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Abstract
In this paper we consider Kirchhoff plates with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases. MSC:35B40, 74K20, 35L70.
Highlights
We consider the following Kirchhoff plates with a memory condition at the boundary: utt + u + a(x)ut =, in × (, ∞), ( . ) ∂u u = =,∂ν on × (, ∞), t–u + g (t – s)B u(s) ds =, on × (, ∞), ∂u +∂ν t g (t – s)B u(s) ds =, on × (, ∞), u(, x) = u (x), ut(, x) = u (x), in, where a ∈ C ( ̄ ) and is an open bounded set of R with a regular boundary
In Section we prove the general decay of the solutions to the Kirchhoff plates with a memory condition at the boundary
3 General decay we show that the solution of the system ( . )-( . ) may have a general decay not necessarily of exponential or polynomial type
Summary
We consider the following Kirchhoff plates with a memory condition at the boundary: utt + u + a(x)ut = , in × ( , ∞),. Cavalcanti and Guesmia [ ] proved the general decay rates of solutions to a nonlinear wave equation with a boundary condition of memory type. Park and Kang [ ] studied the exponential decay for the multi-valued hyperbolic differential inclusion with a boundary condition of memory type. Messaoudi and Soufyane [ ], Santos and Soufyane [ ], and Mustafa and Messaoudi [ ] proved the general decay for the wave equation, von Karman plate system, and Timoshenko system with viscoelastic boundary conditions, respectively. In Section we prove the general decay of the solutions to the Kirchhoff plates with a memory condition at the boundary. In these conditions, we are able to prove the existence of a strong solution.
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