On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity
Xiulan Wu, Yaxin Zhao, Xiaoxin Yang
Abstract
<p>In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.</p>
Topics & Concepts
MathematicsBounded functionMathematical analysisDomain (mathematical analysis)Dirichlet boundary conditionLogarithmLaplace operatorHomogeneousNonlinear systemParabolic partial differential equationSobolev spaceBoundary (topology)Partial differential equationPhysicsCombinatoricsQuantum mechanicsNonlinear Partial Differential EquationsDifferential Equations and Boundary ProblemsAdvanced Mathematical Modeling in Engineering