## Advances in Differential Equations

- Adv. Differential Equations
- Volume 10, Number 9 (2005), 1007-1034.

### Existence and regularity results for solutions to nonlinear parabolic equations

Nathalie Grenon and Anna Mercaldo

#### Abstract

In this paper we prove some existence and regularity results for solutions to a class of nonlinear parabolic equations whose prototype is $$\left\{\begin{array}{lll} \displaystyle\frac{\partial u}{\partial t}-\Delta_p u=f(x,t) &\mbox{ in }Q,\cr u(x,0)=0 &\mbox{ in } \Omega,\cr u(x,t)=0 &\mbox{ on } \Gamma, \end{array}\right. $$ \noindent where $\Omega$ is a bounded open subset of ${{\mathbb R}^ N} $, $N\ge 2$, $Q$ is the cylinder $\Omega \times ]0,T[$, $T>0$, $\Gamma$ the lateral surface $\partial\Omega\times ]0,T[$, $\bigtriangleup _p$ is the so-called $p-$Laplace operator, $ p>1 $ and $f$ belongs to some space $ L^r (0,T;L^q(\Omega )),$ $r\geq 1$, $q\geq 1$.

#### Article information

**Source**

Adv. Differential Equations, Volume 10, Number 9 (2005), 1007-1034.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867815

**Mathematical Reviews number (MathSciNet)**

MR2161757

**Zentralblatt MATH identifier**

1100.35052

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35B65: Smoothness and regularity of solutions 35D10 35K20: Initial-boundary value problems for second-order parabolic equations

#### Citation

Grenon, Nathalie; Mercaldo, Anna. Existence and regularity results for solutions to nonlinear parabolic equations. Adv. Differential Equations 10 (2005), no. 9, 1007--1034. https://projecteuclid.org/euclid.ade/1355867815