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

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.


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