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

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The paper is concerned with linear thermoelastic plate equations in the half-space Rn+={x=(x1,…,xn)|xn>0}: utt+Δ2u+Δθ=0 and θt−Δθ−Δut=0 in Rn+×(0,∞), subject to the boundary condition: u|xn=0=Dnu|xn=0=θ|xn=0=0 and initial condition: (u,Dtu,θ)|t=0=(u0,v0,θ0)∈Hp=W2p,D×Lp×Lp, where W2p,D={u∈W2p|u|xn=0=Dnu|xn=0=0}. We show that for any p∈(1,∞), the associated semigroup {T(t)}t≥0 is analytic in the underlying space Hp. Moreover, a solution (u,θ) satisfies the estimates: ||∇j(∇2u(·,t),ut(·,t),θ(·,t))||Lq(Rn+)≤Cp,qt−\\fracj2−\\fracn2(\\frac1p−\\frac1q)||(∇2u0,v0,θ0)||Lp(Rn+) (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.
社団法人日本数学会の論文
2009-10-01