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Global Existence for Reaction-diffusion Equations with State-Dependent Delay and Fast-growing Nonlinearities

Ruijing Wang

Abstract

This work aims to study the initial-boundary value problem of the reaction-diffusion equation with state-dependent delay $\pa_{t}u-Δu=f(u)+g(u,u(t-τ(t,u_t)))+h(t,x)$ in a bounded domain. We establish the global existence of the problem under suitable dissipative-type structural conditions, allowing both nonlinear terms $f$ and $g$ to have arbitrary polynomial growth rates. Another highlight in this work is that, we significantly relax the continuity assumptions imposed on the delay functions.

Global Existence for Reaction-diffusion Equations with State-Dependent Delay and Fast-growing Nonlinearities

Abstract

This work aims to study the initial-boundary value problem of the reaction-diffusion equation with state-dependent delay in a bounded domain. We establish the global existence of the problem under suitable dissipative-type structural conditions, allowing both nonlinear terms and to have arbitrary polynomial growth rates. Another highlight in this work is that, we significantly relax the continuity assumptions imposed on the delay functions.
Paper Structure (8 sections, 5 theorems, 44 equations)

This paper contains 8 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional spaces
  4. Definition of a weak solution
  5. Global Existence and Estimates of Weak Solutions
  6. Global $L^{q}$ and $H^1$-estimates of Galerkin approximations
  7. Global existence
  8. $L^{\infty}$-smoothing property and global $L^{\infty}$-estimate

Key Result

Lemma 3.1

Assume (H0)-(H2). Let $q_{c}< q< \infty$. Then there exist $\lambda_q,R_q,\lambda_1,\rho_1>0$ and $M_i$ ($i=0,1$) ($M_i's$ are independent of $q$) such that for all $\phi\in \mathscr{X}_1^q$, and

Theorems & Definitions (11)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 1 more