Global Solutions to Some Nonlinear Dissipative Mildly Degenerate Kirchhoff Equations

Marina Ghisi

Abstract: We investigate the evolution problem \begin{eqnarray*} && u_{tt}+\delta\,u_{t}-m\Bigl(\int_{\Omega}|\nabla u|^{2}\,dx\Bigr) \,\Delta u+f(u)=0,
&& u(0,x)=u_0(x),\ u_{t}(0,x)=u_{1}(x),\ x\in\Omega,\ t \geq 0, \end{eqnarray*} where $n\leq 3$, $\Omega\subset\R^n$ is a bounded open set, $\delta>0$, and $m:[0,+\infty[\to[0,+\infty[$ is a locally Lipschitz continuous function, with $m(0)=0$ and $m(r)>0$ in a neighborhood of $0$, and $f(u)u\geq 0$.
We prove that this problem has a unique global solution for positive times, provided that the initial data $(u_{0},u_{1})\in(H_{0}^{1}\cap H^{2})(\Omega)\times H_{0}^{1}(\Omega)$ and $f$ satisfy suitable smallness assumptions and the non-degeneracy condition \hbox{$u_{0} \neq 0$}. We prove also that $(u(t),u'(t),u''(t))\to(0,0,0)$ in $(H_{0}^{1}\cap H^{2})(\Omega)\times H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ as $t\to\infty$.

Keywords: nonlinear hyperbolic equations; degenerate hyperbolic equations; dissipative equations; global existence; asymptotic behaviour; Kirchhoff equations

Classification (MSC2000): 35L80, 35B40.

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