On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system. (English) Zbl 1196.35043
From the text: We consider the initial boundary value problem for a Petrovsky system with nonlinear damping
\[ u_{tt}+\Delta^2u+a|u_t|^{m-2} u_t = b|u|^{p-2} u, \]
in a bounded domain. We showed that the solution is global in time under some conditions without the relation between \(m\) and \(p\). We also prove that the local solution blows-up in finite time if \(p>m\) and the initial energy is non negative. The decay estimates of the energy function and the estimates of the lifespan of solutions are given. In this way, we can extend the result of S. A. Messaoudi [J. Math. Anal. Appl. 265, No. 2, 296–308 (2002; Zbl 1006.35070)].
\[ u_{tt}+\Delta^2u+a|u_t|^{m-2} u_t = b|u|^{p-2} u, \]
in a bounded domain. We showed that the solution is global in time under some conditions without the relation between \(m\) and \(p\). We also prove that the local solution blows-up in finite time if \(p>m\) and the initial energy is non negative. The decay estimates of the energy function and the estimates of the lifespan of solutions are given. In this way, we can extend the result of S. A. Messaoudi [J. Math. Anal. Appl. 265, No. 2, 296–308 (2002; Zbl 1006.35070)].
MSC:
| 35B33 | Critical exponents in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35G31 | Initial-boundary value problems for nonlinear higher-order PDEs |
| 35B45 | A priori estimates in context of PDEs |
