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Improved theory for a nonlinear degenerate parabolic equation. (English) Zbl 0702.35140

Consider the nonlinear degenerate parabolic equation

(1)u t =(a(u)) xx +(b(u)) x ,

where subscripts denote partial differentiation. The functions a and b are hypothesized to belong to C([0,))C 2 (0,) and be such that a ' (s)>0 for s>0, and a '' and b '' are locally Hölder continuous on (0,) and a(0)=0 and b(0)=0. In the present paper the author has established improved existence and uniqueness theorems for the Cauchy problem, the Cauchy-Dirichlet problem, and the first boundary value problem for equation (1). The comparison principles for generalized solutions of equation (1) and the relationship between the results given in this paper and those in earlier publications is also given.

Reviewer: B.G.Pachpatte

MSC:
35K65Parabolic equations of degenerate type
35K55Nonlinear parabolic equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35D05Existence of generalized solutions of PDE (MSC2000)
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)