Some problems of the qualitative theory of nonlinear degenerate second- order parabolic equations. (English. Russian original) Zbl 0642.35047

Kalashnikov, A. S.

Russ. Math. Surv. 42, No. 2, 169-222 (1987); translation from Usp. Mat. Nauk 42, No. 2(254), 135-176 (1987).

A survey on modern theory of degenerate parabolic equations of the form \(u_ t=F(x,t,u,u_ x,u_{xx}).\)
Investigations on such equations are important, because the equation of non-Newtonian filtration, the equation of rapid diffusion, the equations of nonlinear heat conduction and some other ones contain degeneracies.
The principal objects of investigation are generalized solutions. Chapter 1 is discussing problems of their existence and uniqueness. Estimates and regularity problems are considered in chapter 2. Short time and long time behavior of solutions is described in chapters 3 and 4. A list of about 300 references is presented.

Reviewer: I.Dorfman

MSC:

35K65	Degenerate parabolic equations
35K15	Initial value problems for second-order parabolic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
35B40	Asymptotic behavior of solutions to PDEs
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations

Keywords:

bibliography; survey; degenerate; filtration; rapid diffusion; nonlinear heat conduction; generalized solutions; existence; uniqueness; Estimates; regularity; time behavior