## 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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