An upper bound for the waiting time for doubly nonlinear parabolic equations

Abstract

We obtain an upper bound for the waiting time for the doubly nonlinear parabolic equations

depending on the growth of the initial value $u_0$ with parameters $p \geq 2,\ 1 < q < p$, and $|u_0|^{q-1} \in L^1(\mathds{R}^N)$. This upper bound coincides with the lower bound given by Giacomelli-Gr\"{u}n \cite{GG}. Therefore it is optimal. Special cases are the porous medium equation (for $p=2$) for which we obtain the result of Chipot-Sideris \cite{CS} and the parabolic $p$-Laplace equation (for $q=2$).