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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 \(R^ n\).

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
Full Text: DOI
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