TY  - JOUR
ID  - ElOuardy2024
LB  - ElOuardy2024
AU  - El Ouardy, Mounim
AU  - El Hadfi, Youssef
AU  - Sbai, Abdelaaziz
TI  - Degenerate singular parabolic problems with natural growth
ST  - Degenerate singular parabolic problems with natural growth
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2024
VL  - 44
IS  - 4
SP  - 471
EP  - 503
DO  - https://doi.org/10.7494/OpMath.2024.44.4.471
UR  - https://doi.org/10.7494/OpMath.2024.44.4.471
AB  - 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.
KW  - degenerate parabolic equations
KW  - singular parabolic equations
KW  - natural growth term
SN  - 1232-9274
ER  -