## On the $$L_{p}$$ analytic semigroup associated with the linear thermoelastic plate equations in the half-space.(English)Zbl 1184.35062

The authors study the following system of linear equations, which describe a linear thermoelastic plate, \begin{aligned} u_{tt}(x,t)+\Delta^2u(x,t)+\Delta\theta(x,t)&=0,\\ \theta_t(x,t)-\Delta\theta(x,t)-\Delta u_t(x,t)&=0, \end{aligned}\tag{1} where $$(x,t)\in\mathbb R^n_+\times\mathbb R_+$$, $$\mathbb R^n_+:=\{x=(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_n>0\}$$, the unknown function $$u$$ denotes a mechanical variable denoting the vertical displacement of the plate and the unknown function $$\theta$$ denotes a thermal variable. The system (1) is subject to the initial conditions $u(x,0)=u_0(x),\quad u_t(x,0)=v_0(x),\quad \theta(x,0)=\theta_0(x)\tag{2}$ and the boundary conditions $u|_{x_n=0}=D_nu|_{x_n=0}=\theta|_{x_n=0}=0.\tag{3}$ It is proved in the paper that the initial boundary value problem (1)–(3) generates an analytic semigroup in the $$L_p(\mathbb R^n_+)$$ framework, $$1<p<\infty$$, and the corresponding estimates for the solution of the problem are established. In particular, for the solution $$(u,\theta)$$, for $$t>0$$, $\|\nabla^j(\nabla^2u(\cdot,t),D_t u(\cdot,t),\theta(\cdot,t))\|_{L_q(\mathbb R^n_+)}\leq C_{p,q}t^{-\frac j2-\frac n2\left(\frac 1p-\frac 1q\right)}\|(\nabla^2 u_0,v_0,\theta_0)\|_{L_p(\mathbb R^n_+)},$ for $$j=0,1,2$$, provided that $$1<p\leq q\leq\infty$$ when $$j=0,1$$ and that $$1<p\leq q<\infty$$ when $$j=2$$, where $$\nabla^j$$ stands for space gradient of order $$j$$. In order to get the results, the authors, first, study the corresponding resolvent equation and prove the corresponding resolvent estimates.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 74F05 Thermal effects in solid mechanics 47A10 Spectrum, resolvent 47D06 One-parameter semigroups and linear evolution equations
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