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Existence and uniform decay for a nonlinear beam equation with nonlinearity of Kirchhoff type in domains with moving boundary. (English) Zbl 1092.35068
Summary: We prove the exponential decay in the case n>2, as time goes to infinity, of regular solutions for the nonlinear beam equation with memory and weak damping u tt +Δ 2 u-M(u L 2 (Ω t ) 2 )Δu+ 0 t g(t-s)Δu(s)ds+αu t =0 in Q ^ in a noncylindrical domain of n+1 (n1) under suitable hypothesis on the scalar functions M and g, and where α is a positive constant. We establish existence and uniqueness of regular solutions for any n1.
MSC:
35L75Nonlinear hyperbolic PDE of higher (>2) order
35L35Higher order hyperbolic equations, boundary value problems
35B40Asymptotic behavior of solutions of PDE
45K05Integro-partial differential equations