×

zbMATH — the first resource for mathematics

Nonlinear initial-boundary value problems. (English) Zbl 0613.34013
We prove global existence, uniqueness and exponential decay of a global solution, u(t), of a Cauchy problem in a Hilbert space H for an equation whose weak formulation is \[ \frac{d}{dt}(u',v)+\delta (u',v)+\alpha b(u,v)+\beta a(u,v)+(G(u),v)=0 \] where \('=d/dt\), (,) is the inner product in H, b(u,v), a(u,v) are given forms on subspaces \(U\subset W\), respectively, of H, G is the Gateaux derivative of a given convex functional \(J: V\subset H\to [0,\infty),\) u is a test function in V, and \(\alpha\geq 0\), \(\delta\geq 0\), real \(\beta\) are given constants. Application is given to initial-boundary value problems in a bounded domain of \(R^ n\).

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] de Brito, E.H., Decay estimates for the generalized damped extensible string and beam equation, Nonlinear analysis, 8, 1489-1496, (1984) · Zbl 0524.35026
[2] Lions, J.L., Quelques Méthodes de Résolution des problémes aux limites non lineaires, (1969), Dunod Gauthier-Villars Paris · Zbl 0189.40603
[3] Lions, J.L.; Magenes, E., Problèmes aux limites non homogènes et applications, Vol. 1, (1968), Dunod Paris
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.