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Multidimensional viscoelasticity equations with nonlinear damping and source terms. (English) Zbl 1057.74007

The authors investigate the initial-boundary value problem \(u_{tt}-\Delta u_t-\sum_{i=1}^N\frac \partial{\partial x_i}\sigma_i(u_{x_i})+f(u_t)=g(u)\), \(x\in \Omega\), \(t>0,\) \(u(x,0)=u_0(x)\), \(u_t(x,0)=u_1(x)\), \(x\in \Omega\); \(u(x,t)=0\), \(x\in \partial\Omega\), \(t\geq 0,\) where \(\Omega\subset \mathbb R^N\) is a bounded domain. They introduce a potential well \(W=\{u\in W_0^{1,m+1}(\Omega)\mid I(u)>0\), \(J(u)<d\}\cup \{0\}\), where \( I(u)=B_1\| \nabla u\|_{m+1}^{m+1}- B_5\| u\|_{p+1}^{p+1}\), \(J(u)=\frac {B_1}{m+1}\| \nabla u\|_{m+1}^{m+1}- \frac {B_5}{p+1}\| u\|_{p+1}^{p+1},\) \( d=\inf J(u)\) is subject to \(y\in W_0^{1,m+1}(\Omega)\), \(\| \nabla u\|_{m+1}\neq 0\), \(I(u)=0\), and \(\sigma_i, f, g\in C \) satisfy the conditions: \(\sigma_i(0)=0\), \((\sigma_i(s_1)-\sigma_i(s_2))(s_1-s_2)\geq 0\) \(\forall s_1, s_2\in \mathbb R,\) \(B_1| s|^m\leq | \sigma_i(s)| \leq B_2(1+| s|^m )\), \(m\geq 1\), \(i=1,\dots,N\). \(f(s)s\geq 0\), \(B_3 | s|^p\leq | f(s)| \leq B_4(1+| s|^q )\), \(q\geq 1, \) \(g(s)s\geq 0\), \(| g(s)| \leq B_5| s|^p\), \(1\leq m<p<\infty\) if \(m+1\geq N\); \(2\leq m+1<p+1\leq N(m+1)/(N-m-1)\) if \(m+1<N.\) Some new families of potential wells are introduced. Using them, the authors prove new existence theorems on global solutions, and examine the behaviour of vacuum isolation of solutions.

MSC:

74D10 Nonlinear constitutive equations for materials with memory
74H20 Existence of solutions of dynamical problems in solid mechanics
35L20 Initial-boundary value problems for second-order hyperbolic equations
35Q72 Other PDE from mechanics (MSC2000)
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References:

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