Green's function for second order parabolic equations with singular lower order coefficients

Abstract

We construct Green's functions for second order parabolic operators of the form $Pu={\partial }_{t}u-\mathrm{div}(\mathbf{A}\nabla u+\mathbf{b}u)+\mathbf{c}\cdot \nabla u+du$ in $(-\infty ,\infty )\times \Omega$, where $\Omega$ is an open connected set in ${\mathbb{R}}^n$. It is not necessary that $\Omega$ to be bounded and $\Omega ={\mathbb{R}}^n$ is not excluded. We assume that the leading coefficients $\mathbf{A}$ are bounded and measurable and the lower order coefficients $\mathbf{b}$, $\mathbf{c}$, and $d$ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $d-\mathrm{div}\mathbf{b}\ge 0$ and $\mathrm{div}(\mathbf{b}-\mathbf{c})\ge 0$. We show that the Green's function has the Gaussian bound in the entire $(-\infty ,\infty )\times \Omega$.

20 pages

References in corpus (2)

• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• On the Green's matrices of strongly parabolic systems of second order