Solutions of nonlinear parabolic equations without growth restrictions on the data

Boccardo, Lucio ; Gallouët, Thierry ; Vazquez, Juan Luis

Electronic Journal of Differential Equations, Tome 2001 (2001).

Résumé

Summary: The purpose of this paper is to prove the existence of solutions for certain types of nonlinear parabolic partial differential equations posed in the whole space, when the data are assumed to be merely locally integrable functions, without any control of their behaviour at infinity. A simple representative example of such an equation is $$ u_t-\Delta u + |u|^{s-1}u=f, $$ which admits a unique globally defined weak solution $u(x,t)$ if the initial function $u(x,0)$ is a locally integrable function in $\mathbb{R}^N, N\geq 1$, and the second member $f$ is a locally integrable function of $x\in\mathbb{R}^N$ and $t\in [0,T]$ whenever the exponent $s$ is larger than 1. The results extend to parabolic equations results obtained by Brezis and by the authors for elliptic equations. They have no equivalent for linear or sub-linear zero-order nonlinearities.

Détail

Classification : 35K55, 35K65
Keywords: nonlinear parabolic equations, global existence, growth conditions