## Advances in Differential Equations

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

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

#### 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