On the L p analytic semigroup associated with the linear thermoelastic plate equations in the half-space
Yuka Naito, Yoshihiro Shibata · Journal of the Mathematical Society of Japan · 2009
The paper is concerned with linear thermoelastic plate equations in the half-space R + n = { x = ( x 1 , … , x n ) ❘ x n > 0 } : u tt + Δ 2 u + Δ θ = 0 and θ t - Δ θ - Δ u t = 0 R + n × ( 0 , ∞ ) , subject to the boundary condition: u | x n = 0 = D n u | x n = 0 = θ | x n = 0 = 0 and initial condition: ( u , D t u , θ ) | t = 0 = ( u 0 , v 0 , θ 0 ) ∈ H p = W p , D 2 × L p × L p , where W p , D 2 = { u ∈ W p 2 ❘ u | x n = 0 = D n u | x n = 0 = 0 } . We show that for any p ∈ ( 1 , ∞ ) , the associated semigroup { T ( t ) } t ≥ 0 is analytic in the underlying space H p . Moreover, a solution ( u , θ ) satisfies the estimates: ∥ ∇ j ( ∇ 2 u ( · , t ) , u t ( · , t ) , θ ( · , t ) ) ∥ L q ( R + n ) ≤ C p,q t - j2 - n2 ( 1p - 1q ) ∥ ( ∇2 u0 , v0 , θ0 ) ∥ Lp ( 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 .