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 University-Level Research Fund Project in Guizhou University of Finance and Economics (Grant No.2019XYB08).
 Author Name Affiliation Zhongqing LI School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guizhou 550025, P. R. China
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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.