Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

Huimin Tian,Lingling Zhang

doi:10.1515/math-2020-0088

Abstract

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

Highlights

This paper is concerned with the following nonlocal reaction diffusion equations with time-dependent coefficients:(g(u))t = ∇⋅(ρ(|∇u|p)|∇u|p−2 ∇u) + k(t)f (x, u), (x, t) ∈ Ω × (0, t⁎), ∂u ∂n =(x, t) ∈ ∂Ω × (0, t⁎), (1)u(x, 0) = u0(x) ≥ 0, x ∈ Ω, where p ≥ 2 and Ω ⊂ N (N > 2)
For the initial boundary value problem under nonlinear boundary conditions, using the auxiliary functions and modified differential inequality technique, they established some conditions on time-dependent coefficients and nonlinear data to ensure that the solution u(x, t) exists globally and blows up at some finite time
Nonlinear reaction diffusion model plays an important role in the fields of physics, chemistry, biology, and engineering, as its global and blow-up solutions always reflect the stability and instability of heat and mass transport process

Summary

Introduction

This paper is concerned with the following nonlocal reaction diffusion equations with time-dependent coefficients:. Liu and Fang [17] focused on the blow-up phenomena to the following equations with time-dependent coefficients: ut = ∇⋅(h(|∇u|2 )∇u) − k(t)f (u), (x, t) ∈ Ω × (0, t⁎), where Ω ⊂ RN(N ≥ 2) is a bounded star-shaped region domain with smooth boundary ∂Ω. For the initial boundary value problem under nonlinear boundary conditions, using the auxiliary functions and modified differential inequality technique, they established some conditions on time-dependent coefficients and nonlinear data to ensure that the solution u(x, t) exists globally and blows up at some finite time. In [19], the authors dealt with the blow-up phenomena of the following quasilinear reaction diffusion equations with weighted nonlocal source under Robin boundary conditions:.

Blow-up solution and an upper bound for blow-up time

A lower bound for blow-up time

Applications

Conclusions