## Differential and Integral Equations

- Differential Integral Equations
- Volume 28, Number 11/12 (2015), 1155-1172.

### Parabolic equations with singular and supercritical reaction terms

Lucio Boccardo, Miguel Escobedo, and Maria Michaela Porzio

#### Abstract

We prove that, for every $\lambda \gt 0 $ and $u_0 \geq 0$, the following evolution problem has a solution for small value of $T$, \begin{equation} \label{zbep1} \begin{cases} u > 0 \text{ in }\;\Omega_T = \Omega \times (0,T) \;, \\ \displaystyle \; u_t - \div (M(x)\nabla u)= \frac{\lambda}{u^\gamma}+ \mu \, u^{p} \; \text{ in }\;\Omega_T, \\ u=0 \text{ on }\;\partial\Omega \times (0,T),\\ u(0)=u_0 \text{ in }\; \Omega, \end{cases} \end{equation} where $\gamma > 0$, $\mu\geq0$ and $p > 0$. Moreover, we show the existence of a solution for every value of $T$ for suitable small data $\lambda$ and $u_0$ if $p > 1$ and for every data if $0 < p < 1$.

#### Article information

**Source**

Differential Integral Equations, Volume 28, Number 11/12 (2015), 1155-1172.

**Dates**

First available in Project Euclid: 18 August 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1439901045

**Mathematical Reviews number (MathSciNet)**

MR3385138

**Zentralblatt MATH identifier**

1374.35193

**Subjects**

Primary: 35K10: Second-order parabolic equations 35K58: Semilinear parabolic equations 35K67: Singular parabolic equations

#### Citation

Boccardo, Lucio; Escobedo, Miguel; Porzio, Maria Michaela. Parabolic equations with singular and supercritical reaction terms. Differential Integral Equations 28 (2015), no. 11/12, 1155--1172. https://projecteuclid.org/euclid.die/1439901045