On internal elastic membrane with strong damping. (English) Zbl 1249.35217

Araruna, F. D.; Clark, M. R.; Lima, O. A.

Differ. Integral Equ. 24, No. 7-8, 601-618 (2011).

Under some assumptions on \(M,A,F\) the authors prove existence and uniqueness of weak and strong solutions of the initial problem to the equation \(u_{tt}+M(\| u(t)\| ^2)Au(t)+Fu(t)+Au'(t)=0\) in a real Hilbert space, where \(t\in (0,T)\), \(M\) is degenerate real function and \(A,F\) are operators. Furthermore, energy decay \(E(t)\leq C(1+t)^{-(1+1/s)}\) of the initial Dirichlet boundary value problem to the equation \(u_{tt}-\| u(t)\| ^{2s}\Delta u+g(u)-\Delta u_t=0\) is established.

Reviewer: Marie Kopáčková (Praha)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35L20	Initial-boundary value problems for second-order hyperbolic equations

Keywords:

Kirchhoff model; strong damping; asymptotic decay