Seventh Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 17 (2009), pp. 185-195. 

Title: Regularity of solutions to doubly nonlinear diffusion equations

Author: Jochen Merker (Univ. of Rostock, Germany)
 
Abstract: 
  We prove under weak assumptions that solutions $u$ of
  doubly nonlinear reaction-diffusion equations
  $$
    \dot{u}=\Delta_p u^{m-1} + f(u)
  $$
  to initial values $u(0) \in L^a$ are instantly regularized to
  functions $u(t) \in L^\infty$ (ultracontractivity). Our proof is
  based on a priori estimates of $\|u(t)\|_{r(t)}$ for a time-dependent
  exponent $r(t)$. These a priori estimates can be obtained in an
  elementary way from logarithmic Gagliardo-Nirenberg inequalities
  by an optimal choice of $r(t)$, and they do not only imply
  ultracontractivity, but provide further information about the
  long-time behaviour. 

Published April 15, 2009.
Math Subject Classifications: 35K65, 35B35, 46E35, 35B45.
Key Words:  p-Laplacian; doubly nonlinear evolution equations;
            ultracontractive semigroups; 
	    logarithmic Gagliardo-Nirenberg inequalities