On the L_p analytic semigroup associated with the linear thermoelastic plate equations in the half-space

The paper is concerned with linear thermoelastic plate equations in the half-space R₊ⁿ = {x = (x₁,..., x_n) | x_n > 0}: u_tt + Δ²u + Δθ = 0 and θ_t - Δθ - Δθu_t = 0 in R₊ⁿ × (0, ∞), subject to the boundary condition: u|_xn=0 = D_n u|_{x n}=0 = θ|_xn=0 = 0 and initial condition: (u,D_tu,θ)|_t=0 = (u₀,v₀, θ₀) ∈ H_p = W_p,D² × L_P × L_P, where W_p,D² = {u ∈ W_p² | u|_Xn=0 = D _nu|_xn=0 = 0}. We show that for any p ∈ (1, infin;), the associated semigroup {T(t)}_t≥0 is analytic in the underlying space H_p. Moreover, a solution (u, θ) satisfies the estimates: ||∇^j(∇²u(·, t), u _t(·, t), θ(·, t))||_{L q}(R₊ⁿ) ≤ c_p,qt-1\2-n\2(1\p-1\q) ||(∇²u₀,V₀, θ₀)|| _Lq(R₊ⁿ) (t > 0) for j = 0, 1, 2 provided that 1 < p ≤ q ≤ ∞ when j = 0, 1 and that 1 < p ≤ q < ∞ when j = 2, where ∇^j stands for space gradient of order j.