Lp theory for the linear thermoelastic plate equations in bounded and exterior domains

Abstract

The paper is concerned with linear thermoelastic plate equations in a domain Ω: utt + Δ2u + ΔΘ = 0 and Θt - ΔΘ - Δut =0 in Ω × (0, ∞), subject to the Dirichlet boundary condition u|r = Dvu|γ = Θ|γ = 0 and initial condition (u,ut,Θ)|t=0 = (u0,v0,Θ0) ∈ W2p,D(Ω) × Lp × Lp. Here, ω is a bounded or exterior domain in ℝn (n > 2). We assume that the boundary Γ of Ω is a C4 hypersurface and we define W2P,D by the formula W2P,D = {u ∈ W2p: u|γ = Dvu|γ = 0}. We show that, for any p ∈ (l,∞), the associated semigroup {T(t)}t>o is analytic. Moreover, if fi is bounded, then {T(t)}t≥o is exponentially stable.