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Global Existence, Regularity, and Dissipativity of Reaction-diffusion Equations with State-dependent Delay and Supercritical Nonlinearities

Ruijing Wang, Desheng Li

Abstract

This work aims to study the initial-boundary value problem of the reaction-diffusion equation $\pa_{t}u-Δu=f(u)+g(u(t-τ(t,u_t)))+h(t,x)$ in a bounded domain with state-dependent delay and supercritical nonlinearities. We establish the global existence and discuss the regularity and dissipativity of the problem under weaker assumptions. In particular, the existence of a global pullback attractor is proved regardless of uniqueness.

Global Existence, Regularity, and Dissipativity of Reaction-diffusion Equations with State-dependent Delay and Supercritical Nonlinearities

Abstract

This work aims to study the initial-boundary value problem of the reaction-diffusion equation in a bounded domain with state-dependent delay and supercritical nonlinearities. We establish the global existence and discuss the regularity and dissipativity of the problem under weaker assumptions. In particular, the existence of a global pullback attractor is proved regardless of uniqueness.
Paper Structure (11 sections, 15 theorems, 105 equations)

This paper contains 11 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional spaces
  4. Definitions of weak solutions and mild solutions
  5. Global $L^{q}$ and $H^1$-estimates of $u$
  6. Global $L^{q}$ and $H^1$-estimates of $u_k$
  7. The proof of Theorems \ref{['t:x.1.1']} and \ref{['t:1.2x']}
  8. Fundamental properties in fractional power spaces
  9. Global pullback ${\mathscr D}$-attractor of system \ref{['e:1.1']}
  10. Pullback ${\mathscr D}$-attractors of set-valued processes
  11. Global pullback ${\mathscr D}$-attractor of \ref{['e:1.1']}

Key Result

Theorem 1.1

Assume $\tau$ satisfies (H0), $f$ and $g$ satisfy (H1)-(H3). Let $q_0<q<\infty$, and $h\in L^{\infty}(\mathbb{R};L^{q_\alpha/\alpha}(\Omega))\cap C(\mathbb{R};L^{q_\alpha/\alpha}(\Omega))$. If $\phi\in \mathscr{X}_1^q$, then the solutions $u=u(t;\phi)$ of e:1.1 satisfy that Furthermore, there exist $B_i$, $\eta_i$ and $\rho_i>0$ ($i=0,1$) such that

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • ...and 17 more