A semilinear wave equation with space-time dependent
coefficients and a memory boundarylike antiperiodic
condition: Regularity and stability
Út V. Lê
a͒
Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu FIN-90014,
Finland
͑Received 11 April 2010; accepted 31 August 2010; published online 8 October 2010
͒
This paper deals with regularity and stability of solutions to an initial-boundary
value problem of a semilinear wave equation. This equation admits space-time
dependent coefficients and a memory boundarylike antiperiodic condition. For
regularity or existence of a unique strong solution, the Faedo–Galerkin method and
the energy method, associated with the maximal solution of a nonlinear Volterra
integral equation, are applied. Then, as an application of the regularity result, sta-
bility of this unique strong solution is obtained by the energy method. © 2010
American Institute of Physics. ͓doi:10.1063/1.3494574͔
I. INTRODUCTION
Because of being motivated from different models of mechanics and physics, initial-boundary
value problems of wave equations with memory boundary conditions become a very interesting
topic, and hundreds of publications have been appeared. The first interesting result may be from
Tiehu.
27
He investigated the nonlinear initial-boundary value problem,
ץ
2
u
ץ
t
2
−
ץ
ץ
x
ͩ
ץ
u
ץ
x
ͪ
=0, 0Ͻ x Ͻ L, t Ͼ 0, ͑1.1͒
u͑0,t͒ =0, t Ն 0, ͑1.2͒
u͑L,t͒ +
͵
0
t
a͑t − s͒
ͩ
ץ
u
ץ
x
͑L,s͒
ͪ
ds = g͑t͒, t Ն 0, ͑1.3͒
u͑x,0͒ = u
0
͑x͒,
ץ
u
ץ
t
͑x,0͒ = u
1
͑x͒,0Ͻ x Ͻ 1, ͑1.4͒
where
, f, a, g, u
0
, and u
1
are given functions. Boundary condition ͑1.3͒ is the so-called memory
boundary condition regarding the presence of the convolution a ء
͑
ץ
u
ץ
x
͑L,·͒
͒
͑t͒. This problem is
said to be a mathematical model for a nonlinear one-dimensional motion of an elastic bar con-
nected with a viscoelastic element at one end of the bar. Under some physically reasonable
assumptions for the given data, Tiehu showed that the boundary value is dissipative and proved the
existence of a global smooth solution for small initial data, namely, ͑u
0
,u
1
͒ C
3
͓͑0,L͔͒
ϫC
2
͓͑0,L͔͒. Then in Ref. 28, he replaced ͑1.2͒ and ͑1.3͒ by
u͑L,t͒ =0, t Ն 0, ͑1.5͒
a͒
Electronic addresses: levanut@gmail.com and ut.van.le@oulu.fi.
JOURNAL OF MATHEMATICAL PHYSICS 51, 103504 ͑2010͒
51, 103504-10022-2488/2010/51͑10͒/103504/28/$30.00 © 2010 American Institute of Physics