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Existence and uniqueness of the solutions to second-order abstract equations with nonlinear and nonmonotone boundary conditions. (English) Zbl 0822.35096

The paper is devoted to study the existence of solutions to abstract second order (in time) nonlinear differential equations (inequalities) of the form \[ Mu_{tt} (t)+ Au(t)+ AG \partial \Phi G^* Au_ t+ AG f(u(t))\ni {\mathcal F} (u); \qquad t>0; \]
\[ u(t=0) =u_ 0, \qquad u_ t (t=0)= u_ 1, \] where \(\partial \Phi\) is the subdifferential of a convex lower semicontinuous function \(\Phi\), the nonlinear operators \({\mathcal F}\) and \(f\) are assumed to be Fréchet differentiable. The unboundedness and nonmonotonicity of the operator \(AG f\) causes one of the main difficulties of the problem under consideration. The abstract existence result is illustrated by von Kármán plate equations with nonlinear boundary conditions which in such a framework are represented by the mentioned nonlinear unbounded operator \(AG f\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
35R70 PDEs with multivalued right-hand sides
74K20 Plates
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