DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

WELL-POSEDNESS AND BLOW-UP OF SOLUTIONS TO WAVE
EQUATIONS WITH SUPERCRITICAL BOUNDARY SOURCES
AND BOUNDARY DAMPING
IRENA LASIECKA AND LORENA BOCIU
We present local and global existence of finite-energy solutions of the wave equation
driven by boundary sources with critical and supercritical exponents. The results presented
depend on the boundary damping present in the model. In the absence of boundary
“overdamping,” finite-time blow-up of weak solutions is exhibited.
Copyright © 2006 I. Lasiecka and L. Bociu. 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
Let Ω ⊂ R n be a bounded domain with sufficiently smooth boundary Γ. We consider the
following model of semilinear wave equation with nonlinear boundary/interior monotone
dissipation and nonlinear boundary/interior sources
( )
u tt + g 0 ut = Δu + f (u) inΩ × [0,∞),
∂ ν u + cu + g ( )
u t = h(u) inΓ × [0,∞),
u(0) = u 0 ∈ H 1 (Ω), u t (0) = u 1 ∈ L 2 (Ω).
(1.1)
Our main aim is to discuss the well-posedness of the system given by (1.1), with c ≥ 0,
on a finite-energy space H = H 1 (Ω) × L 2 (Ω). This includes existence and uniqueness of
both local and global solutions and also blow-up of the solutions with nonpositive initial
energy. The main difficultyoftheproblemisrelatedtothepresenceoftheboundary
nonlinear term h(u), and it has to do with the fact that Lopatinski condition does not
hold for the Neumann (c = 0) or Robin (c >0) problem, that is, the linear map h →
U(t) = (u(t),u t (t)), where U(t)solves
u tt = Δu in Ω × [0,∞),
∂ ν u + cu = h in Γ × [0,∞),
u(0) = u 0 ∈ H 1 (Ω), u t (0) = u 1 ∈ L 2 (Ω),
(1.2)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 635–643