Global solutions for a nonlinear wave equation with the $$p$$-Laplacian operator.(English)Zbl 0954.35031

The authors investigate the initial-boundary value problem for the nonlinear wave equation: $u_{tt}- \Delta_pu+ (-\Delta)^\alpha u_t+ g(x,u)= f(t,x)\quad\text{in }(0,T)\times \Omega,$
$u= 0\quad\text{on }(0,T)\times \partial\Omega,$
$u(0,x)= u_0\quad\text{and }u_t(0,x)= u_1\quad\text{in }\Omega,$ where $$0< \alpha\leq 1$$, $$p\geq 1$$ and $$\Omega$$ is a bounded domain in $$\mathbb{R}^n$$ with smooth boundary. $$-\Delta_p$$ is the pseudo-Laplacian: $-\Delta_pu=- \sum^n_{i=1} {\partial\over\partial x_i} \Biggl(\Biggl|{\partial u\over\partial x_i}\Biggr|^{p- 2}{\partial u\over\partial x_i}\Biggr),$ and $$g(x,u)$$ is a continuous function satisfying $$|g(x,u)|\leq a|u|^{\sigma- 1}+ b\quad\text{for all }(x,u)\in \Omega\times \mathbb{R}$$, where $$1<\sigma< pn/(n- p)$$ if $$n> p$$ and $$1<\sigma<\infty$$ if $$n\leq p$$.
The authors show that if $$\sigma< p$$, there exists a global solution $$u\in L^\infty(0,T; W^{1,p}_0(\Omega))$$ such that $$u'\in L^\infty(0,T; L^2(\Omega))\cap L^2(0,T; D((-\Delta)^{\alpha/2}))$$ for any $$u_0\in W^{1,p}_0(\Omega)$$, $$u_1\in L^2(\Omega)$$ and $$f\in L^2(0, T;L^2(\Omega))$$, and if $$\sigma> p$$, the same result holds for sufficiently small data $$u_0$$, $$u_1$$, $$f$$. Concerning the asymptotic behaviour at $$t= \infty$$ it is shown that an exponential decay or an algebraic decay holds according as $$p=2$$ or $$p> 2$$, if the additional condition $$g(x,u)u\geq \rho G(x,u)\geq 0$$ is satisfied in the case $$\sigma< p$$, where $$\rho$$ is some positive constant and $$G(x,u)= \int^u_0 g(x,s) ds$$.

MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
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