article

Title: Global existence for parabolic problems involving the $p$-Laplacian and a critical gradient term

Authors: A. Dall'aglio, D. Giachetti and S. Segura de Leon

Issue: Volume 58 (2009), Issue 1, 1-48

Abstract:

$We study existence and regularity of solutions for nonlinear parabolic problems whose model is \begin{equation} \label{P} \begin{cases} u_t - \mbox{Div}(|\nabla u|^{p-2} \nabla u) = \beta(u) |\nabla u|^{p} + f & \mbox{in } \Omega \times \left ] 0, \infty \right [ , \\ u(x,t) = 0, & \mbox{on } \partial\Omega \times \left ] 0, \infty \right [ , \\ u(x,0) = u_0(x), & \mbox{in } \Omega, \end{cases} \end{equation} where $p > 1$ and $\Omega \subset \mathbb{R}^N$ is a bounded open set; as far as the function $\beta$ is concerned, we make no assumption on its sign; instead, we consider three possibilities of growth for $\beta$, which essentially are: (1) constant, (2) polynomial and (3) exponential. In each case, we assume appropriate hypotheses on the data $f$ and $u_0$, depending on the growth of $\beta $, and prove that a solution $u$ exists such that an exponential function of $u$ belongs to the natural Sobolev "energy" space. Since the solutions may well be unbounded, one cannot use sub/supersolution methods. However we show that, under slightly stronger assumptions on the data, the solution that we find is bounded. Our existence results, in the cases (2) and (3) above, rely on new logarithmic Sobolev inequalities.$