×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EMIS EuDML