Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-55wx7 Total loading time: 0.263 Render date: 2021-03-04T14:56:07.075Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Regularity of solutions for quasi-linear parabolic equations

Published online by Cambridge University Press:  22 January 2016

Yoshiaki Ikeda
Affiliation:
Aichi University of Education
Rights & Permissions[Opens in a new window]

Extract

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
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), 81122.CrossRefGoogle Scholar
[2] Aronson, D. G. and Serrin, J. : A maximum principle for non-linear parabolic equations., Annali Scuola Normale Superiore di Pisa, 21 (1967), 291305.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), 544547.Google Scholar
[4] Serrin, J. : Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247302.CrossRefGoogle Scholar
[5] Stampacchia, G. : Le problème de Dirichlet pour les équations elliptiques de second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189258.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 62 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 4th March 2021. This data will be updated every 24 hours.

Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Regularity of solutions for quasi-linear parabolic equations
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Regularity of solutions for quasi-linear parabolic equations
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Regularity of solutions for quasi-linear parabolic equations
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *