JournalsifbVol. 9 , No. 1DOI 10.4171/ifb/157

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

  • Kianhwa C. Djie

    RWTH Aachen, Germany
An upper bound for the waiting time for doubly nonlinear parabolic equations cover

Abstract

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

{(uq2u)tdiv(up2u)=0in \mathdsRN×[0,) ,u(x,0)=u0(x)for all x\mathdsRN ,\left\{\begin{array}{rlll}(|u|^{q-2}u)_t - \textrm{div}(|\nabla u|^{p-2}\nabla u)&=&0&\textrm{in}\ \mathds{R}^N \times [0, \infty)\ ,\\ u(x, 0)&=&u_0(x)&\textrm{for all}\ x \in \mathds{R}^N\ ,\\ \end{array}\right.

depending on the growth of the initial value u0u_0 with parameters p2, 1<q<pp \geq 2,\ 1 < q < p, and u0q1L1(\mathdsRN)|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=2p=2) for which we obtain the result of Chipot-Sideris \cite{CS} and the parabolic pp-Laplace equation (for q=2q=2).