# Pure and Applied Mathematics Quarterly

## Volume 14 (2018)

### 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

Abdelmoujib Benkirane (Laboratory LAMA, Department of Mathematics, Faculty of Sciences, Sidi Mohamed Ben Abdellah University, Fez, Morocco)

Badr El Haji (Laboratory LAMA, Department of Mathematics, Faculty of Sciences, Sidi Mohamed Ben Abdellah University, Fez, Morocco)

Mostafa El Moumni (Department of Mathematics, Faculty of Sciences, University Chouaib Doukkali, El Jadida, Morocco)

#### 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