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# Pure and Applied Mathematics Quarterly

## Volume 14 (2018)

### Number 3-4

### On the existence of solution for degenerate parabolic equations with singular terms

Pages: 591 – 606

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n3.a8

#### Authors

#### Abstract

We are interested in results concerning the solutions to the parabolic problems whose simplest model is the following:

\begin{cases}\frac{\partial u}{\partial t} - \Delta_p u (:= \operatorname{div} ({\lvert \nabla u \rvert}^{p-2} \nabla u)) + B \frac{{\lvert \nabla u \rvert}^p}{u} = f & \textrm{in} & (0, T) \times \Omega , \\u (0, x) = u_0 (x) & \textrm{in} & \Omega , \\u (t, x) = 0 & \textrm{on} & (0, T) \times \partial \Omega , \\\end{cases}

where $T\gt 0$, $N \geq 2$, $B \gt 0$, $u_0$ is a positive function in $L^{\infty} (\Omega)$ bounded away from zero, and $f$ is a nonnegative function that belongs to some Lebesgue space.

#### Keywords

nonlinear parabolic equations, singular parabolic equations, Sobolev space

#### 2010 Mathematics Subject Classification

35K55, 35K65, 35K67

Received 12 January 2019

Received revised 13 May 2019

Accepted 3 June 2019

Published 5 November 2019