\begin{section}*{\it Abstract. } We study existence and regularity of solutions for nonlinear parabolic problems whose model is \begin{equation}\label{P} \left\{ \begin{array}{ll} u_t -{\rm div} ( |\nabla u|^{p-2}\nabla u) = \beta (u)|\nabla u|^{p} +f \qquad &{\rm in} \ \Omega \times ]0,\infty[\\ \\ u(x, t) = 0, &{\rm on} \ \partial \Omega \times ]0,\infty[\\ \\ u(x, 0) = u_0, &{\rm in} \ \Omega \end{array} \right. \end{equation} where $p>1$ and $\Omega \subset {\bf R}^N$ 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. \end{section}
Global existence for parabolic problems involving the p-Laplacian and a critical gradient term / Dall'Aglio, Andrea; Giachetti, Daniela; SEGURA DE LEON, S.. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - STAMPA. - 58, no.1:(2009), pp. 1-48. [10.1512/iumj.2009.58.3409]
Global existence for parabolic problems involving the p-Laplacian and a critical gradient term
DALL'AGLIO, Andrea;GIACHETTI, Daniela;
2009
Abstract
\begin{section}*{\it Abstract. } We study existence and regularity of solutions for nonlinear parabolic problems whose model is \begin{equation}\label{P} \left\{ \begin{array}{ll} u_t -{\rm div} ( |\nabla u|^{p-2}\nabla u) = \beta (u)|\nabla u|^{p} +f \qquad &{\rm in} \ \Omega \times ]0,\infty[\\ \\ u(x, t) = 0, &{\rm on} \ \partial \Omega \times ]0,\infty[\\ \\ u(x, 0) = u_0, &{\rm in} \ \Omega \end{array} \right. \end{equation} where $p>1$ and $\Omega \subset {\bf R}^N$ 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. \end{section}I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.