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A note on parabolic equation with nonlinear dynamical boundary condition. (English) Zbl 1185.35131
The authors consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition related to the so-called Wentzell boundary condition. Existence and uniqueness of global solutions are proved which imply also the existence of a global attractor. Further, a suitable Łojasiewicz-Simon type inequality is derived in order to show the convergence of the global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms $$f,g$$ are real analytic. Moreover, an estimate for the convergence rate is provided.

MSC:
 35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B41 Attractors 35B45 A priori estimates in context of PDEs 35K58 Semilinear parabolic equations
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References:
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