Upper bound estimate for the blow-up time of a class of integrodifferential equation of parabolic type involving variable source

Abita Rahmoune

doi:10.5802/crmath.8

Abstract

Considérons une classe d’équations intégro-différentielles paraboliques comprenant une source variable et avec condition de Dirichlet au bord

Highlights

Let Ω ⊂ Rn (n ≥ 1), be a bounded domain with a smooth boundary Γ = ∂Ω
Many authors, in the different cases of the values of the memory kernel, when g = 0 or g > 0, semilinear parabolic problems with a memory term associated with the Laplace operator and source term with Dirichlet type condition has been considered:
If n > p2 if n ≤ p2 we consider the following semilinear generalized parabolic boundary value problem governed by partial differential equations that describe the evolution of viscoelastic materials with nonlinearities of variable exponent type under Dirichlet type condition:

Summary

Introduction

The boundary Γ of Ω is assumed to be regular, g : R+ → R+ is a bounded C 1 R+ function and T , p are positive constants. If n > p2 if n ≤ p2 we consider the following semilinear generalized parabolic boundary value problem governed by partial differential equations that describe the evolution of viscoelastic materials with nonlinearities of variable exponent type under Dirichlet type condition: u t. Where g : R+ → R+ is a bounded C 1 R+ function, Ω ⊂ Rn (n ≥ 1) is a bounded domain with smooth boundary, T ∈ (0, +∞], and the initial value u0 ∈ H01 (Ω). We introduce some preliminaries and notations, which will be used throughout this paper

Preliminaries

Blow-up and upper bound of blow-up time

Upper bounds for blow-up time