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Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. (English) Zbl 0873.35051

The existence of a strong \(\omega\) anti-periodic solution for the quasilinear wave equation with viscosity \[ u_{tt}-\text{ div } \big(\sigma(|\nabla u|^2)\nabla u\big)-\Delta u_t=f(x,t) \quad\text{ in}\quad\Omega\times \mathbb{R} \] under the Dirichlet boundary condition \(u(t)|_{\partial\Omega}=0\) is proved under the assumption that \(f(x,t)\) is \(\omega\)-anti-periodic in \(t\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with boundary \(\partial\Omega\) and \(\sigma(\cdot)\) is a \(C^2([0,\infty))\) function satisfying for some constants \(k_1,k_2>0\) and some \(p>0\) the conditions: \(0\leq\sigma(v^2)\leq k_1<\infty\), \(|\sigma'(v^2)|\leq k_1<\infty\) and \(\sigma-2|\sigma'|v^2\geq k_2 (\sigma+2|\sigma'|v^2)^p.\) Such is for instance the function \(\sigma(v^2)=1/\sqrt{1+v^2}.\)
Reviewer: I.Ginchev (Varna)

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
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