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

Consider the nonlinear degenerate parabolic equation \[ (1)\quad u_ t=(a(u))_{xx}+(b(u))_ x, \] where subscripts denote partial differentiation. The functions a and b are hypothesized to belong to \(C([0,\infty))\cap C^ 2(0,\infty)\) 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.
Reviewer: B.G.Pachpatte

MSC:

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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