Taiwanese Journal of Mathematics · 2009 · 47 citations · 4 references
Natural SciencesMathematicsPhysicsDifferential EquationsDynamical SystemsFluid DynamicsBoundary ConditionsBoundary-value ProblemsDissipative SystemsExistence TheoryNonlinear DynamicsPartial Differential EquationsBlow-up CriteriaBlow-up PhenomenaDissipative Dynamical SystemsFinite TimeFinite-time StabilityInitial Boundary ConditionsInitial Value Problem FormulationInitial-boundary Value Problems
We consider the initial boundary value problem for a Petrovsky system with nonlinear damping \begin{equation*} u_{tt}+\Delta ^{2}u+a\left| u_{t}\right| ^{m-2}u_{t}=b\left| u\right| ^{p-2}u, \end{equation*} 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 $p\gt m$ and the initial energy is nonngeative. 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 ([6]).
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