Existence and regularity results of parabolic problems with convection term and singular nonlinearity

Mounim El Ouardy; Youssef El Hadfi; Abdelaaziz Sbai

doi:10.4171/zaa/1769

JournalszaaVol. 43, No. 3/4pp. 299–328

Existence and regularity results of parabolic problems with convection term and singular nonlinearity

Mounim El Ouardy
Sultan Moulay Slimane University, Khouribga, Morocco
Youssef El Hadfi
Sultan Moulay Slimane University, Khouribga, Morocco
Abdelaaziz Sbai
Sultan Moulay Slimane University, Khouribga, Morocco

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Abstract

In this work, we investigate the influence of the convection term and the singular lower order term on the existence and regularity of solutions to the following parabolic problem:

⎩ ⎨ ⎧ \frac{\partial u}{\partial t} - div (M (x, t) \nabla u) = - div (u E (x, t)) + \frac{f}{u ^{θ}} u (x, t) = 0 u (x, 0) = u_{0} (x) in Ω \times (0, T), on \partial Ω \times (0, T), in Ω,

where $θ > 0$ , $Ω \subset R^{N} (N > 2)$ is a bounded smooth domain with $0 \in Ω$ , and $f \in L^{m} (Ω \times (0, T))$ with $m \geq 1$ is a non-negative function. The function $u_{0}$ is a non-negative function that belongs to the space $L^{\infty} (Ω)$ such that

\forall ω \subset\subset Ω, \exists c_{ω} > 0, u_{0} \geq c_{ω} in ω .

The main idea of this research explains the combined impact of the convection term and the singular lower order term on the existence and regularity of a solution to the above problem.

Cite this article

Mounim El Ouardy, Youssef El Hadfi, Abdelaaziz Sbai, Existence and regularity results of parabolic problems with convection term and singular nonlinearity. Z. Anal. Anwend. 43 (2024), no. 3/4, pp. 299–328

DOI 10.4171/ZAA/1769