Global solvability for damped abstract nonlinear hyperbolic systems. (English) Zbl 0892.47063

Banks, H. T.; Gilliam, D. S.; Shubov, V. I.

Differ. Integral Equ. 10, No. 2, 309-332 (1997).

The problem of existence of weak Hilbert-space-valued solutions to an initial-value problem for the abstract equation \(w_{tt} + {\mathcal A}_1w +{\mathcal A}_2w_t + {\mathcal N}^*g({\mathcal N} w) = f\) with given operators \({\mathcal A}_1, {\mathcal A}_2, {\mathcal N}, g\) and right-hand side \(f\) is solved via compactness and monotonicity method based on Galerkin approximations and an energy-type a priori estimate. The authors also give sufficient conditions for uniqueness and derive a variation-of-constants formula for the solution. As an example they consider the case where \({\mathcal N} = \Delta \) is the Laplacian in a bounded smooth domain \(\Omega \subset R^m\), \({\mathcal A}_1 = {\mathcal A}_2 = \Delta ^2\) with boundary conditions \(w = \partial w/\partial n = 0\) on \(\partial \Omega \) and \(g\) is a real function.

Reviewer: P.Krejčí (Praha)

MSC:

47J25	Iterative procedures involving nonlinear operators
35G25	Initial value problems for nonlinear higher-order PDEs

Keywords:

abstract second order equations; weak solutions; existence; uniqueness