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Zbl 0642.35047
Kalashnikov, A.S.
Some problems of the qualitative theory of nonlinear degenerate second- order parabolic equations. (English. Russian original)
[J] Russ. Math. Surv. 42, No.2, 169-222 (1987); translation from Usp. Mat. Nauk 42, No.2(254), 135-176 (1987). ISSN 0036-0279

A survey on modern theory of degenerate parabolic equations of the form $u\sb t=F(x,t,u,u\sb x,u\sb{xx}).$ \par 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. \par 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.
[I.Dorfman]

MSC 2000:: *35K65 Parabolic equations of degenerate type
35K15 Second order parabolic equations, initial value problems
35D05 Existence of generalized solutions of PDE
35B40 Asymptotic behavior of solutions of PDE
35K60 (Nonlinear) BVP for (non)linear parabolic equations

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

Cited in: Zbl 0818.35015

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