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The first initial-boundary value problem for quasilinear second order parabolic equations. (English) Zbl 0655.35047
In a fundamental work by J. Serrin [Philos. Trans. R. Soc. Lond. Ser. A 264, 413-496 (1969; Zbl 0181.380)], he obtained very general conditions under which the Dirichlet problem for a quasilinear elliptic equation with arbitrary smooth boundary values is solvable in a given domain; he also showed that these conditions were sharp in that the problem is not solvable for some (infinitely differentiable) boundary values when they are violated. The purpose of this paper is to present sharp parabolic analogs of Serrin’s existence and non-existence results. The elliptic methods carry over to the parabolic setting with only minor changes. This fact is apparent by N. S. Trudinger [Math. J., Indiana Univ. 21, 657-670 (1972; Zbl 0236.35022)] but not by A. V. Ivanov [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 38, 10-32 (1973; Zbl 0345.35054)], so we shall repeat many of the details. Moreover we consider somewhat less smooth initial and boundary values (corresponding to the regularity of boundary values for elliptic problems by the author [Commun. Partial Differ. Equations 6, 437-497 (1981; Zbl 0458.35039)] than would be used in a strict analog of Serrin’s theory.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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