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

  • Kianhwa C. Djie

    RWTH Aachen, Germany

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).

Cite this article

Kianhwa C. Djie, An upper bound for the waiting time for doubly nonlinear parabolic equations. Interfaces Free Bound. 9 (2007), no. 1, pp. 95–105

DOI 10.4171/IFB/157