M Leocata
email: marta.leocata@sns.it

J Vovelle
email: julien.vovelle@ens-lyon.fr

Global solutions to quadratic systems of stochastic reaction-diffusion equations in space-dimension two

published or not. The documents may come

Global solutions to quadratic systems of stochastic reaction-diffusion equations in space-dimension two

Marta Leocata, Julien Vovelle

The species A i , i = 1, . . . , 4 move by diffusion in the periodic domain T d = R d /Z d . Some statements below (entropy estimate, L 2 -estimate by duality) are given in any space dimension d, but our main results (L ∞ -bounds, global-in-time solutions) is restricted to the space dimensions d = 1 or d = 2. The evolution of the concentrations a i (t), i = 1, . . . , 4, of the species A i , i = 1, . . . , 4 is described by the following system of equations

da i -∇ x •(κ i ∇ x a i ) dt = f i (a)dt + σ α i (a)dB α (t), in Q T := T d × (0, T ), a i = 0 on ∂T d × (0, T ), a i (0) = a i0 ≥ 0, in T d ,
(1.2) for i = 1, . . . , 4 where f i (a) = (-1) i (a 1 a 3 -a 2 a 4 ).

(1.

3)

The family {B α ; 1 ≤ α ≤ d W } is a finite family of independent one-dimensional Wiener processes of the filtered probability space (Ω, F, P, (F t )). Summation over the repeated index α is used in (1.2). Our modelling approach, explained in Section 1.4 below, leads us to consider coefficients σ α i (a) of the type σ 1 i (a) = (-1) i √ a 1 a 3 , σ 2 i (a) = (-1) i √ a 2 a 4 , σ α i = 0, ∀α ≥ 3.

(1.4) However, we need the cancellation condition

a i = 0 ⇒ σ α i (a) = 0, ∀i, (1.5)
to ensure that the solutions to (1.2) stay non-negative, so we assume that (1.4) is satisfied asymptotically only: for α ∈ N \ {0}, σ α i : R d → R is a smooth function satisfying (1.5) and the growth condition

α |σ α i (a)| 2 ≤ ν(a 1 a 3 + a 2 a 4 ), (1.6)
where ν is a positive constant. We also assume that inf 1≤i≤4 κ i > 0, and without loss of generality, we will suppose that 1 ≤ inf 1≤i≤4 κ i .

(1.7)

We consider constant diffusion coefficients κ i , but variable (deterministic) coefficients depending on the variable (t, x) may also be considered, as long as the following bounds are satisfied:

sup 1≤i≤4 ∇ x κ i L ∞ (Q T ) + κ i L ∞ (Q T ) + κ -1 i L ∞ (Q T ) < +∞.
(1.8)

We note also that we may as well consider different, independent noises for each equation in (1.2), as in (A.1). Let us remark further that the restriction d W < +∞, i.e. the fact that we work with a finite-dimensional Wiener process, is relevant only in the justification of Theorem A.1 in appendix A.

Remark 1.1 (Boundary conditions). We consider periodic conditions to avoid the compatibility problems that arise in parabolic SPDEs when Dirichlet conditions (even homogeneous Dirichlet conditions) are considered (all the more since, for a system of equations, we would have to consider a system of compatibility conditions, for which existence of solution is not guaranteed without further analysis -at least if one would not consider compactly supported initial data), cf. [START_REF] Flandoli | Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of solutions[END_REF][START_REF] Debussche | A regularity result for quasilinear stochastic partial differential equations of parabolic type[END_REF][START_REF] Gerencsér | Boundary regularity of stochastic PDEs[END_REF]. It would also be very natural to consider homogeneous Neumann boundary conditions, a case for which the adequate references would be missing both for the direct parabolic stochastic problems and for the backward SPDEs used in Section 3.

Our main result and some additional comments and bibliographical references are given in Section 1.3.

Notion of solution

Let us first introduce a notion of weak solution.

Definition 1.1 (Weak solution up to a stopping time). Let a 0 ∈ L2 (T d ; R q )) and let τ be a stopping time, τ > 0 a.s. A predictable L 2 (T d ; R 4 )-valued process (a(t)) t<τ is said to be a weak solution to the system (1.2) up to time τ if 1. P-a.s., for all t ∈ [0, τ ), a i (t) ≥ 0, for all i = 1, . . . , q,

2. for all T > 0, the stopped processes a T i : t → a i (t ∧ T ) satisfy, for all i = 1, . . . , q,

a T i ∈ L 2 (Ω × [0, T ]; H 1 (T d )) ∩ L 2 (Ω; C([0, T ]; L 2 (T d ))), (1.9)
and

f i (a T ) ∈ L 2 (Ω × Q T ), σ α,i (a T ) ∈ L 2 (Ω × Q T ),
(1.10)

3. for all ϕ ∈ H1 (T d ), for all i ∈ {1, . . . , q}, a i satisfies P-almost surely, for all t ∈ [0, τ ),

a i (t), ϕ -a i0 , ϕ + κ i t 0 ∇ x a i (s), ∇ x ϕ ds = t 0 f i (a(s)), ϕ ds + t 0 d W α=1 σ α,i (a(s)), ϕ dB α (s
τ n = inf t ∈ [0, τ ); ess sup x∈T d a(x, t) 2 (R 4 ) > n ,
(1.12) the following properties are satisfied: for all p ∈ [2, ∞), q ∈ (2, ∞), m ∈ N with m ≥ 2 and mp > d + 2, and all T > 0, the stopped processes a τn∧T : t → a(t ∧ τ n ∧ T satisfy

a τn∧T ∈ L q (Ω; C([0, T ]; W m,p 0 (T d ))) ∩ L mq (Ω; C([0, T ]; W 1,mp 0 (T d ))).
(1.13)

A regular solution up the stopping time τ ≡ +∞ is said to be a global-in-time regular solution.

Construction of a regular solution up to a stopping time. Introduce the truncated non-linearities

f n i (a) = |χ n (|a| 2 )| 2 f i (a), σ α,n i (a) = χ n (|a| 2 )σ α i (a), (1.14)
where

|a| 2 = 4 i=1 a 2 i 1 2 , χ n (r) = χ(n -1 r), (1.15)
the function χ : R + → R+ being smooth, non-increasing, and such that χ(r) = 1 if r ≤ 1, χ(r) = 0 if r ≥ 2. We let a [n] denote the global-in-time regular solution to (1.2) where f i is replaced by f n i and σ α i is replaced by σ α,n i . The existence of a [n] is ensured by Theorem A.1 in Appendix A. We then set τn = inf t ≥ 0; ess sup x∈T d a [n] (x, t) 2 > n .

(1.16)

Since f n+1 i (a) = f n i (a), σ α,n+1 i (a) = σ α,n i (a) if |a| 2 ≤ n, the functions t → a [n] (t ∧ τn ∧ τn+1 ) and t → [n+1] (t ∧ τn ∧ τn+1 ) (1.17)
satisfy the same equation. By uniqueness (cf. Theorem A.1) they coincide, with the consequence that τn ≤ τn+1 , and

a [n] = a [n+1] a.s. on [0, τ n ]. We then set τ = lim n→+∞ τn . (1.18)
Then τ is a stopping time such that τ > 0 a.s. and the process (a(t)) defined by

a(t) = a [n] (t) if t ≤ τn 0 if t > τ (1.19)
is a regular solution to (1.2) up to the stopping time τ .

Main result

Theorem 1.1 (Global solutions). Suppose that the coefficients σ α i satisfy (1.5) and (1.6), that the diffusion coefficients κ i are strictly positive, and that the space dimension is There are few works on systems of reaction-diffusion equations with stochastic forcing in the literature. Let us mention [Kun15, DJZ19, BK22, DYZ23] and [START_REF] Hausenblas | Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations[END_REF] however. The papers [Kun15, DJZ19, BK22, DYZ23] are related to our work, but relatively different in nature: the emphasis is more on the existence of weak solutions (the problem of uniqueness is addressed in [Kun15] also). In [START_REF] Hausenblas | Theoretical study and numerical simulation of pattern formation in the deterministic and stochastic Gray-Scott equations[END_REF] a stochastic perturbation of the 2 × 2 Gray-Scott model is considered on the Torus in dimension d ≤ 3, and solutions in a Sobolev space that injects in L ∞ are obtained.

d = 1 or d = 2. Assume also a i0 ∈ C ∞ (T d ), a i0 ≥ 0, i = 1, . . . ,
One the two equations in the Gray-Scott model has a "good" non-linear reaction term, insofar as it is non-positive (when the solutions stay in the class of solutions with non-negative components).

This allows to start with an a priori estimate for the good component, a situation which has no counterpart in the case of the quadratic system (1.2).

Our method of proof uses a L 2 ln(L 2 ) estimate as in [START_REF] Desvillettes | Global existence for quadratic systems of reaction-diffusion[END_REF]. An argument of duality is used to establish this L 2 ln(L 2 ) estimate. In our stochastic context, we have to consider a backward SPDE as dual equation. Such a duality approach via BSPDE was already exploited in [START_REF] Debussche | Diffusion-approximation for a kinetic spray-like system with random forcing[END_REF].

The L ∞ bound on solutions is obtained by the method of De Giorgi, via truncation in the entropy inequality, as performed in [START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF]. In the companion paper [START_REF] Leocata | Supremum estimates for parabolic stochastic partial differential equations[END_REF], this method is analysed in the framework of linear parabolic equations. An alternative proof by duality of supremum estimates in this context is also given, and we still refer to this paper for references about supremum estimates for parabolic stochastic equations.

We conclude this paragraph with some details on the structure of the paper. The following section 1.4 accounts for the stochastic terms in the modelling of chemical reactions. The L 1 ln(L 1 ) entropy estimate, Proposition 2.1, and the L 2 ln(L 2 ) estimate, Theorem 3.1], are derived in Section 2 and Section 3 respectively. The L ∞ estimate on solutions, Theorem 4.1, is then obtained in Section 4.

Diffusion-approximation and modelling of the chemical reaction

In this section, we explain how an asymptotic expansion at the diffusive scale on the generator associated to the Markov description of the reaction (1.1) leads to (1.2). We will neglect the spatial displacements of the reactants in this discussion.

In a stochastic modelling of (1.1), each reaction happens at a random time, given as an exponential random variable. The parameters of these exponential random variables, also called transition rates of the reaction, are respectively of the form

r → = λ → 1 N N 1 N 3 , r ← = λ ← 1 N N 2 N 4 ,
where N is the total number of reactants and N i the number of molecules of type i, [START_REF] Ethier | Markov processes[END_REF]. Note that N stays constant in the evolution. We take it as main parameter and consider the situation where it is large. Using the approach of [EK86, p.455], we consider the Markov process ÂN = (N i (t)) 1≤i≤4 with state space N 4 , given by

ÂN (t) = ÂN (0) + E t 0 N η ÂN (s) N ds , (1.21)
In (1.21), the sum is over the two following indices (where * indicates transposition)

→ = (-1, 1, -1, 1) * , ← = -→ , (1.22)
the processes E are two independent unit Poisson processes and, given a ∈ (R + ) 4 , the functions η (a) are defined by

η → (a) = a 1 a 3 , η ← (a) = a 2 a 4 .
The generator L N of the rescaled process A N (t) = N -1 ÂN (t) acts on functions ϕ : (R + ) 4 → R and is given by

L N ϕ(a) = N η (a)(ϕ(a + N -1 ) -ϕ(a)).
We introduce f (a) := η (a) (it coincides with (1.3)) to expand L N ϕ(a) as follows

L N ϕ(a) = f (a) • D a ϕ(a) + N η (a)(ϕ(a + N -1 ) -ϕ(a) -N -1 D a ϕ(a)) = f (a) • D a ϕ(a) + 1 N η (a) * : D 2 a ϕ(a) + O 1 N 2 , (1.23)
where A : B = i,j A ij B ij . At the order 0, (1.23) gives the generator f • D a ϕ associated to the ODE ȧ = f (a). At order 1, we obtain the generator

L 1 N ϕ(a) = f (a) • D a ϕ(a) + 1 N η (a) * : D 2 a ϕ(a),
associated to the SDE

da(t) = f (a(t))dt + 2 N [η (a(t))] 1/2 dB (t), (1.24)
where the (B ) are independent one-dimensional Wiener processes. When the space variable is neglected, (1.24) corresponds to (1.2) where the sum over α involves two indexes, and

f (a) = (a 1 a 3 -a 2 a 4 ) , σ 1 (a) = ε √ a 1 a 3 , σ 2 (a) = ε √ a 2 a 4 , (1.25)
where ε is a small constant and = → , as given in (1.22).

Entropy estimate

We use the notation

s * = 1 + s, s ∈ [0, +∞). Let Φ(s) = s * ln(s * ) -s * + 1 = (1 + s) ln(1 + s) -s, Φ(s) = Φ(s) + s = (1 + s) ln(1 + s). (2.1)
Proposition 2.1 (Entropy estimate). Suppose that the coefficients σ α i satisfy (1.5) and (1.6). Assume also a i0 ∈ C ∞ (T d ), a i0 ≥ 0, i = 1, . . . , 4. Let a be a global-in-time regular solution to (1.2) or a global-in-time solution to (1.2) with the truncated non-linearities defined in (1.14)-(1.15). Then a satisfies the following estimates: for all p ∈ [1, +∞), for all T > 0,

E 4 i=1 T d Φ(a i (t))dx + E 4 i=1 t 0 T d κ i |∇ x a i | 2 1 + a i dxds p ≤ C(p)E 4 i=1 T d Φ(a i0 )dx p , (2.2) for all t ∈ [0, T ],
where the constant C(p) depends on p, d, on ν, and T . Proof of Proposition 2.1. We will denote by C 1 , C 2 , . . . any constant depending on d and ν only. We write C 1 (p), C 2 (p), . . . if there is also a dependence on p. We will establish (2.2) under the additional assumption that

T ≤ T 1 (p), (2.3)
for a positive time T 1 (p) depending on p, d and ν (cf. (1.6)). The general case will follow by iteration of this result. We consider first the case of a global-in-time regular solution a of (1.2). The case of truncated non-linearities is explained at the end of the proof.

By the Itô formula, we have for all i:

d Φ(a i ) -∇ x •(κ i ∇ x Φ(a i ))dt = Φ (a i )f i (a) -κ i Φ i (a i )|∇ x a i | 2 + 1 2 α |σ α i (a i )| 2 Φ i (a i ) dt + Φ i (a i )σ α i (a)dB α (t). (2.4)
Then by integration in x, we obtain

U Φ(a i )(t)dx = U Φ(a i )(0)dx + t 0 U Φ (a i )f i (a) -κ i Φ i (a i )|∇ x a i | 2 + 1 2 α |σ α i (a i )| 2 Φ i (a i ) dxds + t 0 U Φ i (a i )σ α i (a)dxdB α (s), (2.5) Lemma 2.

(Control of the source terms).

There is a constant C 1 depending on ν only such that

4 i=1 Φ (a i )f i (a) + 1 2 α |σ α i (a i )| 2 Φ i (a i ) ≤ C 1 4 i=1 Φ(a i ), (2.6) for all a 1 , . . . , a 4 ∈ R + .
Summing on i in (2.5) and using the positivity condition (1.7) and the estimate (2.6), we obtain

E(t) + D(t) ≤ C 1 t 0 E(s)ds + M(t) + E(0), (2.7)
where

E(t) = 4 i=1 T d Φ(a i (t))dx, D(t) = E 4 i=1 t 0 T d |∇a ξ i | 2 a ξ * i dxds,
(2.8) and M(t) denotes the martingale

M(t) = t 0 T d α,i σ α i (a i ) (1 + ln(a * i )) dxdB α (s).
(2.9)

Our aim is to get the following estimate:

E [U(t) p ] ≤ C(p)E [E(0) p ] , U(t) := sup 0≤s≤t E(s) + D(t), (2.10) for 0 ≤ t ≤ T 1 (p). The bound C 1 t 0 E(s)ds ≤ C 1 t U(t) (2.11) inserted in (2.7) gives us U(t) ≤ 2E(0) + 2M(t) * , (2.12)
where

M(t) * = sup s∈[0,t] |M(s)|, for 0 ≤ t ≤ T 1 (p), provided C 1 T 1 (p) ≤ 1 2 . Raising (2.12) to the power p yields U(t) p ≤ C 2 (p) [E(0) p + (M(t) * ) p ] . (2.13)
We take expectation in (2.13) and use the Burkholder-Davis-Gundy inequality to get

E [U(t) p ] ≤ C 3 (p) E [E(0) p ] + E M, M p 2 t .
(2.14)

The quadratic variation of M(t) is

M, M t = t 0 α T d i σ α i (a) (1 + ln(a * i )) dx 2 ds.
(2.15)

Set F = a 1 a 3 + a 2 a 4 . By the Cauchy-Schwarz inequality and (1.6), we have

M, M t ≤ t 0   T d α,i F -1/2 |σ α i (a)| 2 (1 + ln(a * i )) dx T d i F 1/2 (1 + ln(a * i )) dx   ds ≤ ν t 0 T d i F 1/2 (1 + ln(a * i )) dx 2 ds ≤ C 4 t 0 T d i,j a i 1 + ln(a * j ) dx 2 ds.
(2.16)

Observe that we have the bound a ≤ Φ(a) and

a * i ln(a * j ) ≤ a * i ln(a * i ) + a * j ln(a * j ), (2.17)
which, exploited in (2.16), yields

M, M t ≤ C 5 t 0 T d i Φ(a i )dx 2 ds ≤ C 5 t sup 0≤s≤t E(s) 2 ≤ C 5 t U(t) 2 .
(2.18)

We insert this last estimate (2.18) in (2.14) to obtain

E [U(t) p ] ≤ C 6 (p) E [E(0) p ] + t p 2 E [U(t) p ] . (2.19)
The desired bound (2.10) follows, under the condition t ≤ T 1 (p).

Proof of Lemma 2.2. Our aim is to control

S = -(a 1 a 3 -a 2 a 4 ) (ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) + 1 2 α,i |σ α i (a i )| 2 a * i .
(2.20)

By the growth condition (1.6), we have

S ≤ -(a * 1 a * 3 -a * 2 a * 4 ) (ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) + ν 4 i=1 a * 1 a * 3 + a * 2 a * 4 a * i + e 1 , ( 2.21)
where

e 1 = (a * 1 + a * 3 -a * 2 -a * 4 ) (ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) . (2.22)
We use the inequality (2.17) to obtain the bound

e 1 ≤ C 2 4 i=1 Φ(a i ).
(2.23)

We also have

4 i=1 a * 1 a * 3 + a * 2 a * 4 a * i ≤ C 3 4 i=1 (Φ(a i ) + a i ) + a * 1 a * 3 a * 2 + a * 1 a * 3 a * 4 + a * 2 a * 4 a * 1 + a * 2 a * 4 a * 3 , (2.24) so S ≤ Θ(a) + e 2 , (2.25)
where e 2 satisfies the same bound (2.23) as e 1 and where

Θ(a) = -(a * 1 a * 3 -a * 2 a * 4 ) (ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) + ν a * 1 a * 3 a * 2 + a * 1 a * 3 a * 4 + a * 2 a * 4 a * 1 + a * 2 a * 4 a * 3 .
(2.26)

We will conclude the proof by showing that

Θ(a) ≤ C 4 4 i=1 Φ(a i ).
(2.27)

Let K > 1 denote a constant that will be fixed later. Let us examine Θ(a) in the three regions

Υ + = {a * 2 a * 4 > Ka * 1 a * 3 }, Υ 0 = {Ka * 1 a * 3 ≥ a * 2 a * 4 ≥ K -1 a * 1 a * 3 }, Υ -= {a * 1 a * 3 > Ka * 2 a * 4 }.
(2.28) For symmetry reasons, it is sufficient to examine the last two regions Υ 0 , Υ -. In Υ 0 , we simply use the sign of the entropy dissipation term,

-(a * 1 a * 3 -a * 2 a * 4 )(ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) ≤ 0, (2.29)
and the bound

a * 1 a * 3 a * 2 + a * 1 a * 3 a * 4 + a * 2 a * 4 a * 1 + a * 2 a * 4 a * 3 ≤ K (a * 4 + a * 2 + a * 3 + a * 1 ) . (2.30)
In Υ -, we can estimate Θ(a) from above as follows: let us write

a * 2 a * 4 = αa * 1 a * 3 with α < K -1 . Then we obtain (a * 1 a * 3 -a * 2 a * 4 )(ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) = a * 1 a * 3 ψ(α), (2.31)
where ψ(α) = -(1 -α) ln(α) is positive decreasing on (0, 1] with ψ(0+) = +∞. In particular, ψ(α) > ψ(K -1 ) and thus

-(a * 1 a * 3 -a * 2 a * 4 )(ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) ≤ -a * 1 a * 3 ψ(K -1 ).
(2.32)

The remaining part of Θ(a) is bounded by

a * 1 a * 3 a * 2 + a * 1 a * 3 a * 4 + K -1 a * 3 + K -1 a * 1 ≤ 2a * 1 a * 3 + K -1 a * 3 + K -1 a * 1 .
(2.33)

We can conclude to (2.27) in Υ -if 2ν < ψ(K -1 ), which is always satisfied for K large enough since ψ(0+) = +∞.

To conclude the proof of Proposition 2.1, we still have to explain why it remains true when the non-linearities f n and σ n defined in (1.14) are considered. This amounts to justify the validity of Lemma 2.2 in this case (with a constant independent on n). Consider thus

S n = f n 1 (a) (ln(a * 1 a * 3 ) -ln(a * 2 a * 4 )) + 1 2 α,i |σ n α,i (a i )| 2 a * i , ( 2
T d i | Φ(a i (t))| 2 dx. (3.1) Assume that E 2 (0) = E [E 2 (0)] < +∞. (3.2)
The main result of this section shows that the initial L2 ln(L 2 )-bound (3.2) is propagated in a bound

E T 0 E 2 (t)ds < +∞. (3.3)
This quadratic estimate is obtained by duality, by considering an appropriate backward parabolic SPDE (see (3.10)). This is an extension to the stochastic framework of the duality method developed in [START_REF] Desvillettes | Global existence for quadratic systems of reaction-diffusion[END_REF] (see also [START_REF] Pierre | Global existence in reaction-diffusion systems with control of mass: a survey[END_REF]).

Theorem 3.1 (L 2 ln(L 2 )-estimate). Suppose that the coefficients σ α i satisfy (1.5) and (1.6). Assume also a i0 ∈ C ∞ (T d ), a i0 ≥ 0, i = 1, . . . , 4. Let a be a global-in-time regular solution to (1.2) or a global-in-time solution to (1.2) with the truncated non-linearities defined in (1.14)-(1.15). Then, under the smallness condition (1.20), a satisfies the following estimate: for all T > 0,

E T 0 E 2 (t)ds ≤ Ce 2C1T E 2 (0), (3.4)
where the constant C depends on d, max 1≤i≤4 κ i only, and C 1 is the constant in introduced in Lemma 2.2.

Proof of Theorem 3.1. The proof breaks into several steps.

Step 1. We start from the equation (2.4) and use (2.6) to obtain

dz -∇ x • i κ i ∇ x z i dt = F dt + i ln(a i )σ α i (a)dB α (t), (3.5)
where

z = i Φ(a i ), (3.6)
and where, for all x ∈ T d , (F (x, t)) is an adapted process, satisfying F (x, t) ≤ C 1 z(x, t) a.s., for all (x, t) ∈ Q T . We rewrite (3.5) as

dz -∆(Kz)dt = F dt + g α dB α (t), (3.7)
where

g α = i (1 + ln(a i ))σ α i (a), (3.8)
and where the coefficient K is given 2 by

K(x, t) = 1 z(x, t) i κ i Φ(a i (x, t)) ∈ [1, max 1≤i≤4 κ i ].
(3.9)

Thus K is measurable in (x, t) and bounded, with K(x, t) ≥ 1 for all (x, t) ∈ Q T , P-a.s., and for all x ∈ T d , (K(x, t)) is adapted. Next, we introduce the solution (w, (q α )) to the backward SPDE dw(t) are P × B(T d )-measurable, and that P-a.s., for all (x, t)

+ ( K(t)∆w(t) + C 1 w(t))dt = -H(t)dt + q α (t)dB α (t), ( 3
∈ Q T , 1 ≤ K(x, t) ≤ max 1≤i≤4 κ i , 0 ≤ H(x, t), (3.13)
and that there exists an integer q > 2 + d/2, a constant C H,K ≥ 0 such that P-a.s., for all (x, t) ∈ Q T , for all multi-index m of length |m| < q

|D m x K(x, t)| + |D m x H(x, t)| ≤ C H,K . (3.14)
Note in particular that there is a modulus γ : [0, +∞) → [0, +∞) (i.e. a continuous and increasing γ with γ(r) = 0 if, and only if r = 0) such that P-a.s., for all t ∈ [0, T ], for all x, y ∈ T d ,

| K(t, x) -K(t, y)| ≤ γ(|x -y|). (3.15)
By [START_REF] Du | Strong solution of backward stochastic partial differential equations in C 2 domains[END_REF] (see Remark 3.1 and Corollary 3.4), the problem (3.10)-(3.11) admits a solution (w, (q α )) in the following sense:

1. (w, (q α )) satisfies the interior regularity

w ∈ L 2 (Ω × (0, T ), P, C 2 (T d )) ∩ L 2 (Ω, C( QT )), q α ∈ L 2 (Ω × (0, T ), P, C 1 (T d )), (3.16)
with the bound

E T 0 α |q α (x, t)| 2 dt < +∞, (3.17)
for all x ∈ T d , 2. the equation (3.10) is satisfied point-wise, for every x ∈ T d , 3. (w, (q α )) satisfies the bound

E T 0 w(t) 2 H 2 (T d ) + α q(t) 2 H 1 (T d ) dt + E sup t∈[0,T ] w(t) 2 H 1 (T d ) ≤ C 0 E T 0 H 2 L 2 (T d ) dt, (3.18)
where the constant C 0 depends on d, T and on the modulus γ in (3.15).

Note that, in [START_REF] Du | Strong solution of backward stochastic partial differential equations in C 2 domains[END_REF] this is the Homogeneous Dirichlet problem that is considered, but the adaptation to the case of periodic boundary conditions is straightforward. We will justify in

Step 2 the Itô formula, which, based on the equations satisfied by z and w, gives

E Q T Hzdxdt = E T d z 0 w(0)dx + E Q T ∆w(K -K)zdxdt + E Q T (F -C 1 z)wdxdt + E Q T g α q α dxdt . (3.19)
In

Step 3, we show that P-a.s., w(x, t) ≥ 0 for all (x, t) ∈ Q T . In Step 4, we establish the following bounds on (w, (q α )):

E w(0) 2 H 1 (T d ) + E T 0 e 2C1t w(t) 2 H 2 (T d ) + α q α 2 H 1 (T d ) dt ≤ R 1 E T 0 e 2C1t H(t) 2 L 2 (T d ) dt, (3.20)
where the constant R 1 depends on d, and max 1≤i≤4 κ i only. In the last, fifth step, we consider the limit of (3.19) when K = K ε is a regularization of K, and conclude our argument.

Step 2. We need to justify the Itô formula for the product z(t), w(t) L 2 (T d ) : at least three possible approaches seem possible in our situation:

1. use the Itô formula for the square of the L 2 -norm (see [START_REF] Krylov | Stochastic evolution equations[END_REF] for instance) of w, z, w + z, 2. consider the equations at fixed x (this is possible since we work with regular enough solutions), use the Itô formula for real-valued processes, and integrate the result over T d , 3. use a spectral decomposition of the processes, apply the Itô formula for real-valued processes, and gather the results. This is the last method that we will employ since, even if regularity of solutions is available in our case, it is less demanding from that point of view than the second approach (and as explained in Remark 1.1, solutions with less regularity may have to be considered if different boundary conditions are assumed). So let (v n ) denote the Fourier basis of L 2 (T d ): -∆v n = λ n v n , λ n = 4π 2 |n| 2 , for all n ∈ Z d . We consider the spectral decomposition

w(t) = n∈Z d ŵn (t)v n , q α (t) = n∈Z d qα n (t)v n z(t) = n∈Z d ẑn (t)v n . (3.21)
The integrability and regularity properties of w, q α and z ensure that the series in (3.21) have at least the following convergence properties:

E T 0 n∈Z d λ 2 n (| ŵn (t)| 2 + |ẑ n (t)| 2 ) < +∞, E T 0 n∈Z d α≥1 λ n |q α n (t)| 2 < +∞. ( 3
-∆u, v n L 2 (T d ) = λ n u, v n L 2 (T d ) , u ∈ H 2 (T d ), (3.27)
to obtain in particular a term

E T 0 |n|≤N K(t)z(t), v n L 2 (T d ) ∆w(t), v n L 2 (T d ) -K(t)∆w(t), v n L 2 (T d ) z(t), v n L 2 (T d ) dt.
(3.28) Since all the terms Kz, ∆w, K∆w, z involved in (3.28) belong to L 2 (Ω × (0, T ) × T d ), we have the convergence when N → +∞ of the quantity (3.28) to the term

E Q T ∆w(K -K)zdxdt . (3.29)
The convergence of the other terms in the sum over |n| ≤ N of (3.26) is similar: we obtain (3.19) in the limit.

Step 3. We claim here that P-a.s., for all (x, t) ∈ Q T , w(x, t) ≥ 0. This can be proved by justification of the Itô formula for the quantity w → w -2 L 2 (T d ) , where w -= max(-w, 0) is the negative part of w. The result also follows directly from [DT12, Theorem 5.1] and (3.11), (3.13).

Step 4. Bounds on (w, (q α )). The basic principle to obtain appropriate estimates on w and q α (see (3.20)) is to "multiply" the equation (3.10) by ∆w. In the deterministic case, we use an integration by parts and the terminal condition (3.11) to obtain

T 0 ∂ t w, ∆w L 2 (T d ) dt = - T 0 ∂ t ∇w, ∇w L 2 (T d ) dt = - T 0 d dt 1 2 ∇w(t) 2 L 2 (T d ) dt = 1 2 ∇w(0) 2 L 2 (T d ) . (3.30)
The analogous result for the solution to (3.10) is more delicate to justify as ∂ t w has no proper sense. We use again, as in Step 2., a spectral decomposition to justify our computations. We start from the equation (3.23) for ŵn (t), use the Itô formula and the terminal condition (3.11) to compute the evolution of the square of t → e C1t ŵn (t). After taking the expectation of the result, we obtain

1 2 E | ŵn (0)| 2 + E T 0 e 2C1t -K(t)∆w(t) -H(t), v n L 2 (T d ) ŵn (t) + 1 2 α |q α n | 2 dt = 0. (3.31)
We multiply (3.31) by λ n , sum the result over |n| ≤ N , use (3.27), and the identity

∇u L 2 (T d ) = n≥1 λ n u, v n L 2 (T d ) 2 , u ∈ H 1 (T d ), (3.32)
to obtain in the limit N → +∞ the following estimate:

1 2 E ∇w(0) 2 L 2 (T d ) + E T 0 e 2C1t K(t)∆w(t), ∆w(t) L 2 (T d ) + 1 2 α ∇q α 2 L 2 (T d ) dt = -E T 0 e 2C1t H(t), ∆w(t) L 2 (T d ) dt. (3.33)
Using the bounds on K in (3.13) and the Cauchy-Schwarz inequality, we deduce from (3.33) that

1 2 E ∇w(0) 2 L 2 (T d ) + E T 0 e 2C1t 1 2 ∆w(t) 2 L 2 (T d ) + 1 2 α ∇q α 2 L 2 (T d ) dt ≤ max 1≤i≤4 (κ i )E T 0 e 2C1t H(t) 2 L 2 (T d ) dt. (3.34) Since u 2 H 1 (T d ) ≤ C(d) ∇u 2 L 2 (T d ) , u 2 H 2 (T d ) ≤ C(d) ∆u 2 L 2 (T d ) , u ∈ H 2 (T d ), (3.35)
where C(d) is a constant depending on d, (3.34) gives (3.20) as desired.

Step 5. Let us conclude the proof of (3.4). We use the fact that F ≤ C 1 z (see (3.6)-(3.7)) and w ≥ 0, the Cauchy-Schwarz inequality and the bounds (3.20) to deduce from (3.19) that

E Q T Hzdxdt 2 ≤ R 2 E z 0 2 L 2 (T d ) + E Q T e -2C1t (K -K) 2 z 2 dxdt +E Q T e -2C1t α |g α | 2 dxdt E T 0 e 2C1t H(t) 2 L 2 (T d ) dt , (3.36)
where the constant R 2 depends on d and max 1≤i≤4 κ i only. We apply (3.36) with K = Kε , where

Kε (x, t) = z(x, t) z(x, t) + ε K(x, t) = 1 z(x, t) + ε i κ i Φ(a i (x, t)).
(3.37)

This coefficient K has the desired regularity properties by regularity of z, and satisfies the bound

0 ≤ (K -Kε )z ≤ ε, (3.38)
so, taking the limit ε → 0 in (3.36) where K = Kε , gives us

E Q T Hzdxdt 2 ≤ R 2 E z 0 2 L 2 (T d ) + E Q T e -2C1t α |g α | 2 dxdt E T 0 e 2C1t H(t) 2 L 2 (T d ) dt .
(3.39)

We replace H with t → e C1t H(t) and, by an argument of density relax the hypothesis of regularity of H to obtain

E Q T H ẑdxdt 2 ≤ R 2 E z 0 2 L 2 (T d ) + E Q T e -2C1t α |g α | 2 dxdt E T 0 H(t) 2 L 2 (T d ) dt , (3.40)
for all non-negative H ∈ L 2 (Ω × (0, T ), P; L 2 (T d )), where ẑ(t) = e -C1t z(t). At last, we estimate the term

E Q T e -2C1t |g α | 2 dxdt . (3.41)
Recall that g α is defined in (3.8). By (1.6), and (2.17) we have

α |g α | 2 ≤ R 3 νz 2 (3.42)
where R 3 is a numerical constant, and so

E Q T e -2C1t |g α | 2 dxdt ≤ R 3 νE Q T |ẑ| 2 dxdt . (3.43)
We insert (3.43) in (3.40) and invoke (1.20) to ensure that R 2 R 3 ν ≤ 1 2 (note well that the constant R 2 R 3 does not depend on time) to conclude that

E Q T |ẑ| 2 dxdt ≤ R 4 E z 0 2 L 2 (T d ) ,
(3.44) which yields (3.4).

L ∞ -estimates

L ∞ -estimate via truncation of the entropy

Various approaches to L ∞ -estimates for (deterministic) quadratic reaction-diffusion systems exist (see [START_REF] Caputo | Global regularity of solutions to systems of reaction-diffusion with sub-quadratic growth in any dimension[END_REF][START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF][START_REF] Caputo | Solutions of the 4species quadratic reaction-diffusion system are bounded and C ∞ -smooth, in any space dimension[END_REF][START_REF] Souplet | Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth[END_REF][START_REF] Fellner | Global classical solutions to quadratic systems with mass control in arbitrary dimensions[END_REF]). Here we follow the same approach à la De Giorgi as in [START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF], using also some of the elements in [START_REF] Leocata | Supremum estimates for parabolic stochastic partial differential equations[END_REF].

Theorem 4.1 (L ∞ -estimate). Let Θ(u) = ln(1 + ln(1 + u)), u ≥ 0. (4.1)
Suppose that the coefficients σ α i satisfy (1.5) and (1.6). Assume also a i0 ∈ C ∞ (T d ), a i0 ≥ 0, i = 1, . . . , 4. Let a be a global-in-time regular solution to (1.2) or a global-in-time solution to (1.2) with the truncated non-linearities defined in (1.14)-(1.15). Assume also that the space dimension d is d = 1 or d = 2 and that the strength of noise is small in the sense of (1.20). Then a satisfies the following estimate:

E Θ sup 1≤i≤4 a i L ∞ (Q T ) ≤ C ∞ , (4.2)
where the constant C ∞ depends on d, ν, T , max 1≤i≤4 κ i and sup 1≤i≤4 a i0 L ∞ (T d ) .

Proof of Theorem 4.1. We will consider only the case d = 2. For a, ξ ≥ 0, set

a ξ = (a -ξ) + , a ξ * = 1 + a ξ , (4.3) and Φ(a; ξ) = Φ(a ξ ) = (1 + a ξ ) ln(1 + a ξ ) -a ξ = a ξ * ln(a ξ * ) -a ξ * + 1. (4.4)
Evolution of the entropy. Note that

∂ ∂a Φ(a; ξ) = ln(a ξ * ) = ln(a ξ * )1 a>ξ , ∂ 2 ∂a 2 Φ(a; ξ) = 1 a>ξ a ξ * , ( 4.5)
so, similarly to (2.7), we have

E(t; ξ) + D(t; ξ) ≤ S(t; ξ) + M(t; ξ) + E(0; ξ), (4.6)
where

E(t; ξ) = 4 i=1 T d Φ(a ξ i (t))dx, D(t; ξ) = E 4 i=1 t 0 T d |∇a ξ i | 2 a ξ * i dxds, ( 4.7)
and

S(t; ξ) = t 0 T d    -(a 1 a 3 -a 2 a 4 ) ln(a ξ * 1 a ξ * 3 ) -ln(a ξ * 2 a ξ * 4 ) + 1 2 α,i |σ α i (a i )| 2 a ξ * i 1 ai>ξ    dxds.
(4.8) The martingale term is

M(t; ξ) = t 0 T d α,i σ α i (a i ) ln(a ξ * i )dxdB α (s). (4.9)
As in (2.10), we introduce the quantity

U(t; ξ) = sup 0≤s≤t E(s; ξ) + D(t; ξ). (4.10) Auxiliary functional. Set Ψ(a) = a 0 Φ(s) s * ds = a 0 (1 + s) ln(1 + s) -s 1 + s ds (4.11)
and

U ψ (t; ξ) = t 0 T d i |Ψ(a ξ i )| 2 dxds 1/2 . (4.12)
We have

∇Ψ(a) = Φ(a) ∇a √ 1 + a , ( 4.13)
so, by Sobolev's embedding (recall that the space dimension is d = 2 here) and Cauchy-Schwarz' inequality,

Ψ(a) L 2 (T d ) ∇Ψ(a) L 1 (T d ) Φ(a) 1 2 L 1 (T d ) (1 + a) -1 2 ∇a L 2 (T d ) , (4.14) which yields U ψ (t; ξ) 2 t 0 E(s; ξ) i (1 + a ξ i ) -1 2 ∇a ξ i 2 L 2 (T d ) ds sup s∈[0,t] E(s; ξ) D(t; ξ), (4.15)
and thus U ψ (t; ξ) U(t; ξ). (4.16)

As in [GV10, Lemma 3.2], we will use the fact that

g(a) ln(a * ) |Ψ(a)| 2 , g(a) := a 3 1 0≤a≤1 + a 2 1 a>1 . (4.17)
Bound on the source term. We will proceed as in the previous section 2, taking into account the additional effects of the truncation at level ξ. In particular, our aim is not a bound like (2.6), but instead, using (4.16)-(4.17), a bound of S(t; ξ) by a power of U(t; ζ) (or more exactly, a power of U ψ (t; ζ)), for a given truncation level ζ < ξ. We begin as in Section 2 however, and use first the growth condition (1.6), to get

S(t; ξ) ≤ t 0 T d -(a ξ * 1 a ξ * 3 -a ξ * 2 a ξ * 4 ) ln(a ξ * 1 a ξ * 3 ) -ln(a ξ * 2 a ξ * 4 ) dxds + ν t 0 T d i a ξ * 1 a ξ * 3 + a ξ * 2 a ξ * 4 a ξ * i 1 ai>ξ dxds + ϕ(t; ξ), ( 4.18)
where

ϕ(t; ξ) = t 0 T d (a ξ * 1 a ξ * 3 -a ξ * 2 a ξ * 4 ) -(a 1 a 3 -a 2 a 4 ) ln(a ξ * 1 a ξ * 3 ) -ln(a ξ * 2 a ξ * 4 ) dxds + ν t 0 T d i (a 1 a 3 + a 2 a 4 ) -(a ξ * 1 a ξ * 3 + a ξ * 2 a ξ * 4 ) a ξ * i 1 ai>ξ dxds. (4.19)
We have

a i a j -a ξ * i a ξ * j = a i a j -a ξ i a ξ j -1 -(a ξ i + a ξ j ). (4.20)
The term a i a j -a ξ i a ξ j is non-positive, so

-1 -(a ξ i + a ξ j ) ≤ a i a j -a ξ * i a ξ * j . (4.21)
From (4.20), we also obtain the bound from below a i a j -a ξ * i a ξ * j ≤ a i a j -a ξ i a ξ j , and since

a i ≤ a ξ i + ξ, we get -1 -(a ξ i + a ξ j ) ≤ a i a j -a ξ * i a ξ * j ≤ ξ 2 + ξ(a ξ i + a ξ j ), (4.22)
which implies in particular

|a i a j -a ξ * i a ξ * j | ≤ ξ 2 + ξ(a ξ i + a ξ j ), (4.23) if ξ ≥ 1. (4.24)
Assuming (4.24) thus, we obtain the first estimate

ϕ(t; ξ) ≤ 2 t 0 T d ξ 2 + ξ i a ξ i ln(a ξ * 1 a ξ * 3 ) + ln(a ξ * 2 a ξ * 4 ) dxds + ν t 0 T d i ξ 2 + ξ j a ξ j a ξ * i 1 ai>ξ dxds. (4.25)
Next, we will use the following inequalities (similar to (2.17)):

a ξ j ln(a ξ * i ) ≤ a ξ i ln(a ξ * i ) + a ξ j ln(a ξ * j ), a ξ j 1 ai>ξ ≤ a ξ i 1 ai>ξ + a ξ j 1 aj >ξ . (4.26)
In the second term in (4.25) we also use the elementary bound (a ξ * i ) -1 ≤ 1 to get

ϕ(t; ξ) t 0 T d i (ξ 2 + ξa ξ i )(ln(a ξ * i ) + 1 ai>ξ )dxds. (4.27)
Let then ζ ∈ (0, ξ). We have a ξ ≤ a ζ and

1 a>ξ ≤ a ζ ξ -ζ α+1 1 a ζ ≤1 + a ζ ξ -ζ α 1 1<a ζ , (4.28)
for some arbitrary exponent α ≥ 0. Taking successively α = 2, 1, and using the fact that ln(a ξ * ) = 0 if a ≤ ξ, we deduce from (4.17)-(4.28) that

ln(a ξ * ) ≤ W 2 (ξ, ζ)|Ψ(a ζ )| 2 , a ξ ln(a ξ * ) ≤ W 1 (ξ, ζ)|Ψ(a ζ )| 2 (4.29)
where

W α (ξ, ζ) = 1 (ξ -ζ) α+1 + 1 (ξ -ζ) α .
(4.30)

We also note

a > ξ ⇒ a ζ * ≥ 1 + (ξ -ζ) ⇒ ln(a ζ * ) ln(1 + (ξ -ζ))
≥ 1, (4.31) so, similarly to (4.29), we have

1 a>ξ ≤ W 2 (ξ, ζ) ln(1 + (ξ -ζ)) |Ψ(a ζ )| 2 , a ξ 1 a>ξ ≤ W 1 (ξ, ζ) ln(1 + (ξ -ζ)) |Ψ(a ζ )| 2 . (4.32)
Finally, we deduce from (4.27), (4.29), (4.32) and (4.16) that

ϕ(t; ξ) 1 1 ∧ ln(1 + (ξ -ζ)) ξ 2 W 2 (ξ, ζ) + ξW 1 (ξ, ζ) U ψ (t; ξ) 2 , (4.33)
and obtain therefore a bound from above on the last term in the right-hand side of (4.18). The other two terms in the right-hand side of (4.18) can be gathered to form a quantity very similar to the the function Θ(a) in (2.26). The procedure followed to prove (2.27) can be adapted to establish We can estimate the right-hand side of (4.41) in two different ways. First, using (4.32), we get, for ζ < ξ,

t 0 T d -(a ξ * 1 a ξ * 3 -a ξ * 2 a ξ * 4 ) ln(a ξ * 1 a ξ * 3 ) -ln(a ξ * 2 a ξ * 4 ) dxds + ν t 0 T d i a ξ * 1 a ξ * 3 + a ξ * 2 a ξ * 4 a ξ * i 1 ai>ξ dxds t 0 T d i,j a ξ *
T d i,j a i ln(a ξ * j )dx ξW 2 (ξ, ζ) + W 1 (ξ, ζ) ln(1 + (ξ -ζ)) i T d |Ψ(a ζ i )| 2 dx. (4.42)
We also have

a ξ i ln(a ξ * i ) ≤ 2Φ(a ξ i ) ≤ 2Φ(a ζ i ) and ξ ln(a ξ * i ) ≤ ξ a ζ ξ -ζ ln(a ζ * i ) ≤ 2 ξ ξ -ζ Φ(a ζ i ), (4.43) so sup 0≤s≤t T d i,j a i ln(a ξ * j )dx ξ ξ -ζ sup 0≤s≤t E(s; ζ) ξ ξ -ζ U(t; ζ). (4.44)
Using both (4.42) and (4.44) in (4.40), we obtain

M(ξ), M(ξ) t ξ ξ -ζ ξW 2 (ξ, ζ) + W 1 (ξ, ζ) ln(1 + (ξ -ζ)) U(t; ζ)U Ψ (t; ζ) 2 , (4.45)
and from (4.16), we deduce that

M(ξ), M(ξ) t ξ ξ -ζ ξW 2 (ξ, ζ) + W 1 (ξ, ζ) ln(1 + (ξ -ζ)) U(t; ζ) 3 . (4.46) 1≤i≤4 a i0 L ∞ (T d ) ≤ ξ. (4.47)
Let also ρ be a given exponent satisfying

1 < ρ < 1 + ρ 2 = 3 2 . (4.48)
Let δ ∈ (0, 1) be a given parameter (which will eventually depend on ξ, see (4.92)). We set

ξ k = (1 -2 -k-1
)ξ, k ∈ N and we examine the occurrence of the bound

U k ≤ δ ρ k , U k := U(T ; ξ k ). (4.49)
Assume first that (4.49) is satisfied at rank k. Then, using (4.16) and the inequality (4.37) with ζ = ξ k and ξ = ξ k+1 , we observe that

U k+1 ≤ C 1 1 1 ∧ ln(1 + 2 -k-2 ξ) W k δ ρρ k + C 1 M(t; ξ k+1 ) * , ( 4.50)
where

W k = ξ 2 k+1 W 2 (ξ k+1 , ξ k ) + ξ k+1 W 1 (ξ k+1 , ξ k ) 8 k ξ + 4 k , ( 4
.51) so (4.49) will be satisfied at rank k + 1 if

C 1 W k 1 ∧ ln(1 + 2 -k-2 ξ) δ (ρ-ρ)ρ k ≤ 1 2 , ( 4
H = k≥0 H k = k≥0 H k , H k := k j=0 H j . (4.54)
We will estimate k≥0 p k -p k+1 , where p k = P(H k ), and then, in the next step, evaluate p 0 . Assume that (4.52) is satisfied for all k ≥ 0 (we will see that an appropriate choice of parameter δ ensures this). Then

H k ∩ E k+1 ⊂ H k+1 , so p k -p k+1 ≤ P(H k ) -P(H k ∩ E k+1 ) = P(H k ∩ E c k+1 ) (4.55)
We use the exponential martingale inequality3

P (M * ∞ ≥ a + b M, M ∞ ) ≤ e -2ab (4.56)
with M t = M(t ∧ T ; ξ k+1 ) and some deterministic numbers a = V k , b = V -1 k that will be fixed later (see (4.61)), to get

P(H k ∩ E c k+1 ) ≤ e -2V k V -1 k + P(B k ), (4.57)
where B k is the event

B k = H k ∩ 1 2 δ ρ k+1 ≤ V k + V -1 k M(ξ k+1 ), M(ξ k+1 ) T . (4.58)
We use the estimate (4.46) on the quadratic variation of M(ξ) to obtain

M(ξ k+1 ), M(ξ k+1 ) T ≤ C 2 2 k W k ξ ln(1 + 2 -k-2 ξ) U 3 k , (4.59)
and thus

M(ξ k+1 ), M(ξ k+1 ) T ≤ C 2 2 k W k ξ ln(1 + 2 -k-2 ξ) δ 3ρ k , (4.60)
if H k is realized. We choose V k and Vk as follows:

V k = 1 8 δ ρ k+1 , V -1 k = V k C 2 2 k W k ξ ln(1 + 2 -k-2 ξ) δ 3ρ k -1 . (4.61)
With this choice of the parameters, B k has probability 0 and (4.57) yields

p k -p k+1 ≤ exp - ξ ln(1 + 2 -k-2 ξ) C 3 2 k W k δ -ρ k (3-2ρ) . (4.62)
Let K denote the index such that 2 -K ξ ≈ 1 (we are interested in large values of ξ, so large values of K as well). For k ≤ K, we have W k 4 k by (4.51) and 1 ln

(1 + 2 -k-2 ξ), so (4.52) is realized if C 4 4 k δ (ρ-ρ)ρ k ≤ 1 2 , ( 4.63)
while (4.62) implies

p k -p k+1 ≤ exp - 1 C 5 4 -k δ -ρ k (3-2ρ) . (4.64) The map t → 4 t δ (ρ-ρ)ρ t is non-increasing on R + if ln(4) ≤ ln(ρ)(ρ -ρ)| ln(δ)| so (4.63) is satisfied if ln(4) ≤ ln(ρ)(ρ -ρ)| ln(δ)|, C 4 δ ρ-ρ ≤ 1 2 , ( 4
p k -p k+1 ≤ C 7 Φ (0) ≤ C 7 exp -C -1 5 δ -(3-2ρ) . (4.71) If k ≥ K, then W k 8 k ξ -1 by (4.51) and 2 -k ξ ln(1 + 2 -k-2 ξ), so (4.52) is satisfied if C 4 16 k ξ 2 δ (ρ-ρ)ρ k ≤ 1 2 , (4.72) whereas (4.62) implies p k -p k+1 ≤ exp - ξ 3 2 -5k C 3 δ -ρ k (3-2ρ) . (4.73)
Since ξ ≥ 1 (see (4.47)), the first criterion (4.72) can be reduced to

C 4 16 k δ (ρ-ρ)ρ k ≤ 1 2 , ( 4
p k -p k+1 ≤ C 8 Φ (K), Φ (t) := exp - ξ 3 2 -5t C 3 δ -ρ t (3-2ρ) . (4.76)
The rough estimate Φ (K) ≤ Φ (0) ≤ exp -C -1 3 ξ 3 , which yields k≥K p k -p k+1 ≤ C 8 exp -C -1 3 ξ 3 , (4.77) will be sufficient for our purpose. Indeed, using (4.71) and (4.77), we obtain

P(H) ≥ p 0 -C 7 exp -C -1 5 δ -(3-2ρ) -C 8 exp -C -1 3 ξ 3 . (4.78)
We will take δ ≈ (ln(ξ)) -1 2 in (4.92) below, so the two last terms in (4.78) have a very fast decay in ξ. We will see below how to estimate p 0 (and actually will get a decay in ξ which is much slower, see (4.89)). Initial estimate. We wish to estimate the probability p 0 of the event The estimate of E U ψ (T ; ξ/4) 2 is done as follows: we observe that Ψ is non-increasing, so that and suppose that F i : R q → R and g i,α : R q → R are functions of class C m , bounded, with all their derivatives up to order m bounded. Then (A.1) admits a unique weak solution which belongs to the space L q (Ω; C([0, T ]; W m,p (T d ))) ∩ L mq (Ω; C([0, T ]; W 1,mp (T d ))). (A.4)

|Ψ(a ξ i )| 2 ≤ |Ψ(a i )| 2 1 ai>ξ ≤ |Ψ(a i )| 2 ln(1 + a i ) ln(1 + ξ) |Φ(a i ) 2 | ln(1 + ξ) . ( 4
P [U 0 ≥ δ] 1 δ ln(1 + ξ) E 2 (0) + 1 δξ 1/2 E(0) 1/2 E 2 (0) 1/2 ≤ C(B 0 ) 1 δ ln(1 + ξ) , ( 4
Proof of Theorem A.1. The statement of Theorem A.1 is essentially the statement of Theorem 2.1 in [START_REF] Hofmanová | Strong solutions of semilinear stochastic partial differential equations[END_REF]. The extension to systems does not raise particular difficulties. The essential point is a generalization of the estimate

G(h) W m.p ≤ C (1 + h m W 1,mp + h W m,p ) (A.5)
in [Hof13, Proposition 3.1], given for h real-valued and G defined on R, to the case where h takes values in R q and G is defined on R q . This is obtained simply by generalizing the chain-rule formula

D β G(h(x)) = |β| l=1 α1+•••+α l =β αi =0 C β,l,α1,••• ,α l G (l) (h(x))D α1 h(x) • • • D α l h(x) (A.6) to D β G(h(x)) = |β| l=1 α1+•••+α l =β αi =0 C β,l,α1,••• ,α l × |γ|=l D γ G(h(x)) δ1+•••+δ l =γ |δi|=1 D α1 h δ1 (x) • • • D α l h δ l (x), (A.7)
where, given δ ∈ N q of length |δ| = 1, h δ denote the component h i , where i is the only index such that δ i = 0.

  .10) with terminal condition w(x, T ) = 0, x ∈ T d . (3.11) Let P denote the predictable σ-algebra of Ω × [0, T ]. In (3.10), we assume that K, H : Ω × [0, T ] × T d → R (3.12)

{U 0 =

 0 U(T ; ξ/2) ≤ δ} . (4.79) By the Markov inequality and the inequality (4.37) used with ζ = ξ/4, we haveP [U 0 ≥ δ] ≤ δ -1 E [U(T ; ξ/2)] δ -1 E U ψ (T ; ξ/4) 2 + δ -1 E [M(T ; ξ/2) * ] .(4.80)In the right-hand side of (4.80), the term related to the martingale part can be estimated relatively easily. We apply (4.45) with ζ = 0 and use (4.47) which implies 1 ln(1 + ξ/2) to obtain M(ξ/2), M(ξ/2) T 1 ξ U(T )U Ψ (T ; 0) 2 . (4.81)By the Burkholder-Davis-Gundy inequality,E [M(T ; ξ/2) * ] 1 ξ 1/2 E U(T ) 1/2 U Ψ (T ; 0) ,and thus, by the Cauchy-Schwarz inequality and the entropy estimate (2.2) with p = 1,E [M(T ; ξ/2) * ] 1 ξ 1/2 {E [E(0)]} 1/2 E U Ψ (T ; 0) 2 1/2 . (4.82) To bound from above the last factor in (4.82), we use the estimate |Ψ(a)| 2 |Φ(a)| 2 , (4.83) which follows from (4.86) below. We exploit then the L 2 -estimate (3.4) to get E [M(T ; ξ/2) * ] 1 ξ 1/2 {E [E(0)]} 1/2 {E 2 (0)} 1/2 . (4.84)

  .85) In the last inequality of (4.85), we use the bound |Ψ(a)| 2 ln(1 + a) |Φ(a)| 2 , (4.86) which is deduced from the obvious estimate Ψ(a) ≤ a 0 ln(1 + s)ds ≤ a ln(1 + a).Then (4.85) impliesE U ψ (T ; ξ/4) 2 E 2 (0) ln(1 + ξ) . (4.87) Set B 0 = sup 1≤i≤4 a i0 L ∞ (T d ). By (4.80), (4.84) and (4.87), we obtain

( 0 F

 0 Weak solution to (A.1)). A process a ∈ L 2 (Ω × [0, T ]; P; H 1 (T d )) (where P is the predictable σ-algebra) is said to be a weak solution to (A.1) if1. F i (a), g i,α (a) ∈ L 2 (Ω × Q T ), 2. a i ∈ L 2 (Ω; C([0, T ]; L 2 (T d ))),3. for all t ∈ [0, T ], for all ϕ ∈ H 1 (T d ), P-a.s.,a i (t), ϕ = a i0 , ϕ κ i -t 0 ∇a i (s), ϕ ds + t i (a), ϕ ds + d W α=1 g i,α (a), ϕ dW i α (t). (A.2)We can then state the following result. Theorem A.1 (Semilinear stochastic parabolic systems). Let p ∈ [2, ∞), q ∈ (2, ∞) and let m be an integer ≥ 2 such that mp > d + 2. Assume u 0 ∈ W m,p (T d ) ∩ W 1,mp (T d ), (A.3)

with the convention τn = τ if ess sup x∈T d a(x, t)

(R 4 ) ≤ n for all t ∈ [0, τ )

if a(x, t) = 0, we set K(x, t) = 1

give ref.

Conclusion. Let

(4.90)

We have B T ≤ ξ if H is realized, so (4.78) and (4.89) imply the tail estimate

which satisfies the condition (4.66), to obtain (under the condition (4.47))

). The desired result (4.2) then follows from (4.93) and the expression

(4.94)

Existence of global-in-time regular solutions

The proof of Theorem 1.1 is now straightforward. We consider the regular solution a constructed in Section 4.2, based on the sequence of solution (a n ) of the problem with truncated nonlinearities. By (4.2) and the Markov inequality, we have

which yields P(τ ≤ T ) = 0 at the limit n → +∞. This being true for every T , we have τ = +∞ a.s.

A Regular solutions to semilinear stochastic parabolic systems

We consider the following system of SPDEs: for 1 ≤ i ≤ q and 1 ≤ α ≤ d W , let (W i α (t)) be a family of one-dimensional Wiener processes such that, for each i, W i 1 , . . . , W i d W are jointly independent. For 1 ≤ i ≤ q, let F i : R q → R and g i,α : R q → R be some given functions and let κ i be some (strictly) positive coefficients. We consider the system

for i = 1, . . . , q, where a = (a i ) 1,q , a 0 = (a i0 ) is a given function T d → R q and T > 0. Weak solutions are defined as follows.