Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 60, pp. 1-20.
Title: Solutions of nonlinear parabolic equations without growth
restrictions on the data
Authors: Lucio Boccardo (Univ. di Roma 1, Roma, Italy)
Thierry Gallouet (Univ. de Marseille I, France)
Juan Luis Vazquez (Univ. Autonoma de Madrid, Spain)
Abstract:
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.
Submitted July 17, 2001. Published September 12, 2001.
Math Subject Classifications: 35K55, 35K65.
Key Words: Nonlinear parabolic equations; global existence;
growth conditions.