Abstract
Abstract In this article, we study the periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions. By using the theory of Leray-Schauder degree, we obtain the existence of a nontrivial nonnegative time periodic solution.
Highlights
The aim of this work is to consider the following periodic problem for a quasilinear parabolic equation:∂u ∂t – Di aij(x, t, u)Dju = m – [u] u, (x, t) ∈ QT, ( . ) ∂u =, (x, t) ∈ ∂ × (, T),∂n u(x, ) = u(x, T), x ∈, where is a bounded domain in Rn with smooth boundary ∂∂ ∂n denotes the outward normal derivative on ∂, QT = ×(, T), aij satisfies some suitable smoothness and structure conditions
The term Di(aij(x, t, u)Dju) models a tendency to avoid high density in the habitat, m – [u] describes the ways in which a given population grows and shrinks over time, as controlled by birth, death, emigration or immigration, and the Neumann boundary condition models the trend of the biology population who survive on the boundary
Linear parabolic equations with nonlocal terms have been investigated by numerous researchers [ – ]
Summary
1 Introduction The aim of this work is to consider the following periodic problem for a quasilinear parabolic equation: Results on the quasilinear periodic parabolic equations with nonlocal terms and Neumann boundary conditions are few. By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions.
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