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.


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)
Full Text: DOI


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