Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 20, pp. 1-19.

Title:  Recent results and open problems on parabolic equations
        with gradient nonlinearities
        
Author: Philippe  Souplet (Univ.  de Picardie, France)
         
Abstract: 
 We survey recent results and present a number of open problems
 concerning the large-time behavior of solutions of semilinear parabolic
 equations with gradient nonlinearities.
 We focus on the model equation with a dissipative gradient term
 $$u_t-\Delta u=u^p-b|\nabla u|^q,$$
 where $p$, $q>1$, $b>0$, with homogeneous Dirichlet boundary conditions.
 Numerous papers were devoted to this equation in the last ten years,
 and we compare the results with those known for the case of the pure
 power reaction-diffusion equation ($b=0$). In presence of
 the dissipative gradient term a number of new phenomena appear
 which do not occur when $b=0$. The questions treated
 concern: sufficient conditions for blowup, behavior of blowing up
 solutions, global existence and stability, unbounded
 global solutions, critical exponents, and stationary states.
 
Submitted February 19, 2001. Published March 26, 2001.
Math Subject Classifications: 35K55, 35B35, 35B40, 35B33, 35J60.
Key Words: nonlinear parabolic equations; gradient term;
           finite time blowup;  global existence.