Opuscula Math. 44, no. 4 (2024), 471-503

Opuscula Mathematica

Degenerate singular parabolic problems with natural growth

Mounim El Ouardy
Youssef El Hadfi
Abdelaaziz Sbai

Abstract. In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term \[\begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^{q})|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^{p}}{u^{\gamma}}=f & \text{ in } Q,\\ u(x,t)=0 & \text{ on } \Gamma, \\ u(x,t=0)=u_{0}(x) & \text{ in } \Omega, \end{cases}\] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^{N}\), \(N\gt 2\), \(Q\) is the cylinder \(\Omega \times (0,T)\), \(T\gt 0\), \(\Gamma\) the lateral surface \(\partial \Omega \times (0,T)\), \(2\leq p\lt N\), \(a(x,t)\) and \(b(x,t)\) are positive measurable bounded functions, \(q\geq 0\), \(0\leq\gamma\lt 1\), and \(f\) non-negative function belongs to the Lebesgue space \(L^{m}(Q)\) with \(m\gt 1\), and \(u_{0}\in L^{\infty}(\Omega)\) such that \[\forall\omega\subset\subset\Omega\, \exists D_{\omega}\gt 0:\, u_{0}\geq D_{\omega}\text{ in }\omega.\] More precisely, we study the interaction between the term \(u^{q}\) (\(q>0\)) and the singular lower order term \(d(x,t)|\nabla u|^{p}u^{-\gamma}\) (\(0\lt\gamma\lt 1\)) in order to get a solution to the above problem. The regularizing effect of the term \(u^q\) on the regularity of the solution and its gradient is also analyzed.

Keywords: degenerate parabolic equations, singular parabolic equations, natural growth term.

Mathematics Subject Classification: 35A25, 35B45, 35B09, 35D30, 35K65, 35K67.

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  • Mounim El Ouardy (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-9742-0754
  • Sultan Moulay Slimane University, National School of Applied Sciences Khouribga, BP 77 Bd Beni Amir, Khouribga 25000, Morocco
  • Communicated by J.I. Díaz.
  • Received: 2023-05-07.
  • Revised: 2023-11-22.
  • Accepted: 2024-01-08.
  • Published online: 2024-04-29.
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Cite this article as:
Mounim El Ouardy, Youssef El Hadfi, Abdelaaziz Sbai, Degenerate singular parabolic problems with natural growth, Opuscula Math. 44, no. 4 (2024), 471-503, https://doi.org/10.7494/OpMath.2024.44.4.471

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