Electron. J. Diff. Eqns., Vol. 2001(2001), No. 20, pp. 1-19.

Recent results and open problems on parabolic equations with gradient nonlinearities

Philippe Souplet

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.

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Philippe Souplet
Departement de Mathematiques, Universite de Picardie
INSSET, 02109 St-Quentin, France
Laboratoire de Mathematiques Appliquees
UMR CNRS 7641, Universite de Versailles
45 avenue des Etats-Unis, 78035 Versailles, France.
e-mail: souplet@math.uvsq.fr

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