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Zbl 0702.35140
Gilding, B.H.
Improved theory for a nonlinear degenerate parabolic equation. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 16, No.2, 165-224 (1989). ISSN 0391-173X

Consider the nonlinear degenerate parabolic equation $$ (1)\quad u\sb t=(a(u))\sb{xx}+(b(u))\sb x, $$ where subscripts denote partial differentiation. The functions a and b are hypothesized to belong to $C([0,\infty))\cap C\sp 2(0,\infty)$ and be such that $a'(s)>0$ for $s>0$, and $a''$ and $b''$ are locally Hölder continuous on (0,$\infty)$ 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.
[B.G.Pachpatte]

MSC 2000:: *35K65 Parabolic equations of degenerate type
35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE)
35D05 Existence of generalized solutions of PDE
35B05 General behavior of solutions of PDE

Keywords: existence and uniqueness theorems; Cauchy problem; Cauchy-Dirichlet problem; boundary value problem; comparison principles; generalized solutions

Cited in: Zbl 0722.35049

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