Advances in Differential Equations

Doubly degenerate parabolic equations with variable nonlinearity I: Existence of bounded strong solutions

S. Antontsev and S. Shmarev

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the homogeneous Dirichlet problem for the anisotropic parabolic equation with double degeneracy \begin{align*} \frac{d}{dt} (|v|^{m(x,t)} \operatorname{sign}\,v ) = \sum_{i=1}^{n}D_{i} ( a_{i}(x,t)|D_{i} v|^{p_{i}(x,t)-2}D_{i} v ) +b(x,t)|v|^{\sigma(x,t)-2}v+g(x,t). \end{align*} The exponents of nonlinearity $m(x,t)>0$, $p_{i}(x,t)>1$, and $\sigma(x,t)>1$ are given bounded continuous functions. It is proved that the problem has a bounded solution in a variable-exponent Sobolev space. The main existence result is local in time and is established under minimal restrictions on the low-order terms. It is shown that under further restrictions on $b$ and $\sigma(x,t)$ the constructed solution can be continued to the arbitrary time interval. The energy estimates are derived.

Article information

Source
Adv. Differential Equations Volume 17, Number 11/12 (2012), 1181-1212.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355702941

Mathematical Reviews number (MathSciNet)
MR3013415

Zentralblatt MATH identifier
06120664

Subjects
Primary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 35K67: Singular parabolic equations

Citation

Antontsev, S.; Shmarev, S. Doubly degenerate parabolic equations with variable nonlinearity I: Existence of bounded strong solutions. Adv. Differential Equations 17 (2012), no. 11/12, 1181--1212. https://projecteuclid.org/euclid.ade/1355702941.


Export citation