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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 $$R^ n$$.

##### 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
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##### References:
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