Influence of weight functions to a nonlocal p-Laplacian evolution equation with inner absorption and nonlocal boundary condition

Zhong Bo Fang,Jianyun Zhang

doi:10.1186/1029-242x-2013-301

Zhong Bo Fang, Jianyun Zhang

Open Access

https://doi.org/10.1186/1029-242x-2013-301

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Abstract

In this paper, we investigate an initial boundary value problem for a nonlocal p-Laplacian evolution equation with an inner absorption and weighted linear nonlocal boundary and initial conditions. By using the modified comparison principle and the method of upper-lower solution, we find the influence of weighted function on determining blow-up or not of nonnegative solutions.

Highlights

We consider a nonlocal p-Laplacian evolution equation with an inner absorption term ut – div |∇u|p– ∇u = um un(y, t) dy – αur, (x, t) ∈ × (, +∞), ( . )subject to the weighted linear nonlocal boundary and initial conditions u(x, t) = φ(x, y)u(y, t) dy, (x, t) ∈ ∂ × (, +∞), u(x, ) = u (x) ≥, x ∈, where p >, m ≥, n >, r ≥, α >, and ⊂ RN (N ≥ ) is a bounded domain with smooth boundary
In the diffusion system of some biological species with human-controlled distribution, u(x, t), div(|∇u|p– ∇u), um(y, t) dy, and –α represent the density of the species at location x and time t, the mutation, the humancontrolled distribution, and the decrement rate of biological species, respectively
The evolution of the species at a point of space depends on the density of species in a partial region and in the total region because of the nonlocal boundary condition that arises from the spatial inhomogeneity

Summary

Introduction

There have been many researchers dealing with blow-up solutions to the initial boundary value problems of equations with or without nonlocal boundary conditions; see [ – ] and references therein. For the studies of the initial boundary value problem for a local parabolic equation with weighted nonlocal boundary condition

Results

Conclusion