IJPAM: Volume 30, No. 4 (2006)

WELL-POSEDNESS OF ABSTRACT HYPERBOLIC
MODELS WITH TIME DEPENDENT DAMPING

Michael Pedersen
Department of Mathematics
Technical University of Denmark
Kgs. Lyngby, DK 2800, DENMARK
e-mail: M.Pedersen@mat.dtu.dk


Abstract.In this paper we study the abstract hyperbolic model with time dependent damping given by

\begin{displaymath} \langle u''(t),v\rangle_{V',V}+d(t;u'(t),v)+a(t;u(t),v)=\langle f(t),v\rangle_{V',V} \,, \end{displaymath}

where $V\subset V_D\subset H\subset V'_D\subset V'$ are Hilbert spaces with continuous and dense injections, here $H$ is identified with its dual and $\langle\cdot,\cdot\rangle$ denotes the various duality products. We show that this model allows a unique solution under natural conditions on the time-dependent sesquilinear forms $a(t;\cdot,\cdot):V\times V\to C$ and $d(t;\cdot,\cdot):V_D\times V_D\to C$ and that the solution depends continuously on the data of the problem. This ensures good convergence properties of approximating Galerkin schemes for the numerical solution of the problem. The problem above is the ultimative weak formulation of the ``strong" problem

\begin{displaymath} u''(t,x) +D(t,x)u'(t,x) +A(t,x)u(t,x)=f(t,x), \end{displaymath}

for $ 0<t<T<\infty, x\in\Omega\subset R^n$, modeling very abstract differential operator problems including plate, beam and shell equations with numerous kinds of damping.

Received: July 13, 2006

AMS Subject Classification: 35L10, 35L90

Key Words and Phrases: partial differential equations, weak solutions, well-posedness

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 30
Issue: 4