Global existence and blowup of \(W^{k,p}\) solutions for a class of nonlinear wave equations with dispersive term (Q2486620)

The authors considered the following problem \[ \begin{gathered} u_{tt} -\alpha u_{xxt}-\beta u_{xx}-u_{xxtt} =\sigma (u_{x})_{x},\quad \alpha ,\beta \geq 0,\;x\in \Omega= (0,1),\;t>0\\ u(x,0) =u_{0}(x),\quad u_{t}(x,0)=u_{1}(x)\quad x\in\Omega\tag{1} \\ u(0,t) =u(1,t)=0,\quad \forall t\geq 0.\end{gathered} \] with the conditions \(\sigma \in \,\,c^{k}(\Omega ),\,k\geq 2,\,\sigma ^{\prime }\geq C_{0}\) for some constant \(C_{0}\) and \newline \(| \sigma _{1}(s)| \leq c_{1}\int_{0}^{s}\sigma _{1}(\tau )\,d\tau +c_{2}\), where \[ \sigma _{1}(s)=\sigma (s)-k_{0}s-\sigma (0),\quad k_{0}=\min (C_{0},0). \] They first give a brief history then they showed, by using some \(L^{P}\) estimates and the properties of Green function, for the auxiliary problem \[ u-\triangle u =f,\quad x\in\Omega, \quad u_{| \partial \Omega } =0, \] that (1) with \(\beta =0\) has a unique local solution. They also prove that this solution is global in time. As they mention, these existence results extend to problem (1) with \(\beta >0.\) For the asymptotic behavior, it is crucial to take \(\alpha >0\) and consequently, the authors show that the natural energy of the solution decays exponentially. Finally, by replacing \(\sigma \) by \(\sigma _{1}\) in (1) and assuming some appropriate growth conditions on \(\sigma ,\) the authors established a finite-time blowup result for solutions with specific initial energy.