T ε ⊂ [0, T ] with L 1 ([0, T ]\T ε ) < ε such that the restriction of F to T ε × Y is continuous. Recall that F : [0, T ] × Y → Z is called a Carathéodory map if t → F (t, y) is Lebesgue measurable for all y ∈ Y, y → F (t, y) is continuous for a.e. t ∈ [0, T ].
From the above theorem we deduce the very useful result below. Proposition 2.9. Under assumptions (H A ) and (H f ) there exists a set N ⊂ R + , of Lebesgue measure zero, such that for every

(t 0 , x 0 , u 0 ) ∈ R + \ N × H θ × U, the maximal solution X(•) = X(•; t 0 , x 0 , u 0 ) of (2.33), with the constant control u(•) = u 0 , satisfies lim h→0+ 1 h t 0 +h t 0 e A(t 0 +h-s) f (s, X(s), u 0 )ds = f (t 0 , x 0 , u 0 ).
Proof. Having fixed any T 0, we will construct a set (of full measure) M T ⊂ [ 0, T ] such that the conclusion holds on M T × H θ × U . Then, to obtain our result, it suffices to consider the union, say M, of such sets for a sequence T i ↑ ∞ and take

N = R + \ M. Define ε = 1 i , T = [0, T ] , Y = H θ × U , Z = H.
Let M i := T 1/i be as in Theorem 2.8 and set L j (•) = L(•, j) for all j 1, where the latter is the function in (H f )-(b). Consider the subset M i ⊂ M i of all points which are both Lebesgue density point for M i and Lebesgue points for L j χ [0,T ]\M i and L j for every integer j 1. Then L

1 ( M i ) = L 1 (M i ). Set M T = ∪ i 1 M i . Then M T is of full measure in [0, T ].
Fix any (t 0 , x 0 , u 0 ) ∈ M T ×H θ ×U and consider the solution X(•) of (2.33) with u(•) ≡ u 0 defined on some interval [t 0 , t 0 + δ], where δ > 0. Setting R := max s∈[t 0 ,t 0 +δ] |X(s)| θ , for any integer j > R we have that

t 0 +h t 0 e (t 0 +h-s)A (f (s, X(s), u 0 ) -f (s, x 0 , u 0 ))ds t 0 +h t 0 L j (s)|X(s) -x 0 | θ ds. Furthermore, 1 h t 0 +h t 0 L j (s)|X(s) -x 0 | θ ds max s∈[t 0 ,t 0 +h] |X(s) -x 0 | θ 1 h t 0 +h t 0 L j (s)ds. Hence lim h→0+ 1 h t 0 +h t 0 e (t 0 +h-s)A f (s, X(s), u 0 ) -f (s, x 0 , u 0 ) ds = 0. Let i 1 be such that t 0 ∈ M i . Then for all j 1 lim h→0+ 1 h t 0 +h t 0 L j (s)χ [0,T ]\M i (s)ds = 0 and therefore lim h→0+ 1 h t 0 +h t 0 e (t 0 +h-s)A f (s, x 0 , u 0 )ds = lim h→0+ [t 0 ,t 0 +h]∩M i e (t 0 +h-s)A f (s, x 0 , u 0 )ds. By the continuity of f (•, x 0 , u 0 ) on [t 0 , t 0 + h] ∩ M i , we have that lim h→0+ 1 h [t 0 ,t 0 +h]∩M i e (t 0 +h-s)A f (s, x 0 , u 0 )ds = lim h→0+ 1 h [t 0 ,t 0 +h]∩M i e (t 0 +h-s)A f (t 0 , x 0 , u 0 )ds = lim h→0+ 1 h [t 0 ,t 0 +h]∩M i f (t 0 , x 0 , u 0 )ds. Next, notice that 1 h [t 0 ,t 0 +h]∩M i f (t 0 , x 0 , u 0 )ds = 1 h L 1 ([t 0 , t 0 + h] ∩ M i ) f (t 0 , x 0 , u 0 )
which implies, by the choice of t 0 and i, that

lim h→0+ 1 h [t 0 ,t 0 +h]∩M i f (t 0 , x 0 , u 0 )ds = f (t 0 , x 0 , u 0 ).
Hence, we deduce that lim

h→0+ 1 h [t 0 ,t 0 +h]∩M i e (t 0 +h-s)A f (s, x 0 , u 0 )ds = f (t 0 , x 0 , u 0 )
completing the proof.

We would like to underline the importance of Proposition 2.9, which lies in the fact that the negligible set N is independent of the choice of x 0 and u 0 . By recalling (2.25) and appealing to Proposition 2.9 in the special case of f (t, x, u) = f (t, x), one deduces the following differentiability result.

Corollary 2.10. Under assumptions (H A ) and (H f ) there exists N ⊂ [ 0, ∞[, of Lebesgue measure zero, such that for every

(t 0 , x 0 ) ∈ R + \ N × D(A) the maximal solution X(•; t 0 , x 0 ) of problem (2.1) is differentiable at t 0 and d dt X(t; t 0 , x 0 ) |t=t 0 = Ax 0 + f (t 0 , x 0 ).
(2.36)

Invariance

In all the results of this section, we assume without further notice that:

• (H A ) and (H f ) are satisfied, and

• K ⊂ H is a nonempty closed set.
We denote by θ ∈ [0, 1] the number given by (H f ). We know that, for any (t 0 , x 0 ) ∈ R + × H θ , problem (2.1) has a unique solution X(•; t 0 , x 0 ) and we denote by [t 0 , T (t 0 , x 0 )[ its interval of existence. We begin by defining invariance under (2.1).

Definition 3.1. We say that K ∩ H θ is invariant under (2.1) if, for all t 0 0 and x 0 ∈ K ∩ H θ , we have that

X(t; t 0 , x 0 ) ∈ K ∩ H θ ∀t ∈ [t 0 , T (t 0 , x)[. Denote by d K (x) the distance of x from K, that is, d K (x) = inf y∈K |x -y|, ∀ x ∈ H. (3.1)
We recall that d K is Lipschitz continuous (with constant 1) on H and the lower Dini derivative of d K at x ∈ H in the direction v ∈ H is given by

D -d K (x) v = lim inf λ↓0 d K (x + λv) -d K (x) λ .
For any δ > 0 we set

K δ = x ∈ H \ K : d K (x) < δ . (3.2)
3.1. Sufficient Conditions for Invariance. We first provide sufficient conditions for the invariance of K, which apply to rather general settings.

Theorem 3.2. Suppose there exists a set T ⊂ R + , of Lebesgue measure zero, and a number δ > 0 such that, for all t ∈ R + \ T and all x ∈ D(A) ∩ K δ ,

D -d K (x) (Ax + f (t, x)) C(t, |x| θ ) d K (x), (3.3)
where

C : R + × R + → R + satisfies r → C(t,
Then K ∩ H θ is invariant under (2.

1).

Proof. Fix any (t 0 , x 0 ) ∈ R + × (K ∩ H θ ) and denote by X(•) the maximal solution X(•; t 0 , x 0 ) of (2.1). Let us argue by contradiction assuming there is

T ∈]t 0 , T (t 0 , x 0 )[ such that X(T ) / ∈ K. Set t = max t ∈ [t 0 , T ] : X(t) ∈ K and x = X( t).
We will show that t = T , which contradicts the assumption X(T ) / ∈ K. Suppose t < T and observe that

X(t) / ∈ K ∀t ∈] t, T ]. (3.5)
In light of Proposition 2.6, there exists

E = t ∈ [ t, T ] : X (t) = AX(t) + f t, X(t) .
Owing to (3.3), for all t ∈ E ∩ F \ { t} we have that

D -d K (X(t)) (AX(t) + f (t, X(t))) C(t, 2|x| θ + 1) d K (X(t)). (3.6)
Now, since d K is Lipschitz, the function

φ(t) := d K X(t) (t ∈ [ t, T ]) is absolutely continuous, hence differentiable on a set D of full measure in [ t, T ]. Then, in view of (3.6), for all t ∈ D ∩ E ∩ F \ { t, T } we have that φ (t) = lim h↓0 1 h d K X(t + h) -d K X(t) = lim h↓0 1 h d K X(t) + h AX(t) + f (t, X(t)) -d K X(t) C(t, 2|x| θ + 1)φ(t).
Since D ∩ E ∩ F has full measure in [ t, T ] and φ( t) = 0, Gronwall's lemma ensures that φ(t) ≡ 0, or X(t) ∈ K for all t ∈ [ t, T ], in contrast with (3.5). So, t = T , as claimed, and we have reached the announced contradiction.

When K has further geometric properties, the above sufficient condition can be given in different forms that are easier to handle in specific situations. For any x ∈ H we denote by Π K (x) the set of projections of x onto K, that is, the (possibly empty) subset of K at which the infimum in (3.1) is attained. We recall that K is said to be proximally smooth, if Π K (x) is a singleton for all x ∈ K δ and some δ > 0 or, equivalently, if d K is continuously Fréchet differentiable on K δ . In this case, we have that

Dd K (x) = x -Π K (x) d K (x) ∀x ∈ K δ .
Consequently, if K is proximally smooth, then

D -d K (x)v = Dd K (x), v ∀x ∈ K δ , ∀v ∈ H.
So, Theorem 3.2 yields the following.

Corollary 3.3. Let K be proximally smooth and fix any δ > 0 such that Π K is single-valued on K δ . Furthermore, suppose that, for a.e. t ∈ R + and every

x ∈ D(A) ∩ K δ , x -Π K (x), Ax + f (t, x) C(t, |x| θ ) d 2 K (x), (3.7)
where

C : R + × R + → R + satisfies (3.4). Then K ∩ H θ is invariant under (2.1).
In our next result, we propose a condition that needs to be satisfied only at boundary points. For any x ∈ K we denote by N P K (x) the proximal normal cone to K at x. We recall that a vector p ∈ H belongs to N P K (x) if and only if, for some λ > 0, we have that

p, y -x |p| 2λ |y -x| 2 ∀y ∈ K. (3.8)
Observe that 0 ∈ N P K (x) for all x ∈ K. Whenever (3.8) holds for some p = 0, we say that p is realized by a ball of radius λ. Indeed, in this case, one has that

B λ (x + λp/|p|) ⊂ H \ K. Conversely, if p ∈ H is such that for some λ > 0 we have B λ (x + λp/|p|) ⊂ H \ K, then p ∈ N P K (x), cf. [12, Definition 2.

and Proposition 2.4].

Recall that the contingent cone

T K (x) to K at x ∈ K is the set of all v ∈ H such that there exist h i → 0+ and H v i → v when i → ∞ satisfying x + h i v i ∈ K for all i. Then (3.8) implies that N P K (x) ⊂ T K (x) -, (3.9)
where T K (x) -stands for the negative polar cone of T K (x).

Theorem 3.4. Let K be proximally smooth and fix any δ > 0 such that

Π K is single-valued on K δ . Suppose Π K D(A) ∩ K δ ⊂ D(A) and p, Ax + f (t, x) 0 ∀ p ∈ N P K (x) ∩ D(A), (3.10)
for a.e. t ∈ R + and every x ∈ ∂K ∩ D(A).

Then K ∩ H θ is invariant under (2.1).

Remark 3.5. We would like to emphasize that the assumption Π K D(A) ∩ K δ ⊂ D(A) is essential in the above theorem. Indeed, the interested reader can find in [9, Exercise 3.2.1] an Example demonstrating that a subspace may be invariant by A without being semigroup invariant, the above assumption being not satisfied.

Proof. Let x ∈ D(A) ∩ K δ and set x = Π K (x). Since x -x ∈ N P K (x) ∩ D(A), (3.10) yields x -x, Ax + f (t, x)
0 for a.e. t ∈ R + . Recalling (2.4), by (2.9) with = 2ω 1-θ we then obtain

x -x, Ax + f (t, x) = x -x, Ax + f (t, x) + x -x, A(x -x) + x -x, f (t, x) -f (t, x) -|x -x| 2 1 + ω 1-θ |x -x| 2 θ + L t, M (|x| θ ) 2 4ω 1-θ |x -x| 2 L t, M (|x| θ ) 2 4ω 1-θ d 2 K (x). Hence (3.7) is satisfied. So, K ∩ H θ is invariant in view of Corollary 3.3.
Example 3.6. In the Hilbert space H = L 2 (O), consider the closed convex cone

H + = x ∈ H : x(ξ) 0, ξ ∈ O a.e. .
Denoting by x + and x -the positive and negative parts of x ∈ H, respectively, we have that any x ∈ H can be represented as x = x + -x -. So,

Π H + (x) = x + and d H + (x) = |x -| ∀x ∈ H.
Consequently, H + is proximally smooth and we can study its invariance under the flow associated with (2.10) using Corollary 3.3. Suppose assumptions (2.11), (2.12), (2.13), and (2.14) are satisfied with ϕ 0 ≡ 0 and define A and f as in (2.15) and (2.18), respectively. Then, appealing to Corollary 3.3 we conclude that (3.7) is a sufficient condition for the invariance of H + ∩ H 1 under (2.10). In order to check (3.7), observe that, for all x ∈ D(A) \ H + , integrating by parts we have that

x -Π H + (x), Ax + f (t, x) (3.11) = - O x -(ξ) ∆x(ξ) + F t, ξ, x(ξ) dξ = - O |∇x -(ξ)| 2 dξ - O x -(ξ)F t, ξ, -x -(ξ) dξ
because x + and x -vanish on the intersection of their supports. Now, use (2.13) (with ϕ 0 ≡ 0) and the Sobolev inequality (2.17) to derive

- O x -(ξ)F t, ξ, -x -(ξ) dξ C 0 O |x -(ξ)| |x -(ξ)| p dξ C 0 |x -| O |x -(ξ)| 2p dξ 1 2 C 0 C 2p (O) p |x -| |x -| p 1 |x -| 2 1 + C 2 0 C 2p (O) 2p 4 |x -| 2(p-1) 1 |x -| 2 .
Finally, combine the last inequality with (3.11), to obtain (3.7) with

C(t, r) = C 2 0 C 2p (O) 2p 4 r 2(p-1) (t, r 0), thus yielding the claimed invariance of H + ∩ H 1 .
The above analysis shows that, without assuming ϕ 0 ≡ 0 in (2.13), a sufficient condition for the invariance of

H + ∩ H 1 under (2.10) is that, for a.e. (t, ξ) ∈ R + × O, F (t, ξ, v) Cv ∀v 0 (3.12)
for some constant C 0.

Necessary Conditions for Invariance.

In this section, we show that the above sufficient conditions become also necessary for the invariance of K ∩ H θ under a further assumption, (H ρ ), which connects D(A) with K.

Theorem 3.7. Suppose there exists a number ρ > 0 and a nondecreasing function M : R + → R + , satisfying M (s) s for all s ∈ R + , such that: (H ρ ) for all x ∈ D(A) ∩ K ρ and all h > 0 one can find

x h ∈ K ∩ H θ with |x h -x| < d K (x) + h and |x h | θ M (|x| θ ).
If K ∩H θ is invariant under (2.1), then there exists a set T ⊂ R + of Lebesgue measure zero and a function C : R + × R + → R + satisfying (3.4) such that (3.3) holds true for all t ∈ R + \ T and all x ∈ D(A) ∩ K ρ .

Proof. Let be the function given by Proposition 2.6 and observe that

(t, r) (t, r ) for a.e. t ∈ R + , ∀r ∈ R + . (3.13)
For every integer n 1, denote by P n the set of Lebesgue points of the function (•, n) and set Q n = R + \ P n and Q = ∪ n 1 Q n . Since P n is a set of full measure, Q n is a negligible set and so is Q. Then, T := N ∪ Q, where N is given by Corollary 2.10, is negligible too.

Suppose K ∩H θ is invariant under (2.1). Fix t 0 ∈ R + \T, x 0 ∈ D(A)∩K ρ , and h > 0. Invoke assumption (H ρ ) to construct

x h ∈ K ∩ H θ such that |x h -x 0 | (1 + h 2 )d K (x 0 ) and |x h | θ M (|x 0 | θ ). ( 3
h d K (X(t 0 + h)) -d K (x 0 ) = 1 h d K (X(t 0 + h)) -d K (X h (t 0 + h)) -d K (x 0 ) 1 h |X(t 0 + h) -X h (t 0 + h)| - |x 0 -x h | 1 + h 2 1 h e t 0 +h t 0 (s,M (|x 0 | θ ))ds - 1 1 + h 2 |x 0 -x h | 1 h e t 0 +h t 0 (s, M (|x 0 | θ ) )ds - 1 1 + h 2 |x 0 -x h | for all 0 < h T -t 0 . Hence, again by (3.14), we obtain 1 h d K (X(t 0 + h)) -d K (x 0 ) (3.15) e t 0 +h t 0 (s, M (|x 0 | θ ) )ds -1 h + h 1 + h 2 (1 + h 2 )d K (x 0 ) for all 0 < h T -t 0 . Since d K is Lipschitz
and, in view of the choice of t 0 , X satisfies (2.36), D -d K (x 0 )(Ax 0 + f (t 0 , x 0 )) coincides with the lower limit as h ↓ 0 of the left side of (3.15). Moreover,

lim h↓0 1 h t 0 +h t 0 (s, M (|x 0 | θ ) )ds = (t 0 , M (|x 0 | θ ) )
because t 0 is a Lebesgue point of (•, r ) for any r > 0. We have thus obtained (3.3) with C(t, r) = (•, M (r) ).

Remark 3.8. Whenever assumption (H ρ ) of Theorem 3.7 is satisfied, condition (3.3) is necessary and sufficient for the invariance of K ∩ H θ . Theorem 3.7 and Corollary 3.3 yield the following. Corollary 3.9. Assume that:

(a) K is proximally smooth and let δ > 0 be such that

Π K is single-valued on K δ ; (b) Π K D(A) ∩ K δ ⊂ H θ and |Π K (x)| θ M (|x| θ ) ∀x ∈ D(A) ∩ K δ . (3.16)
where M : R + → R + is nondecreasing and M (s) s for all s ∈ R + . Then K ∩ H θ is invariant under (2.1) if and only if there exists a function C : R + × R + → R + , satisfying (3.4), such that (3.7) holds true for a.e. t ∈ R + and every x ∈ D(A) ∩ K δ .

Finally, we show the necessity of condition (3.10) for boundary points. Proof. Sufficiency follows from Theorem 3.4. In order to prove necessity, observe that if K ∩ H θ is invariant under (2.1), then (3.7) holds true by Corollary 3.9. Let x ∈ ∂K ∩ D(A) and p ∈ N P K (x) ∩ D(A). Then x λ := x + λp ∈ D(A) \ K and Π K (x λ ) = x for λ > 0 sufficiently small. Thus (3.7) yields

x λ -x, Ax λ + f (t, x λ ) C(t, |x + λp| θ ) λ 2 |p| 2 for a.e. t ∈ R + and every x ∈ ∂K ∩ D(A). So, dividing by λ, p, Ax + λAp + f (t, x λ ) C(t, |x + λp| θ ) λ|p| 2 .
Passing to the limit as λ ↓ 0 we obtain (3.10).

If, in Example 3.6, we have used the sufficient condition given by Corollary 3.3 to obtain invariance, the fact that (3.7) is also a necessary condition for invariance (Corollary 3.9) can help to reconstruct structural properties of the data from the behaviour of solutions, as we show in our next example. (3.17)

Here, F ∈ C 1 (R) is supposed to satisfy assumptions (2.13) and (2.14) with ϕ 0 ≡ 1 ≡ ϕ 1 . Defining

D(A) = x ∈ H 2 (O) : ∂x ∂ν | ∂O = 0 Ax = ∆x -x ∀x ∈ D(A) (3.18)
we have that (H A ) is satisfied and .19) satisfies in turn hypothesis (H f ). Furthermore, assumptions (a) and (b) of Corollary 3.9 hold true with δ = ∞. So, we have that (3.7) is necessary (and sufficient) for the invariance of H + ∩ H 1 under (3.17). By the same computations as in Example 3.6, one has that (3.7) can be rewritten as

H 1 = H 1 (O). Moreover, f (x)(ξ) := x(ξ) + F x(ξ) ∀x ∈ H 1 , ξ ∈ O a.e. ( 3
- O x -(ξ)F -x -(ξ) dξ O |∇x -(ξ)| 2 dξ + C(|x| 1 ) O |x -| 2 dξ
for all x ∈ D(A) \ H + . Now, restricting the above inequality to the constant functions x(ξ) ≡ v 0 gives vF (v) C(0)v 2 . Thus, we deduce that, if H + ∩ H 1 is invariant under (3.17), then, for some constant C 0,

F (v) Cv ∀v 0. (3.20)
Since it is easy to check that the above condition implies (3.7), we conclude that (3.20) is necessary and sufficient for the invariance of

H + ∩ H 1 .
Observe that the necessity of (3.20)-much in the spirit of inverse problems-cannot not be deduced from the maximum principle.

3.3. The case of θ = 1. In this section we show that, when (H f ) is satisfied with θ = 1, the invariance of D(A) under Π K can be dropped from the assumptions of Theorem 3.10. We begin by adapting Corollary 3.9. Proposition 3.12. Assume that:

(a) K is proximally smooth and let δ > 0 be such that

Π K is single-valued on K δ ; (b 1 ) Π K H 1 ∩ K δ ⊂ H 1 , Π K : H 1 ∩ K δ → H 1 is continuous, and |Π K (x)| 1 M (|x| 1 ) ∀x ∈ H 1 ∩ K δ , (3.21)
where M : R + → R + is nondecreasing and M (s) s for all s ∈ R + .

Then K ∩ H 1 is invariant under (2.1) if and only if there exists a function

C : R + × R + → R + , satisfying (3.4) such that x -Π K (x), f (t, x) -(-A) 1/2 (x -Π K (x)), (-A) 1/2 x) C(t, |x| 1 ) d 2 K (x) (3.22)
for a.e. t ∈ R + and every x ∈ H 1 ∩ K δ .

Proof. Sufficiency follows directly from Corollary 3.9 because (3.22) implies (3.7) with θ = 1. In order to prove necessity, fix any x ∈ H 1 ∩ K δ and let x j = e A/j x for all integers j 1. Then x j ∈ D(A), |x j | 1 |x| 1 for all j 1, and x j → x in H 1 as j → ∞. Therefore, x j ∈ D(A) ∩ K δ for j sufficiently large and so, in view of (3.7) and Corollary 3.9, we have that for a.e.

t ∈ R + C(t, |x| 1 ) d 2 K (x j ) C(t, |x j | 1 ) d 2 K (x j ) x j -Π K (x j ), Ax j + f (t, x j ) = x j -Π K (x j ), f (t, x j ) -(-A) 1/2 (x j -Π K (x j )), (-A) 1/2 x j .
By taking the limit as j → ∞, one recovers (3.22).

We now turn to the analogue of Theorem 3.10.

Proposition 3.13. Let assumptions (a) and (b 1 ) of Proposition 3.12 be satisfied. Then K ∩ H 1 is invariant under (2.1) if and only if

p, f (t, x) -(-A) 1/2 p, (-A) 1/2 x) 0 ∀p ∈ N P K (x) ∩ H 1 (3.23)
for a.e. t ∈ R + and every x ∈ H 1 ∩ ∂K.

We omit the proof of the above result that can be reconstructed from the one of Theorem 3.10, by replacing Corollary 3.9 with Proposition 3.12.

Level set preserving bilinear controls

In this section, we apply our invariance result to a class of nonlinear parabolic equations that generalize the one considered in Example 2.3. In order to motivate our analysis, we begin with a viability problem for the heat equation under the action of a bilinear control.

(i) φ(0) = φ (0) = 0 (ii) 0 < λ -φ (s) λ + ∀s ∈ R (4.1)
for some constants λ -, λ + > 0. Observe that, owing to the above assumptions, the following estimates are true:

λ + 2 s 2 φ(s) λ - 2 s 2 ∀s ∈ R (4.2) and sφ (s) φ(s) + λ - 2 s 2 λ -s 2 ∀s ∈ R. (4.3)
Moreover, (4.1) and ( 4.3) yield

λ 2 -s 2 |φ (s)| 2 λ 2 + s 2 ∀s ∈ R. (4.4)
Let O ⊂ R n be a bounded domain with boundary of class C 2 . In view of (4.2), for any c > 0 the level set

Λ φ (c) = x ∈ L 2 (O) : O φ x(ξ) dξ = c (4.5)
is nonempty and closed in the Hilbert space H = L 2 (O). For a fixed c > 0 and a given x 0 ∈ Λ φ (c) ∩ H 1 0 (O) we seek a time T > 0 and a control g ∈ L ∞ (0, T ) such that the solution, say X g , of the initialboundary value problem

     ∂X ∂t (t, ξ) = ∆X(t, ξ) + g(t)X(t, ξ) (t > 0, ξ ∈ O) X = 0, on (0, ∞) × ∂O X(0, ξ) = x 0 (ξ), ξ ∈ O. (4.6) satisfies X g (t, •) ∈ Λ φ (c) ∀t ∈ [0, T ]. Now, let g ∈ L ∞ (0, T ) be any control with such a property. Since O φ (X g ) ∂X g ∂t dξ = d dt O φ(X g ) dξ = 0,
multiplying the equation in (4.6) by φ (X g ) and integrating by parts we find that g = G(X g ) where

G(x) = O φ (x(ξ))|∇x(ξ)| 2 dξ O x(ξ)φ (x(ξ))dξ ∀x ∈ H 1 0 (O) \ {0}. (4.7)
Notice that the denominator in the above quotient is strictly positive in view of (4.3) and the fact that x = 0.

4.2.

Invariance of level sets. We now show how to apply the abstract results of this paper to study the invariance of all level sets Λ φ (c)∩H 1 0 (O), with c > 0, under the flow associated with the initial-boundary value problem

     ∂X ∂t = ∆X + G(X)X in (0, ∞) × O X = 0 on (0, ∞) × ∂O X(0, ξ) = x 0 (ξ), ξ ∈ O (4.8)
where G is given by (4.7). For this purpose, we recast the problem in abstract form as in Example 2.3, defining operator A as in (2.15). In this case, it is well-known that H 1 = H 1 0 (O) and we define f :

H 1 → H by f (x)(ξ) = G(x) x(ξ) ∀x ∈ H 1 0 (O) \ {0} 0 if x = 0 (ξ ∈ O a.e.)
Then, one can check that f satisfies assumption (H f )-(b) with θ = 1. So, for any x 0 ∈ H 1 , problem (4.8) has a unique maximal solution. Now, applying Proposition 3.13 we obtain the following.

Proposition 4.1. Assume (4.1). Then, for any c > 0, the set Λ φ (c) ∩ H 1 is invariant under (4.8).

We first prove the following lemma.

Lemma 4.2. Assume (4.1) and let µ ∈ R be such that |µ| 1/(2λ + ). Then, for any x ∈ H = L 2 (O) the equation

x(ξ) = y(ξ) + µ φ y(ξ) (ξ ∈ O a.e.) (4.9)
has a unique solution y x ∈ H. Moreover,

y x ∈ H 1 if x ∈ H 1 , the map Ψ : H 1 → H 1 defined by Ψ(x) = y x is continuous, and |Ψ(x)| 1 M |x| 1 (4.10)
for some constant M > 0 depending only on φ and O.

Proof. Define ψ : R → R by ψ(s) = s+µφ (s) for all s ∈ R. Observe that ψ is of class C 1 (R) and surjective. Since |µ| 1/(2λ + ), ψ (s) = 1+µφ (s) 1/2 for all s ∈ R. Therefore, for all such µ's, ψ is invertible, ψ -1 ∈ C 1 (R), and (ψ -1 ) is bounded independently of µ. Consequently, for any x ∈ H the unique solution of (4.9) is given by y x = ψ -1 (x), which belongs to H and, for x ∈ H 1 , satisfies

∇y x (ξ) = (ψ -1 ) x(ξ) ∇x(ξ) (ξ ∈ O a.e.)
Moreover, since both ψ and ψ -1 vanish at zero, we have that y x has null trace on ∂O. This shows that y x ∈ H 1 for x ∈ H 1 and (4.10) holds true.

As for the continuity of the map x → y x in the H 1 norm, fix any sequence x j → x in H 1 . Without loss of generality, we can assume that x j → x a.e. in O. Then

∇y x j -∇y x = (ψ -1 ) (x j ) ∇x j -∇x + (ψ -1 ) (x j ) -(ψ -1 ) (x) ∇x,
where we realize that lim

j→∞ (ψ -1 ) (x j ) ∇x j -∇x = 0 in L 2 (O; R n ), because (ψ -1 ) is bounded and x j → x in H 1 , and lim j→∞ (ψ -1 ) (x j ) -(ψ -1 ) (x) ∇x = 0 in L 2 (O; R n )
by the dominated convergence theorem. Therefore, y x j → y x in H 1 .

Proof. (of Proposition 4.1) Consider the functional Φ : H → R defined by

Φ(x) = O φ x(ξ) dξ ∀x ∈ H. Since φ ∈ C 2 (R), we have that Φ is Fréchet differentiable and DΦ(x)(ξ) = φ x(ξ) ∀x ∈ H, ξ ∈ O a.e. Moreover, Φ ∈ C 1,1 (H) because, by (4.1)
,

|DΦ(x) -DΦ(y)| λ + |x -y| ∀x, y ∈ H.
Furthermore, for every x ∈ Λ φ (c) with c > 0, (4.4) and (4.3) ensure that

|DΦ(x)| 2 = O φ x(ξ) 2 dξ λ 2 - O x(ξ) 2 dξ 2λ 2 - λ + O φ x(ξ) dξ = 2λ 2 -c λ + .
This shows that DΦ satisfies the lower bound in (6.2) of Proposition 6.1 below. The upper bound in (6.2) follows by a similar argument. Indeed, by (4.2) and (4.4) we have that

|DΦ(x)| 2 λ 2 + O x(ξ) 2 dξ 2λ 2 + λ -O φ x(ξ) dξ = 2λ 2 + c λ - .
Therefore, owing to Proposition 6.1 we conclude that Λ φ (c) is proximally smooth for any c > 0. So, condition (a) of Proposition 3.13 holds true for K = Λ φ (c). We now check that condition (b 1 ) of the same proposition is also satisfied. Still denoting Λ φ (c) by K, we claim that

N P K (x) ⊂ {µDΦ(x) : µ ∈ R} ∀x ∈ K. (4.11)
Indeed, by [2, Proposition 4.3.7],

T K (x) = {ξ ∈ H : DΦ(x)(ξ) = 0}.
This and the separation theorem imply that

T K (x) -= {µDΦ(x) : µ ∈ R}
and (4.11) follows from (3.9). Suppose δ > 0 is such that any point x ∈ K δ admits a unique projection onto K and call x such a projection. Then, for any x ∈ H 1 ∩ K δ we have that x -x ∈ N P K (x) and so, by (4.11),

x = x + µDΦ(x) with |µ| = d K (x) |DΦ(x)| . (4.12)
Hence, Lemma 4.2 guarantees that x ∈ H 1 and Π K :

H 1 ∩ K δ → H 1 is
continuous whenever δ > 0 is sufficiently small. Furthermore, (3.21) follows from (4.10). Therefore, by Proposition 3.13 we conclude that condition (3.23) is necessary and sufficient for the invariance of K ∩ H 1 . Since

DΦ(x), G(x)x -(-A) 1/2 DΦ(x), (-A) 1/2 x = O φ (x)|∇x| 2 dξ O x φ (x) dξ O x φ (x) dξ - O ∇φ (x) • ∇x dξ = 0 for all x ∈ K ∩ H 1 , (3.23) is satisfied and K ∩ H 1 is invariant.
Remark 4.3. We observe that an ad hoc computation can be used to prove invariance under (4.8) for the level set Λ φ (c) ∩ H 1 (c > 0). Indeed, it suffices to check that the time derivative of Φ along the maximal solution of (4.8) vanishes for a. e. t. Nevertheless, the goal of this section is to illustrate the interplay between our abstract conditions for invariance and the specific features of a concrete example.

Example 4.4. Consider the heat equation with a nonlocal term we presented in Example 2.3. One can easily check that, taking φ(s) = s 2 , such an equation reduces to (4.8) provided that X(t, •) ∈ Λ φ (1). Therefore, Proposition 4.1 ensures that the unit sphere of L 2 (O) is invariant under the equation

∂X ∂t (t, ξ) = ∆X(t, ξ) + O |∇X(t, ξ )| 2 dξ X(t, ξ) (t > 0, ξ ∈ O)
with homogeneous Dirichlet boundary conditions. The question remains whether the maximal solution of the above equation, with initial condition

X(0, •) = x 0 ∈ Λ φ (1) ∩ H 1 0 (O), is global (that is, T (0, x 0 ) = ∞)
or not. We return to this problem in the next section. 4.3. Maximal interval of existence. Finally, we apply the above invariance results to give a lower bound for the maximal interval of existence of the solution of (4.8). For simplicity, we suppose t 0 = 0 but the same result can be obtained-by exactly the same reasoning-for any t 0 0. Also, we assume x 0 = 0 since, when x 0 = 0, the null solution is trivially global. Theorem 4.5. Assume (4.1). Then, for any x 0 ∈ H 1 0 (O) \ {0} we have that

T (0, x 0 )      c 0 λ - 2λ + (λ + -λ -)|x 0 | 2 1 if λ -< λ + = ∞ if λ -= λ + , (4.13)
where

c 0 = O φ x 0 (ξ) dξ. (4.14)
Proof. First, observe that c 0 > 0 and, obviously, x 0 ∈ Λ φ (c 0 ) ∩ H 1 . Therefore, abbreviating T (0, x 0 ) to T 0 , Theorem 4.1 ensures that

X(t, •) ∈ Λ φ (c 0 ) ∀t ∈ [0, T 0 [ (4.15)
where X denotes the maximal solution of (4.8

κ = 2 λ + c 0 λ + λ - -1 .
Then, the comparison principle for ordinary differential equation yields

E(t) E(0) 1 -t κ E(0) ∀t ∈ 0, 1 κE(0)
.

This shows that the maximal solution of (4.8) is bounded in

H 1 0 (O) on every interval [0, T ] such that 2 T λ + c 0 λ + λ - -1 O |∇x 0 (ξ)| 2 dξ < 1.
Therefore, T 0 satisfies (4.13).

Example 4.6. We return to Example 2.3 with φ(s) = s 2 and K = Λ φ (1).

Since λ + = λ -, the maximal solution of (2. [START_REF] Shi | Viability theorems for a class of differential-operator inclusions[END_REF]) is global by Theorem 4.5. In other terms, the flow associated with (2.19) preserves the L 2 energy: this result was first observed in [START_REF] Caffarelli | Nonlocal heat flows preserving the L 2 energy[END_REF], where it was applied to study the convergence of a family of singularly perturbed systems of nonlocal parabolic equations.

Viability of a Semilinear Control System

Given a nonempty closed set K ⊂ H, we consider the semilinear control system (2.33) under the state constraint X(s) ∈ K (s t 0 ).

We recall that K δ (δ > 0) is defined in (3.2) and we list below the assumptions that will be imposed in this section.

(H A ) A : D(A) ⊂ H → H satisfies (H A ) and e tA is a compact linear operator on H for all t > 0. (H f ) f : R + × H θ × U → H satisfies assumptions (H f ) for some θ ∈ [ 0, 1 ], with (2.6) replaced by (2.34) for some function c ∈ L 2 loc (R + ), and f (t, x, U ) is closed and convex for a.e. t 0 and every x ∈ H θ . (H ρ ) There exists ρ > 0 and a nondecreasing function M : R + → R + such that, for all x ∈ K ρ ∩ H θ and all h > 0, one can find

x h ∈ K ∩ H θ satisfying |x h -x| < d K (x) + h and |x h | θ M (|x| θ ).
Under assumptions (H A ) and (H f ), in light of Proposition 2.7, for every initial condition (t 0 , x 0 ) ∈ R + × H θ and control u : [t 0 , ∞) → U , (2.33) has a unique global solution X(•; t 0 , x 0 , u). Moreover, arguing exactly as in [START_REF] Cannarsa | Value function and optimality conditions for semilinear control problems[END_REF], one can prove the following compactness result. Lemma 5.1. Under assumptions (H A ) and (H f ), for all (t 0 , x 0 ) ∈ R + × H θ and all T > t 0 the set

S [t 0 ,T ] (x 0 ) := X(•; t 0 , x 0 , u) : u : [t 0 , T ] → U is Lebesgue measurable is compact in C([t 0 , T ]; H θ ).
The set K ∩ H θ is called viable under the control system (2.33) if for every initial condition (t 0 , x 0 ) ∈ R + × (K ∩ H θ ) there exists a (Lebesgue measurable) control function u : [t 0 , ∞) → U such that X(t; t 0 , x 0 , u) ∈ K ∩ H θ for all t t 0 .

In what follows, we provide sufficient conditions for the viability of K ∩H θ in terms of Clarke's derivative of d K which, for any point x ∈ H and any direction v ∈ H, is given by

d 0 K (x)v = lim sup y→x, λ↓0 d k (y + λv) -d k (y) λ .
Notice that, if K is proximally smooth, then for all x ∈ H \ K, sufficiently close to K, one has that Proof. Fix any t 0 ∈ R + , x 0 ∈ K ∩ H θ .

d 0 K (x)v = Dd K (x), v
Step 1. Define, for all t t 0 , R(t) := X(t; t 0 , x 0 , u) : X ∈ S [t 0 ,t] (x 0 ) ⊂ H θ , g(t) := inf |x t -y| : x t ∈ R(t), y ∈ K .

Claim 1. For every t > t 0 there exists x t ∈ R(t) such that

g(t) = d K (x t ). (5.3) Indeed, it is clear that g(t) = inf z∈R(t) d K (z).
Consider y i ∈ R(t) such that g(t) = lim i→∞ d K (y i ). Since R(t) is compact we may assume that the sequence y i converges to some x t ∈ R(t). The continuity of d K (•) yields (5.3). For t = t 0 we take x t 0 = x 0 .

Claim 2. g is continuous.

Indeed, by our assumptions, this is obvious at t 0 . We first show that g is lower semicontinuous on ]t 0 , ∞[. Consider any t > t 0 , and a sequence t i converging to t such that lim inf s→t g(s) = lim i→∞ g(t i ). For each i pick x t i ∈ R(t i ) such that g(t i ) = d K (x t i ). Fix T > t and let X i ∈ S [t 0 ,T ] (x 0 ) be such that X i (t i ) = x t i . By Lemma 5.1, there exists a subsequence X i j converging uniformly to some X ∈ S [t 0 ,T ] (x 0 ). Therefore lim j→∞ X i j (t i j ) = X(t) ∈ R(t). From the continuity of d K (•) we deduce that g(t) d K (X(t)) = lim j→∞ g(t i j ). We show next that g is upper semicontinuous on ]t 0 , ∞[. Fix any t > t 0 and let x t ∈ R(t) be such that g(t) = d K (x t ). For any fixed u ∈ U consider the solution X(•) = X(•; t, x t , u). It is not difficult to realize that for every ε > 0, there exists ρ > 0 such that for all h ∈ [0, ρ],

|e hA x t -x t | ε 2 , t+h t e (t+h-s)A f (s, X(s), u)ds ε 2 .
Hence |X(t + h) -x t | ε and therefore

g(t + h) d K (X(t + h)) d K (x t ) + ε.
Consequently, g(•) is upper semicontinuous from the right at t. In order to prove that g is upper semicontinuous from the left at t, let X ∈ S [t 0 ,t] (x 0 ) be such that X(t) = x t . Then lim h→0+ X(t -h) = X(t). Since

g(t -h) d K (X(t -h)) d K (x t ) + |X(t -h) -x t | = g(t) + |X(t -h) -x t |,
we conclude that g is upper semicontinuous from the left at t. Claim 3. g ≡ 0.

Observe first that, for any y 0 ∈ H\K,

lim sup h→0+ 1 h d K (e hA y 0 ) -d K (y 0 ) 0. (5.4)
Indeed, let y 0 / ∈ K. Fix h > 0 and let y h ∈ K be such that

|y 0 -y h | d K (y 0 ) + h 2 .
Since e tA is contractive, we get |e hA y 0 -e hA y h | |y 0 -y h |. Recalling that e hA y h ∈ K for all h 0, we obtain

d K (e hA y 0 )-d K (y 0 ) |e hA y 0 -e hA y h |-d K (y 0 ) d K (y 0 )+h 2 -d K (y 0 ) = h 2 .
Dividing by h > 0 the above inequality and taking the limit yields (5.4). Next, observe that, if the set {t > t 0 : g(t) > 0} is empty, then g ≡ 0. Otherwise, let t := inf t > t 0 : g(t) > 0 .

Since g is continuous, g( t) = 0 and there exists ε > 0 such that for any t ∈] t, t+ε] with g(t) > 0 and x t as in (5.3), we have x t ∈ K δ . Let t ∈] t, t+ε] be such that g(t ) > 0 and t = sup{t ∈ [ t, t ] : g(t) = 0}. Then, g > 0 on ]t , t ]. Fix T > t and let M = [0, T ] \ N, with N as in Proposition 2.9. Pick any t ∈]t , t [∩M such that (5.2) holds true, and let ū ∈ U be such that

d 0 K (x t )f (t, x t , ū) 2C(t, |x t | θ )d K (x t ). Observe further that |x t | θ C T (1 + |x 0 | θ ) ∀t ∈ [t , t ],
where C T is the positive constant given by (2.35), to conclude that

d 0 K (x t )f (t, x t , ū) C(t)d K (x t ).
(5.5)

where we have set C(t) = 2C(t, C T (1 + |x 0 | θ )). Then we have that

D ↑ g(t) := lim inf h↓0 g(t + h) -g(t) h lim sup h↓0 d K (X(t + h)) -d K (e hA x t ) h + lim sup h↓0 d K (e hA x t ) -d K (x t ) h .
where X(•) = X(•; t, x t , ū). On the other hand, by our assumptions and the choice of t, X(t+h) = e hA x t + t+h t e (t+h-s)A f (s, X(s), ū)ds = e hA x t +hf (t, x t , ū)+o(h) and lim h→0+ e hA x t = X(t). Hence, from (5.4), (5.5), and the growth bound (2.35) we deduce that

D ↑ g(t) d 0 K (x t )f (t, x t , ū) C(t)d K (x t ) = C(t)g(t), (5.6)
Let us introduce the set valued map [t , t ] P (t) = {g(t)} + R + , which has closed values. We claim that P (•) is locally left absolutely continuous on [t , t ]. This means that, for every ε > 0 and every compact set Q ⊂ R, there exists δ > 0 such that, for every partition

t t 1 < τ 1 ... t j < τ j < .... t
satisfying Σ(τ i -t i ) δ, we have that Σe(P (t i ) ∩ Q, P (τ i )) ε, where e(C, D) = inf{λ > 0 : C ⊂ D + λ[-1, 1]} is the excess of C ⊂ R with respect to D ⊂ R. To prove this claim fix t ∈]t , t ] and h > 0 with t -h t . Consider x t-h ∈ R(t -h) such that g(t -h) = d K (x t-h ) and let y h ∈ K ∩ H θ be such that Step 2. To simplify notations we consider only the case t 0 = 0. Let x 0 ∈ K ∩ H θ . Fix an integer j 1. By Claim 3, using the fact that R( 1 2 j ) is compact we can find a solution of (2.33), X j , such that X j ( 1 2 j ) ∈ K. We now proceed by induction: suppose that, for some p ∈ {1, . . . , 2 j -1}, we have already constructed a solution to (2.33), corresponding to some control u(•), such that X j ( k 2 j ) ∈ K for every integer 0 k p. By Step 1 we can extend X j to a solution of (2.33) with t 0 = p 2 j satisfying X j ( p+1 2 j ) ∈ K. In this way we construct a solution X j (•) of (2.33) on [0, 1] such that X j ( k 2 j ) ∈ K for every k ∈ {0, . . . , 2 j }.

|y h -x t-h | g(t -h) + h and |y h | θ M (|x t-h | θ ). Fix any ū ∈ U and let X h (•) = X(•; t -h, x t-h , ū). Then g(t) d K (X h (t)) X h (t) -e hA y h = e hA x t-h + t t-h e (t-s)A f (s, X h (s), ū)ds -e hA y h |x t-h -y h | + t t-h e (t-s)A f (s, X h (s), ū) ds ≤ g(t -h) + t t-h 1 + e (t-s)A f (s, X h (s), ū) ds. Since t ∈ [t ,
By Lemma 5.1 the sequence X j has a subsequence X j i which converges uniformly to a solution X(•) of (2.33) defined on [0, 1]. Fix an integer n 1. Then for all large i, X j

i ( k 2 n ) ∈ K for k ∈ {0, ..., 2 n }. Therefore X( k 2 n ) ∈ K for k ∈ {0, ..., 2 n }.
Since this holds true for every n and X(•) is continuous, we deduce that X(t) ∈ K for all t ∈ [0, 1].

Step 3. Assume that for some integer i 1 and every integer 1 k i we have constructed solutions X k (•) of (2.33) defined on [0, k] such that the restriction of X i to [0, k] is equal to X k and X i ([0, i]) ⊂ K.

Consider the interval [i, i + 1]. Applying the same arguments as in Step 2, with t 0 = 0 replaced by t 0 = i and x 0 by X i (i), we extend X i on the time interval [i, i + 1] as a solution of (2.33) satisfying X([0, i + 1]) ⊂ K. Then, using an induction argument we complete the proof. where O ⊂ R n (n 3) is a bounded domain with boundary of class C 2 . Here, we denote by y(t, ξ) = (y 1 (t, ξ), y 2 (t, ξ)) the vector-valued solution of system (5.7) subject to control u(t, ξ) = (u 1 (t, ξ), u 2 (t, ξ)), that is, a measurable map u : R + × O → V × V , where V is a bounded closed subset of a separable Banach space. We assume that F i : R + × O × R 2 × V → R 2 satisfies the following for i = 1, 2:

       z → F i (t, ξ, z, v) is C 1 (R 2
) for a.e. (t, ξ) ∈ R + × O and every v ∈ V, v → F i (t, ξ, z, v) is continuous for a.e. (t, ξ) ∈ R + × O and every z ∈ R 2 , (t, ξ) → F i (t, ξ, z, v) is Lebesgue measurable for all v ∈ V and z ∈ R 2 .

Moreover, we suppose that, for some given functions

ϕ 0 ∈ L 2 loc (R + ; L 2 (O)), ϕ 1 ∈ L 2 loc (R + ; L ∞ (O)
), the growth conditions (i = 1, 2)

|F i (t, ξ, z, v)| C 0 ϕ 0 (t, ξ) + |z| , ∂F i ∂z (t, ξ, z, v) ϕ 1 (t, ξ)
are satisfied for a.e. (t, ξ) ∈ R + × O, every v ∈ V , every z ∈ R 2 , and some constant C 0 0. Finally, we impose that, for a.e. (t, ξ) ∈ R + × O and every z ∈ R 2 , the following is a closed convex set

F 1 (t, ξ, z, v 1 ), F 1 (t, ξ, z, v 2 ) : v 1 , v 2 ∈ V ⊂ R 2 .
Arguing as in Example 2.2, one can show that the above problem can be recast as a semilinear control system like (2.33). For this purpose, one just needs to take H = L 2 (O; R Then Λ(c 0 ) is proximally smooth.

Proof. Let us abbreviate Λ(c 0 ) by Λ 0 . We want to prove that there exists δ > 0 such that any x ∈ H with d Λ 0 (x) < δ has a unique projection onto Λ 0 . We prove the existence of the projection first, then its uniqueness.

Existence. Fix any x ∈ H \ Λ 0 and suppose that Φ(x) < c 0 . Since d Λ 0 is Fréchet differentiable on a dense subset of H by a well-known result due to Preiss [START_REF] Preiss | Differentiability of Lipschitz functions on Banach spaces[END_REF], we can find a sequence {x n } n , converging to x, such that Π Λ 0 (x n ) = {y n }. Moreover, we have that Finally, for Φ(x) > c 0 the reasoning is exactly the same as above, with the only difference that, in this case, (6.3) is replaced by

x n = y n -d Λ 0 (x n ) DΦ(y n ) |DΦ(y n )| ( n
x n = y n + d Λ 0 (x n ) DΦ(y n ) |DΦ(y n )| (n 1).
Uniqueness. Let x ∈ H be such that 0 < d Λ 0 (x) < δ, with δ as in (6.4), and suppose that x , x ∈ Π Λ 0 (x). Then

Theorem 3 . 10 .

 310 In addition to assumptions (a) and (b) of Corollary 3.9, suppose that Π K D(A)∩K δ ⊂ D(A). Then K ∩H θ is invariant under (2.1) if and only if (3.10) holds true for a.e. t ∈ R + and every x ∈ ∂K ∩ D(A).

Example 3 . 11 .

 311 With the notation of Example 3.6, let us study the invariance of the closed convex cone H + under the flow associated with the semilinear initial-boundary value problem + F (X) in ]0, ∞[×O ∂X ∂ν = 0 on ]0, ∞[×∂O X(0, ξ) = x 0 (ξ) ξ ∈ O a.e.

4. 1 .

 1 A bilinear viability problem. Let φ : R → R be a nonnegative function of class C 2 (R) such that

Example 5 . 3 . 1 ∂t = ∆y 1 + 2 ∂t = ∆y 2 +

 531122 Let us apply Theorem 5.2 to study a viability problem for the following parabolic systemF 1 (t, ξ, y, u 1 ) in ]0, ∞[×O ∂y F 2 (t, ξ, y, u 2 ) in ]0, ∞[×O y 1 = 0 = y 2 on ]0, ∞[×∂O y 1 (0, ξ) = x 1 (ξ) , y 2 (0, ξ) = x 2 (ξ) ξ ∈ O a.e.(5.7)

2 Proposition 6 . 1 .

 261 2 ) andD(A) = H 2 (O; R 2 ) ∩ H 1 0 (O; R 2 ) Ax = (∆x 1 , ∆x 2 ) ∀x = (x 1 , x 2 ) ∈ D(A).Then, assumption (H A ) follows from the positivity and compactness of the heat semigroup. Next, let us denote by U the family of all measurable maps u :O → V × V . Defining f : R + × H × U → H by f (t, x, u)(ξ) = F 1 t, ξ, x(ξ), u 1 (ξ) , F 2 t, ξ, x(ξ), u 2 (ξ) ,(5.8)for a.e. (t, ξ) ∈ R + × O, every x ∈ H, and every u = (u 1 , u 2 ) ∈ U , we have that assumptions (H f ) hold true with θ = 0. So, by applying Theorem 5.2 we conclude that the closed convex coneH + = x ∈ H : x i (ξ) 0, ξ ∈ O a.e. (i = 1, 2) .is viable under the flow associated with (5.7) provided that (5.2) is satisfied. Now, observe that H + is proximally smooth andΠ H + (x) = x + and d H + (x) = |x -| ∀x ∈ H,where we have setx ± = (x ± 1 , x ± 2 ). for all x ∈ D(A) \ H + . So, condition (5.2) reduces to inf u∈U x -Π H + (x), f (t, x, u) C(t, |x|) |x -| Let Φ ∈ C 1,1(H) and let c 0 ∈ R be such that Λ(c 0 ) = ∅. Suppose that there exist positive constants σ and S such that σ |DΦ(x)| S for every x ∈ Λ(c 0 ).(6.2)

2 .

 2 [START_REF] Aubin | Viability theory[END_REF].(6.3) Set µ n = d Λ 0 (x n )/|DΦ(y n )|. In view of (6.2), {µ n } is bounded in R. So, we can extract a convergent subsequence-still denoted by {µ n }. Since, for all sufficiently large integers m and n, we have thaty n -y m = x n -x m + d Λ 0 (x n ) DΦ(y n ) |DΦ(y n )| -d Λ 0 (x m ) DΦ(y m ) |DΦ(y m )| ,we deduce that|y n -y m | 2 = x n -x m , y n -y m + d Λ 0 (x n ) DΦ(y n ) |DΦ(y n )| -DΦ(y m ) |DΦ(y m )| , y n -y m + d Λ 0 (x n ) -d Λ 0 (x m ) DΦ(y m ) |DΦ(y m )| , y n -y m .Notice that (6.2) yields the following inequalityDΦ(x) |DΦ(x)| -DΦ(y) |DΦ(y)| 2 Lip(DΦ) σ |x -y| for all x, y ∈ Λ 0 . Hence, |y n -y m | 2 2|x n -x m | |y n -y m | + d Λ 0 (x n ) 2 Lip(DΦ) σ |y n -y m | Since d Λ 0 (x n ) → d Λ 0 (x), by taking δ > 0 such that d Λ 0 (x) < δ we conclude that {y n } is a Cauchy sequence in H. Consequently, its limit y ∈ Λ 0 satisfies|x -y| = lim n→∞ |x n -y n | = lim n→∞ d Λ 0 (x n ) = d Λ 0 (x),thus qualifying as a projection of x onto Λ 0 .

x

  + µ DΦ(x ) = x = x + µ DΦ(x )for real numbers µ and µ such that µ µ > 0 and|µ | |DΦ(x )| = d Λ 0 (x) = |µ | |DΦ(x )| .Therefore, on account of (6), we conclude that|x -x | d Λ 0 (x) = DΦ(x ) |DΦ(x )| -DΦ(x ) |DΦ(x )| 2 Lip(DΦ) σ |x -x |,which in turn yields x = x because 2d Λ 0 (x)Lip(DΦ)/σ < 1 by (6.4).

  Remark 2.1. Observe that, for a.e. t 0 and every r 0, (2.7) yields

			(2.5)
	t → L(t, r) locally square-summable for all r 0,
	such that, for every r 0 and a.e. t 0,
	f (t, x)	L(t, r)	(2.6)

r) nondecreasing for a.e. t 0 and f (t, x) -f (t, y) L(t, r)|x -y| θ (2.7) for all x, y ∈ H θ with |x| θ , |y| θ r.

  (|x 0 | θ ). Since X h (t 0 + h) ∈ K ∩ H θ for all h ∈ [0, T -t 0 ]because we assume invariance, by (2.27),(3.14), and (3.13) we get 1

.14)

Owing to Proposition 2.6 the maximal solutions X(•) := X(•; t 0 , x 0 ) and X h (•) := X(•; t 0 , x h ) of (2.1) are defined on some common interval [t 0 , T ], on which they satisfy (2.26) for R = M

  ). Fix any T ∈ [0, T 0 [, multiply by X both sides of the equation in(4.8), and integrate by parts to obtain

	1 2	d dt O	|X(t, ξ)| 2 dξ = -	O	|∇X(t, ξ)| 2 dξ +G(X(t, •))	O	|X(t, ξ)| 2 dξ (4.16)
	for all t ∈ [0, T ]. Similarly, multiplying the equation by ∂X ∂t we have that
	0	O	∂X ∂t	(t, x)	2	dξ = -	1 2	d dt O	|∇X(t, ξ)| 2 dξ
										+ G(X(t, •))	1 2	d dt O	|X(t, ξ)| 2 dξ. (4.17)
	By combining (4.16) and (4.17) we conclude that
	d dt O	|∇X(t, ξ)| 2 dξ
				2 G(X(t, •)) G(X(t, •))	|X(t, ξ)| 2 dξ -	|∇X(t, ξ)| 2 dξ . (4.18)
										O	O
	Now, in view of (4.2), (4.3), and (4.15) we have the estimates
				G(X(t, •))		λ + O |∇X(t, ξ)| 2 dξ O φ(X(t, ξ)) dξ	=	λ + c 0 O	|∇X(t, ξ)| 2 dξ
	and
			G(X(t, •))	O	|X(t, ξ)| 2 dξ	λ + O |∇X(t, ξ)| 2 dξ λ -O |X(t, ξ)| 2 dξ	O	|X(t, ξ)| 2 dξ
										=	λ + λ -O	|∇X(t, ξ)| 2 dξ.
	So, returning to (4.18) we obtain
				d dt O		|∇X(t, ξ)| 2 dξ 2	λ + c 0	λ + λ -	-1	O	|∇X(t, ξ)| 2 dξ	2
	for all t ∈ [0, T ]. Setting
										E(t) =	|∇X(t, ξ)| 2 dξ,
										O
	we can recast the above inequality as E (t) κE 2 (t) with

  ∀v ∈ H. Theorem 5.2. Assume (H A ), (H f ), and (H δ ). Suppose that for a.e. t ∈ R + and every x ∈ K δ ∩ H θ . Then K ∩ H θ is viable under (2.33).

		e tA K ⊂ K	∀t 0	(5.1)
	and there exists a function C : R + × R + → R * + , satisfying (3.4), such that
	inf u∈U	d 0 K (x)f (t, x, u) C(t, |x| θ ) d K (x)	(5.2)

  t ] is arbitrary, this inequality and our assumptions on f imply that P (•) is left absolutely continuous on [t , t ].

	By [10, Theorem 4.2] the solution of the differential equation
	z (t) = C(t)z(t), z(t ) = 0
	satisfies z(t) ∈ P (t) for all t ∈ [t , t ]. But z ≡ 0 and therefore g(t) 0 for
	all t ∈ [t , t ]. The derived contradiction yields our claim.
	On the other hand, recalling that
	Graph(P ) = (t, z) : t ∈ [t , t ], z ∈ P (t) ,
	we realize that (5.6) says that, for a.e. t ∈ [t , t ], (1, C(t)g(t)) belongs to
	the contingent cone T Graph(P ) (t, g(t)) to Graph(P ) at (t, g(t)). This fact and

the continuity of g imply that, for a.e. t ∈ [t , t ] and every y ∈ P (t), (1, C(t)y) ∈ T Graph(P ) (t, y).

Acknowledgement. The authors are grateful to the two anonymous referees for the constructive comments on the original submission that helped to improve the presentation of the results.

This work was completed while The first author wishes to express his gratitude to the Institut de Mathématiques de Jussieu -Paris Rive Gauche for supporting his visit through a CNRS Poste Rouge. He also thankfully acknowledges support from the following institutions/research programs: MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP: E83C18000100006; Mission Sustainability 2017 Program of the University of Rome Tor Vergata, CUP: E86C18000550005; Istituto Nazionale di Alta Matematica, within the LIA COPDESC network and EuLaMS Project. The third author acknowledges the support from the Air Force Office of Scientific Research under award number FA9550-18-1-0254.

for a.e. t ∈ R + , every x ∈ H, and some function C : R + × R + → R * + satisfying (3.4). In view of (5.8), the above condition reads as

which can in turn be reduced to

by appealing to the Lipschitz continuity of z → F i (t, ξ, z, v).

It is easy to give pointwise conditions which ensure the validity of (5.9). For instance, we can assume that, for some constant C > 0 and a.e. (t, ξ) ∈ R + × O the following holds true:

(5.10) Indeed, owing to well-known results in set-valued analysis (see, for instance, [2, Theorem 8.2.9]), from (5.10) we deduce that for a.e. t ∈ R + and every x ∈ H there exist measurable maps

for a.e. ξ ∈ O. So, taking u = (u 1 , u 2 ) we have that (5.9) is satisfied.

Appendix

In this Appendix, we prove the proximal smoothness of level sets of smooth functionals on a real Hilbert space H. We denote by C 1,1 (H) the Banach space of all maps Φ : H → R that are continuously Fréchet differentiable at every point of H with a Lipschitz continuous gradient DΦ : H → H, and we set The following result establishes the proximal smoothness of "nondegenerate" level sets.