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Regularity of solutions for quasi-linear parabolic equations

Published online by Cambridge University Press:  22 January 2016

Yoshiaki Ikeda
Yoshiaki Ikeda*
Affiliation:
Aichi University of Education
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Let Ω be a bounded domain in n-dimensional Euclidian space En (n ≧ 2), and consider the space-time cylinder Q = Ω × (0, T] for some fixed T > 0. In this paper we deal with the Cauchy and Dirichlet problem for a second order quasi-linear equation

(1.1) ut — div A(x, t, u, ux) + B(x, t, u, ux) = 0 for (x, t) ∈ Q,

(1.2) u(x, 0) = (ϕ)(x) in Ω and u(x, t) = tψ(x, t) for (x, t) ∈ Γ = ∂Ω × (0, T] ,

where ∂Ω is a boundary of Ω which satisfies the following condition (A) : Condition (A). There exist constants ρ0 and »0 both in (0,1) such that, for any sphere K(ρ) with center on ∂Ω and radius ρ ≦ ρ0, the inequality meas [K(ρ) ∩ Ω] ≦ (1 — λ0) × meas E(ρ) holds, where meas E means the measure of a measurable set E.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 60 , February 1976 , pp. 151 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

References

[1] Aronson, D. G. and Serrin, J. : Local behavior of solutions of quasi-linear parabolic equations, Archive for Rational Mechanics and Analysis, 25 (1967), 81–122.Google Scholar
[2] Aronson, D. G. and Serrin, J. : A maximum principle for non-linear parabolic equations., Annali Scuola Normale Superiore di Pisa, 21 (1967), 291–305.Google Scholar
[3] Ladyzenskaya, O. A. and Ural’ceva, N. N.: A boundary value problem for linear and quasi-linear parabolic equations, Dokl. Akad. Nauk. SSSR, 139 (1964), 544–547.Google Scholar
[4] Serrin, J. : Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247–302.Google Scholar
[5] Stampacchia, G. : Le problème de Dirichlet pour les équations elliptiques de second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189–258.Google Scholar