Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 17 (2009), pp. 185-195.

Regularity of solutions to doubly nonlinear diffusion equations

Jochen Merker

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.

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Jochen Merker
University of Rostock, D-18051 Rostock, Germany
email: jochen.merker@uni-rostock.de

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