On a Convection Diffusion Equation with Absorption Term

Abstract

The paper studies the posedness of the convection diffusion equation

$$\begin{aligned} u_{t}=\text {div}\left( \left| \nabla u^{m}\right| ^{p-2}\nabla u^{m}\right) +\sum _{i=1}^{N}\frac{\partial b_{i}\left( u^{m}\right) }{\partial x_{i}}-u^{mr}. \end{aligned}$$

with homogeneous boundary condition and with the initial value $u_0(x)\in L^{q-1+\frac{1}{m}}(\Omega )$. By considering its regularized problem, using Moser iteration technique, the local bounded properties of the $L^{\infty }$-norm of $u_{k}$ and that of the $L^{p}$-norm of the gradient $\nabla u_{k}$ are got, where $u_{k}$ is the solution of the regularized problem of the equation. By the compactness theorem, the existence of the solution of the equation itself is obtained. By using some techniques in Zhao and Yuan (Chin Ann Math A 16(2):179–194, 1995), the stability of the solutions is obtained too.