The existence of R ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given byu t t + Δ 2 u + Δ θ =f 1 , in Ω × ( 0 , ∞ ) ,θ t − Δ θ − Δ u t =f 2 , in Ω × ( 0 , ∞ ) ,with the initial condition u | ( t =0) = u 0 , u t | ( t =0) = u 1 , and θ | ( t =0) = θ 0 in Ω and the boundary condition u = ∂ ν u = θ =0 on Γ, where u = u ( x , t ) denotes a vertical displacement at time t at the point x =( x 1 ,⋯, x n )∈Ω, while θ = θ ( x , t ) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of R ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C 0 analytic semigroup and the maximal L p ‐ L q ‐regularity of time‐dependent problem are derived.