Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions
Hameed et al. Boundary Value Problems 2013, 2013:34
http://www.boundaryvalueproblems.com/content/2013/1/34
RESEARCH
Open Access
Periodic solution of a quasilinear parabolic
equation with nonlocal terms and Neumann
boundary conditions
Raad Awad Hameed1,2 , Boying Wu1 and Jiebao Sun1*
*
Correspondence:
1
Department of Mathematics,
Harbin Institute of Technology,
Harbin, 150001, P.R. China
Full list of author information is
available at the end of the article
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.
1 Introduction
The aim of this work is to consider the following periodic problem for a quasilinear
parabolic equation:
∂u
– Di aij (x, t, u)Dj u = m – [u] u,
∂t
∂u
= , (x, t) ∈ ∂ × (, T),
∂n
u(x, ) = u(x, T), x ∈ ,
(x, t) ∈ QT ,
(.)
(.)
(.)
∂
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. This model can be used to describe the models for some interesting phenomena in mathematical biology, fisheries and wildlife management. The function u(x, t)
gives the number of individuals (per unit area) of the species at position x and time t, where
x represents the spatial variable and t represents the time. The term Di (aij (x, t, u)Dj u) 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.
In last decades, linear parabolic equations with nonlocal terms have been investigated
by numerous researchers [–]. A typical model was submitted by Allegretto and Nistri []
and they proposed the following equation:
∂u
– u = f x, t, m, [u], u ,
∂t
with the Dirichlet boundary conditions. Also, according to the actual needs, many authors
© 2013 Hameed et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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in any medium, provided the original work is properly cited.
Hameed et al. Boundary Value Problems 2013, 2013:34
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Page 2 of 11
divert attention to nonlinear diffusion equations with nonlocal terms such as the porous
equation [, ] with a typical form
∂u
= um + m – [u] u,
∂t
(.)
and the p-Laplacian equation [] with a typical form
∂u
= div |∇u|p– ∇u + m – [u] u.
∂t
(.)
Equation (.) is degenerate if m > and singular if < m < . Equation (.) is degenerate if p > and singular if < p < . Only the cases m > and p > are considered with
a few exceptions. All these equations are considered with the Dirichlet boundary condition which describes that the boundary is lethal to the species. Moreover, the methods
in these papers are all based on the theory of Leray-Schauder degree. However, results on
the quasilinear periodic parabolic equations with nonlocal terms and Neumann boundary
conditions are few. In a recent paper [], Wang and Yin considered the following periodic
Neumann boundary value problem:
∂u
– um = m – [u] u, (x, t) ∈ QT ,
∂t
∂u
= , (x, t) ∈ ∂ × (, T),
∂n
u(x, ) = u(x, T), x ∈ ,
where m > . By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions.
Inspired by the work of [], we consider the periodic solutions of the Neumann boundary value problem of a quasilinear parabolic equation with nonlocal terms. Compared with
the Dirichlet boundary condition, the Neumann boundary condition causes an additional
difficulty in establishing some a priori estimates. On the other hand, different from the
cases of the Dirichlet boundary condition, an auxiliary problem for (.)-(.) is considered for using the theory of Leray-Schauder degree. We prove that this problem (.)-(.)
admits a nontrivial nonnegative periodic solution, that is, the following theorem.
Theorem If assumptions (A), (A), (A) hold, then problem (.)-(.) admits a nontrivial nonnegative periodic solution u ∈ L (, T; H ()) ∩ CT (QT ).
The article is organized in the following way. In Section , we give some necessary preliminaries including the auxiliary problem. In Section , we establish some necessary a priori estimations of the solutions of the auxiliary problem. Then we show the proof of the
main result of this paper.
2 Preliminaries
In the paper, we assume that
(A) aij (·, ·, u) = aji (·, ·, u) ∈ CT (QT ) and there exist two constants < λ ≤ γ such that
λ|ξ | ≤ aij (x, t, u)ξi ξj ≤ γ |ξ |
for all (x, t) ∈ QT , u ∈ R+ and ξ ∈ Rn ,
Hameed et al. Boundary Value Problems 2013, 2013:34
http://www.boundaryvalueproblems.com/content/2013/1/34
Page 3 of 11
where CT (QT ) is the class of functions which are continuous in × R and
T-periodic with respect to t. Furthermore, aij (·, ·, u) is continuous with respect
to u.
(A) [·] : L+ () → R+ is a bounded continuous functional satisfying
[u] ≤ CuL () ,
where C is a positive constant independent of u, R+ = [, +∞),
L+ () = {u ∈ L ()|u ≥ , a.e. in }.
(A) m(x, t) ∈ CT (QT ) and satisfies that
essinf
x∈
T
T
m(x, t) dt > γ λ ,
where λ is the first eigenvalue of the Laplacian equation on ω with zero boundary
and φ (x) is the corresponding eigenfunction.
Since the regularity follows from a quite standard approach, we focus on the discussion
of weak solutions in the following sense.
Definition A function u is said to be a weak solution of problem (.)-(.), if u ∈
L (, T; H (T)) ∩ CT (QT ) and satisfies
∂ϕ
–u
+ aij (x, t, u)Di uDj ϕ – m – [u] uϕ dx dt = ,
∂t
QT
(.)
for any ϕ ∈ C (QT ) with ϕ(x, ) = ϕ(x, T).
In order to use the theory of Leray-Schauder degree, we introduce a map by considering
the following auxiliary problem:
∂uε
– Di ( – τ )γ Di uε + τ aij (x, t, u)Dj uε + εuε = f ,
∂t
∂uε
= , (x, t) ∈ ∂ × (, T),
∂n
uε (x, ) = uε (x, T), x ∈ ,
(x, t) ∈ QT ,
(.)
(.)
(.)
where ε is a sufficiently small positive constant, τ ∈ [, ] is a parameter and f ∈ CT (QT ).
Then we can define a map uε = G(τ , f ) with G : [, ]×CT (QT ) → CT (QT ). Applying classical estimates (see []), we can see that uε L∞ (QT ) is bounded by f L∞ (QT ) , and uε is Hölder
continuous in QT . Then, by the Arzela-Ascoli theorem, the map G is compact. So, the map
G is a compact continuous map. Let f (uε ) = (m – [uε ])u+ε , where u+ε = max{uε , }, we can
see that the nonnegative solution uε of problem (.)-(.) (...truncated)
