DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ON QUENCHING FOR SEMILINEAR PARABOLIC EQUATIONS
WITH DYNAMIC BOUNDARY CONDITIONS
JOAKIM H. PETERSSON
We present a quenching result for semilinear parabolic equations with dynamic boundary
conditions in bounded domains with a smooth boundary.
Copyright © 2006 Joakim H. Petersson. 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
For semilinear partial differential equations, a particular type of blow-up phenomenon
may arise when the nonlinearity has a pole at a finite value of the solution. Namely, the
solution itself remains bounded while some derivative of it blows up. This is referred
to by saying that the solution “quenches.” The study of quenching phenomena for parabolic
differential equations with singular terms and classical boundary conditions (of
Dirichlet- or Neumann-type) was initiated by Kawarada [7] in connection with the study
of electric current transients in polarized ionic conductors, and has attracted much attention
since then (see the survey [8]). Recently, parabolic differential equations with dynamic
boundary conditions aroused the interest of several researchers (see [1, 3, 6]). The
question of the occurrence of quenching phenomena in the case of dynamic boundary
conditions comes up naturally. In this paper, we provide a simple example.
In Section 2, we present some recent results on parabolic equations with dynamic
boundary conditions in bounded domains with a smooth boundary and we prove some
needed facts about the behavior of the solutions. Section 3 is devoted to a simple criterion
(positivity of the nonlinearities) for the appearance of quenching for certain semilinear
equations of this type.
2. Preliminary considerations
We will consider the semilinear problem with dynamic boundary conditions
u t − Δu = f (x,u), t>0, x ∈ Ω,
u t + u ν = g(x,u), t>0, x ∈ ∂Ω,
u(0,x) = u 0 (x), x ∈ Ω,
(2.1)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 935–941