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