Global existence of solutions and smoothing effects for classes of reaction–diffusion equations on manifolds
Gabriele Grillo, Giulia Meglioli, Fabio Punzo
2021Virtual Community of Pathological Anatomy (University of Castilla La Mancha)13 citationsDOIOpen Access PDF
Abstract
We consider the porous medium equation with a power-like reaction term, posed on Riemannian manifolds. Under certain assumptions on p and m in (1.1), and for small enough nonnegative initial data, we prove existence of global in time solutions, provided that the Sobolev inequality holds on the manifold. Furthermore, when both the Sobolev and the Poincaré inequalities hold, similar results hold under weaker assumptions on the forcing term. By the same functional analytic methods, we investigate global existence for solutions to the porous medium equation with source term and variable density in Rn.
Topics & Concepts
MathematicsSmoothingSobolev spaceTerm (time)Manifold (fluid mechanics)Forcing (mathematics)Reaction–diffusion systemRiemannian manifoldDiffusionApplied mathematicsPure mathematicsMathematical analysisPhysicsStatisticsThermodynamicsMechanical engineeringEngineeringQuantum mechanicsNonlinear Partial Differential EquationsAdvanced Mathematical Physics ProblemsAdvanced Mathematical Modeling in Engineering