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Some properties of shallow shells with thermal effects. (English) Zbl 1309.35166

The authors consider a 2D dynamic Marguerre-Vlasov shallow shell problem with thermal effects and written as \(U_{tt}-\mathrm{div}([B_{ij}])=0\), \( w_{tt}+\Delta ^{2}w-\Delta w_{tt}+\Delta \theta -\mathrm{div}([B_{ij}]\nabla w)+K_{1}B_{11}+K_{2}B_{22}=0\), \(\theta _{t}-\Delta \theta -\Delta w_{t}=0\) in \(M\times (0,\infty )\) where \(M\) is a smooth, oriented and compact surface. Here \(U=( \begin{matrix} u \\ v \end{matrix} )\) (resp. \(w\),\(\theta \)) means the longitudinal point displacement (resp. transversal displacement, temperature) on the middle surface of the shell. The clamped boundary conditions \(U=( \begin{matrix} 0 \\ 0 \end{matrix} )\), \(w=\frac{\partial w}{\partial \nu }=0\), \(\theta =0\) are imposed on \( \partial M\times (0,\infty )\) and initial data are given at \(t=0\). \([B_{ij}]\) is a \(2\times 2\) symmetric matrix whose coefficients depend on \(u\), \(v\) and \(w\) and on their first-order partial derivatives with respect to the space variables. In the first part of their paper, the authors consider a perturbed Marguerre-Vlasov problem changing the first above-indicated equation by \(\varepsilon U_{tt}-\mathrm{div}([B_{ij}^{\varepsilon }])=0\), for some positive \(\varepsilon \) and replacing \([B_{ij}]\) by a perturbed matrix \( [B_{ij}^{\varepsilon }]\) also in the second equation. They describe the asymptotic behavior of the solution when \(\varepsilon \) goes to 0. The main result of the paper indeed proves that under regularity properties of the data \(w^{\varepsilon }\) and \(w_{t}^{\varepsilon }\) (resp. \(\theta ^{\varepsilon }\)) weakly converge in \(L^{\infty }(0,\infty ;H^{2}(M))\) (resp. \(L^{\infty }(0,\infty ;L^{2}(M))\)) to some \(z\) and \(z_{t}\) (resp. \(\varphi \)) and they write the problem is satisfied by these limit, which is a Timoshenko-type problem with thermal effects. They also describe the asymptotic behaviour of the longitudinal displacement and of the energy associated to the perturbed problem. The second main result of the paper describes the asymptotic behavior of the energy of the original Marguerre-Vlasov problem when \(t\) goes to \(\infty \). Again assuming regularity properties of the data, the authors prove that the energy \(E(t)\) decays exponentially \(E(t)\leq C_{1}\exp (-C_{2}t)\), for every initial data such that \(E(0)\leq R\), with constants \(C_{1}\) and \(C_{2}\) possibly depending on \(R\). For the proof of this exponential decay property, the authors draw direct computations and they quote results from G. Avalos and I. Lasiecka [SIAM J. Math. Anal. 29, No. 1, 155–182 (1998; Zbl 0914.35139)].

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74K25 Shells
35Q83 Vlasov equations
35R01 PDEs on manifolds
74F05 Thermal effects in solid mechanics

Citations:

Zbl 0914.35139
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Full Text: Euclid