Electronic Journal of Differential Equations (Apr 2009)
Regularity of solutions to doubly nonlinear diffusion equations
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.
