Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations

In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions utt+Δ2u+αΔθ=f1(t,u,θ),t⩾0,x∈Ω,θt−βΔθ−αΔut=f2(t,u,θ),t⩾0,x∈Ω,θ=u=Δu=0,t⩾0,x∈∂Ω, where α≠0, β>0, Ω is a sufficiently regular bounded domain in RN (N⩾1) and fe1,fe2:R×L2(Ω)2→L2(Ω) define by fe(t,u,θ)(x)=f(t,u(x),θ(x)), x∈Ω, are continuous and locally Lipschitz functions. First, we prove that the linear system (f1=f2=0) generates an analytic strongly continuous semigroup which decays exponentially to zero. Second, under some additional condition we prove that the nonlinear system has a bounded solution which is exponentially stable, and for a large class of functions f1,f2 this bounded solution is almost periodic. Finally, we use the analyticity of the semigroup generated by the linear system to prove the smoothness of the bounded solution.