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