Bounded Weak Solutions to a Class of Parabolic Equations with Gradient Term and $L^r{(0,T;L^q(\Omega))}$ Sources

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.