On internal elastic membrane with strong damping. (Q417008)

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.