Properties of solutions for a reaction\u2013diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition

Jian Wang,Hui Yang

doi:10.1186/s13661-018-1069-9

Abstract

In this paper, the authors consider the reaction–diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition. They prove that the solution either exists globally or blows up in finite time depending on the initial data, the weighting function on the border, and nonlinear indexes in the equation by using the comparison principle.

Highlights

In this paper, we consider the initial boundary value problem for the following nonlocal reaction–diffusion equation with nonlinear absorption: ut = u + aup uq(y, t) dy – bum, x ∈, 0 < t < T, (1.1) ∂u =k(x, y)ul(y, t) dy, x ∈ ∂, 0 < t < T, (1.2)∂ν u(x, 0) = u0(x), x ∈, (1.3)where p, q, m, l > 0, is a bounded domain in Rn (n ≥ 1) with smooth boundary ∂, ν is unit outward normal on ∂
We find that if p + q = m > 1, problem (1.1)–(1.3) has blow-up solutions in finite time as well as global solutions
4 Conclusion In this paper, we considered the properties of solutions for the reaction–diffusion equation with nonlinear absorption and with nonlinear nonlocal Neumann boundary condition and proved that the solution either exists globally or blows up in finite time depending on the initial data, the weighting function on the border, and nonlinear indexes in the equation by using the comparison principle

Summary

Introduction

1 Introduction In this paper, we consider the initial boundary value problem for the following nonlocal reaction–diffusion equation with nonlinear absorption: ut = u + aup uq(y, t) dy – bum, x ∈ , 0 < t < T, (1.1) This paper will extend the above work to the reaction–diffusion equation (1.1) with nonlinear nonlocal Neumann boundary condition and obtain the corresponding results. 3, by using the comparison principle and supersubsolution method, we establish the conditions for blow-up in finite time and global existence.

Results

Conclusion