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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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