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

A survey on modern theory of degenerate parabolic equations of the form ${u}_{t}=F\left(x,t,u,{u}_{x},{u}_{xx}\right)·$

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 Parabolic equations of degenerate type 35K15 Second order parabolic equations, initial value problems 35D05 Existence of generalized solutions of PDE (MSC2000) 35B40 Asymptotic behavior of solutions of PDE 35K60 Nonlinear initial value problems for linear parabolic equations