Green's function for second order parabolic equations with singular lower order coefficients
Seick Kim, Longjuan Xu · Communications on Pure & Applied Analysis · 2021
We construct Green's functions for second order parabolic operators of the form \begin{document}$ Pu = \partial_t u-{\rm div}({\mathbf A} abla u+ {\mathbf b}u)+ {\mathbf c} \cdot abla u+du $\end{document} in \begin{document}$ (-\infty, \infty) \times \Omega $\end{document} , where \begin{document}$ \Omega $\end{document} is an open connected set in \begin{document}$ \mathbb{R}^n $\end{document} . It is not necessary that \begin{document}$ \Omega $\end{document} to be bounded and \begin{document}$ \Omega = \mathbb{R}^n $\end{document} is not excluded. We assume that the leading coefficients \begin{document}$ \mathbf A $\end{document} are bounded and measurable and the lower order coefficients \begin{document}$ \boldsymbol{b} $\end{document} , \begin{document}$ \boldsymbol{c} $\end{document} , and \begin{document}$ d $\end{document} belong to critical mixed norm Lebesgue spaces and satisfy the conditions \begin{document}$ d-{\rm div} \boldsymbol{b} \ge 0 $\end{document} and \begin{document}$ {\rm div}(\boldsymbol{b}-\boldsymbol{c}) \ge 0 $\end{document} . We show that the Green's function has the Gaussian bound in the entire \begin{document}$ (-\infty, \infty) \times \Omega $\end{document} .