DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

MAXIMUM PRINCIPLES AND DECAY ESTIMATES
FOR PARABOLIC SYSTEMS UNDER ROBIN
BOUNDARY CONDITIONS
M. MARRAS AND S. VERNIER PIRO
We investigate nonlinear parabolic systems when Robin conditions are prescribed on the
boundary. Sufficient conditions on data are imposed to obtain decay estimates for the
solution. In addition, a maximum principle is proved for an auxiliary function, from
which we deduce an exponential decay estimate for the gradient.
Copyright © 2006 M. Marras and S. Vernier Piro. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly
cited.
1. Introduction
Reaction diffusion parabolic systems are studied with interest as they provide models
for various chemical and biological problems. Recently some qualitative properties of
their solutions like blow-up and time decay estimates have been studied in [7, 9](forthe
applications, see also the references therein). The aim of this paper is to investigate these
properties for the following system with Robin boundary conditions:
Δu + f 1 (v) = ∂u
∂t
Δv + f 2 (u) = ∂v
∂t
∂u
∂n + αu = 0
∂v
∂n + αv = 0
u(x,0)= h 1 (x)
in Ω × (t>0),
in Ω × (t>0),
on∂Ω × (t>0),
on∂Ω × (t>0),
inΩ,
(1.1)
v(x,0)= h 2 (x)
inΩ,
where Ω is a bounded domain in R 2 , α>0, and for i = 1,2, f i (s) areC 1 functions which
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 767–773