Henriques de Brito, Eliana Nonlinear initial-boundary value problems. (English) Zbl 0613.34013 Nonlinear Anal., Theory Methods Appl. 11, 125-137 (1987). 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 \(R^ n\). Cited in 2 ReviewsCited in 20 Documents 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 Keywords:global existence; exponential decay; global solution; Cauchy problem; Hilbert space PDFBibTeX XMLCite \textit{E. Henriques de Brito}, Nonlinear Anal., Theory Methods Appl. 11, 125--137 (1987; Zbl 0613.34013) 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: Dunod Gauthier-Villars Paris · Zbl 0189.40603 [3] Lions, J. L.; Magenes, E., Problèmes aux Limites non Homogènes et Applications, Vol. 1 (1968), Dunod: Dunod Paris · Zbl 0165.10801 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.