Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation
The authors study the following initial boundary value problem for nonlinear wave equation of Kirchhoff type with a dissipative term: \[ \begin{gathered} u_{tt}- M(\|\nabla u\|^2)\Delta u+ g(u_t)= f(u),\quad (t,x)\in\Omega\times [0,\infty),\\ u(x,0)= u_0(x),\;u_t(0,x)= u_1(x),\quad x\in\Omega,\\ u(x,t)= 0,\quad (t,x)\in(0,\infty)\times \partial\Omega,\end{gathered} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with boundary \(\partial\Omega\) so that divergence theorem can be applied, \(\|\cdot\|\) denotes the usual \(L^2\) norm in \(\Omega\), \(M\) is a nonnegative locally Lipschitz function, \(f\) is a nonlinear function typified by \(f(u)= |u|^{p-2}u\) with \(p> 2\) and \(g(u_t)\) is the strong dissipative term \(-\Delta u_t\) or the linear dissipative term \(u_t\) or nonlinear dissipative term \(|u_t|^{m-2} u_t\) with \(m> 2\). Then they prove the existence of a local solution for \(g(u_t)= u_t\) and \(g(u_t)= -\Delta u_t\) with \(p\leq(2N- 4)/(N- 4)\) for \(N> 4\) and \(p<\infty\) for \(N\leq 4\), and for \(|u_t|^{m-2} u_t\) with \(p\leq (2N- 6)/(N- 4)\) for \(N> 4\) and \(p<\infty\) for \(N\leq 4\). Moreover, the authors prove the existence of a blow-up solution for \[ {sf(s)\over 2\int^s_0 f(r)\,dr}\geq 1+\delta\geq {sM(s)\over \int^s_0 M(r)\,dr} \] with \(s> 0\) and \(\delta> 0\) under some suitable assumptions to the initial energy.
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