Decay rates for solutions of degenerate parabolic systems

Abstract

Explicit decay rates for solutions of systems of degenerate parabolic equations in the whole space or in bounded domains subject to homogeneous Dirichlet boundary conditions are proven. These systems include the scalar porous medium, fast diffusion and $p$-Laplace equation and strongly coupled systems of these equations. For the whole space problem, the (algebraic) decay rates turn out to be optimal. In the case of bounded domains, algebraic and exponential decay rates are shown to hold depending on the nonlinearities. The proofs of these results rely on the use of the entropy functional together with generalized Nash inequalities (for the whole space problem) or Poincare inequalities (for the bounded domain case).

Keywords

• Explicit decay rates
• long-time behavior of solutions
• algebraic decay
• exponential decay
• degenerate parabolic equations
• Nash inequality.