Global existence and asymptotic behaviour of the solution of some quasilinear hyperbolic equation with linear damping. (English) Zbl 0702.35165
We are interested in considering the existence and asymptotic behavior of solutions of the equation
\[
(*)\quad u_{tt}(x,t)- a(\int_{\Omega}| \nabla u(y,t)|^ 2dy)\Delta u(x,t)+2\gamma u(x,t)=f(x,t),
\]
\(\Omega\subset {\mathbb{R}}^ n\), \(\gamma >0\), with null- Dirichlet boundary for arbitrary data of finite energy (i.e. \((u_ 0,u_ 1,f)\in H^ 1_ 0(\Omega)\cap H^ 2(\Omega)\times H^ 1(\Omega)\times L^ 2(0,\infty;\) \(H^ 1(\Omega))\equiv D)\), which are open provided that the data are not small nor analytic. Let \(U(x,t)\) be the global solution of (*) with (large) analytic data \((U_ 0,U_ 1,\bar f)\), which exists and decays as \(t\to \infty\) if \(\bar f\) tends to zero as \(t\to \infty\). Let fix \((U_ 0,U_ 1,\bar f)\) and U. We show that if \((u_ 0,u_ 1,f)\) belongs to some (small) neighborhood of \((U_ 0,U_ 1,\bar f)\) in D, then there exists a unique solution u of (*) with the data \((u_ 0,u_ 1,f)\) saisfying u-U decays. Thus we have obtained the global existence of solutions of (*) for “many” data which are not necessarily small nor analytic.
Reviewer: Kenji Nishihara
MSC:
35L70 | Second-order nonlinear hyperbolic equations |
35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
35B40 | Asymptotic behavior of solutions to PDEs |
74H45 | Vibrations in dynamical problems in solid mechanics |
35B45 | A priori estimates in context of PDEs |