Global existence and blowup of solutions for a class of nonlinear higher-order wave equations
The authors considered a class of nonlinear higher-order wave equation with nonlinear damping \(u_{tt}+(-\Delta)^m u+a|u_t|^{p-2}u_t=b|u|^{q-2}u\) in a bounded domain \(\Omega\subset\mathbb{R}^N\) (\(m, N \geq 1\) are natural numbers, \(a,b>0\) and \(p,q>2\)), with boundary conditions \(D^\alpha u(x,t)=0\) for \(|\alpha|\leq m-1\). They showed that the solution is global in time under some conditions without the relation between \(p\) and \(q\), and they also showed that the local solution blows up in finite time if \(q > p\) with some assumptions on the initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blow-up solution are also given. This extends the recent results of \textit{Y. Ye} [J. Inequal. Appl. 2010, Article ID 394859, 14 p. (2010; Zbl 1190.35161)].
scientific article; zbMATH DE number 6062091