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.