Bounded Weak Solutions to a Class of Parabolic Equations with Gradient Term and $L^r{(0,T;L^q(\Omega))}$ Sources 
Received:April 16, 2021 Revised:June 21, 2021 
Key Word:
parabolic equations lower order gradient term $L^\infty$ estimate bounded solutions

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11901131) and the UniversityLevel Research Fund Project in Guizhou University of Finance and Economics (Grant No.2019XYB08). 

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Abstract: 
We consider a class of nonlinear parabolic equations whose prototype is $$\begin{cases}u_t\Delta u=\overrightarrow{b}(x,t)\cdot\nabla u+\gamma\nabla u^2\text{div}{\overrightarrow{F}(x,t)}+f(x,t), &(x,t)\in \Omega_T,\\ u(x,t)=0,&(x,t)\in\Gamma_T,\\ u(x,0)=u_0(x), &x\in\Omega, \end{cases}$$ where the functions $\overrightarrow{b}(x,t)^2,\overrightarrow{F}(x,t)^2,f(x,t)$ lie in the space $L^r{(0,T;L^q(\Omega))}$, $\gamma$ is a positive constant. The purpose of this paper is to prove, under suitable assumptions on the integrability of the space $L^r{(0,T;L^q(\Omega))}$ for the source terms and the coefficient of the gradient term, a priori $L^\infty$ estimate and the existence of bounded solutions. The methods consist of constructing a family of perturbation problems by regularization, Stampacchia's iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process. 
Citation: 
DOI:10.3770/j.issn:20952651.2022.03.006 
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