Stability of a class of nonlinear reaction–diffusion equations and stochastic homogenization

Abstract

of a class of nonlinear reaction-diffusion equations and stochastic homogenization.Abstract. We establish a convergence theorem for a class of nonlinear reaction-diffusion equations when the diffusion term is the subdifferential of a convex functional in a class of functionals of the calculus of variations equipped with the Mosco-convergence. The reaction term, which is not globally Lipschitz with respect to the state variable, gives rise to bounded solutions, and cover a wide variety of models. As a consequence we prove a homogenization theorem for this class under a stochastic homogenization framework.that the derivative du dt of the solution belongs to L 2 (0, T * , X). Nevertheless √ t du dt ∈ L 2 (0, T * , X) (see [8, Theorem 17.2.5] or [13, Theorem 3.6]). Hence, for 0 < δ < T * , du dt belongs to L 2 (δ, T * , X). Set E := {T > δ : ∃u ∈ C ([0, T ], X) solution of (P)} .Since T * ∈ E, we have E = ∅. We define the maximal time in R + by T Max := sup E and denote by u the maximal solution of (P) in C ([0, T Max ), X). We have the following alternative:Theorem 2.2 (Global existence or blow-up in finite time). Assume that F satisfies (C 1 ), (C 2 ), then we have the blow-up alternative (G1) T Max = +∞ (existence of a global solution); (G2) T Max < +∞. In this case lim T →+T Max u C([0,T ],X) = +∞ (blow-up in finite time).Moreover, for all T , 0 < T < T Max , the restriction of u to [0, T ] satisfies assertions (L 1 ) and (L 2 ), and furthermore satisfies (Proof. We assume that T Max < +∞ and show that lim T →+T Max u C([0,T ],X) = +∞. We argue by contradiction. Assume that u does not fulfill lim T →+T Max u C([0,T ],X) = +∞, then there exist G > 0 and a sequence (T n ) n∈N in E such that T n → T Max and u C([0,Tn],X) ≤ G.Step 1. We show that lim t→T Max u (t) exists in X.Let n ∈ N. For a.e. t ∈ (0, T n ) we have

Convergence of nonlinear integrodifferential reaction-diffusion equations via Mosco×Γ-convergence

“…. This convergence is a straightforward consequence of the weak convergences 𝑟 𝑛 ⇀ 𝑟, 𝑞 𝑛 ⇀ 𝑞 and the pointwise convergence 𝑔 𝑛 → 𝑔 together with the uniform bound of 𝑔 𝑛 (for a complete proof refer to [2]).…”

“…Note that when (𝑢, 𝑣) ∈ dom(𝜕Ψ) 2 , then (2.1) yields 𝜕Ψ(𝑢) − 𝜕Ψ(𝑣), 𝑢 − 𝑣 ≥ 𝛼 Ψ 𝑢 − 𝑣 2 𝑉 , hence, since 𝜕Ψ(0) = {0}, for all 𝑢 ∈ dom(𝜕Ψ), we infer that 𝜕Ψ(𝑢), 𝑢 ≥ 𝛼 Ψ 𝑢 2 𝑉 . We assume that the subdifferentials 𝜕Φ and 𝜕Ψ are connected via the following coercivity condition on 𝜕Φ • 𝜕Ψ −1 : there exist two constants 𝛼 Φ,Ψ > 0 and 𝛽 Φ,Ψ ≥ 0 such that for all 𝑢 * ∈ 𝑅 𝜕Φ (𝜕Ψ), 𝜕Φ((𝜕Ψ) −1 (𝑢 * )), 𝑢 * ≥ 𝛼 Φ,Ψ 𝑢 * 2 𝑋 − 𝛽 Φ,Ψ .…”

“…𝑢 ∈ (𝜕Ψ) −1 (𝑢 * )) and also 𝑢 ∈ dom(𝜕Φ), and for any 𝜉 * ∈ 𝜕Φ(𝑢) we have 𝜉 * , 𝑢 * ≥ 𝛼 Φ,Ψ 𝑢 * 2 𝑋 − 𝛽 Φ,Ψ . In short, 2.2 is equivalent to 𝜕Φ(𝑢), 𝜕Ψ(𝑢) ≥ 𝛼 Φ,Ψ 𝜕Ψ(𝑢) 2 𝑋 − 𝛽 Φ,Ψ for all 𝑢 ∈ dom(𝜕Φ), with the following notation convention: let 𝐴 and 𝐵 be two sets of Hilbert space (𝑋, • 𝑋 ), then…”

“…In the spirit of [2,4], we investigate the compactness or the stability in terms of convergence, of integrodifferential reaction-diffusion problems defined in 𝐿 2 (0, 𝑇, 𝑋) by under suitable variational convergences on the classes of functionals Φ and Ψ (see Theorem 5.1). As usually the integral in the first member is taken in the sense of Bochner.…”

Stochastic homogenization of nonlocal reaction–diffusion problems of gradient flow type

“…Remark 2.12. Under additional assumptions on the structure of F , we automatically have F p¨, uq P AC pr0, T s; L 2 pOqq (see [AHMM19] or [AHMM22, §2.2.2, pp. 27] and the example treated in Section 5).…”