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

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: \begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}(M(x,t)\nabla u) =-\operatorname{div}(uE(x,t))+\frac{f}{u^{\theta}}&\text{in }\Omega\times(0,T),\\ u(x,t)=0&\text{on }\partial \Omega\times (0,T),\\ u(x,0)=u_{0}(x)&\text{in }\Omega,\end{cases} where \theta>0 , \Omega\subset \mathbb{R}^{N}\ (N>2) is a bounded smooth domain with 0\in \Omega , and f\in L^{m}(\Omega\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}(\Omega) such that \forall \omega\subset\subset \Omega,\ \exists c_{\omega}>0,\quad u_{0}\geq c_{\omega}\text{ in }\omega. 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.