Green's function for second order parabolic systems with Neumann boundary condition (Q1943477)

This paper is concerned with Green's function for parabolic systems subject to Neumann boundary condition in \(Q=\Omega\times (-\infty,\infty)\), where \(\Omega\) is a domain in \(\mathbb R^n\), \(n\geq 1\). The authors prove that if the domain \(\Omega\) satisfies the multiplicative embedding inequality \[ \|u\|_{L^{2(n+2)/n}(\Omega)}\leq C \|Du\|^{n/(n+2)}_{L^2(\Omega)} \|u\|^{2/(n+2)}_{L^2(\Omega)} \] for some universal constant \(C>0\) and all \(u\in \widetilde W^{1,2}(\Omega)\) and if any weak solution satisfies an interior Hölder continuity estimate, then the Neumann Green's function exists and satisfies a natural growth estimate near the pole. As remarked by the authors, the multiplicative embedding inequality holds if \(\Omega\) is a Sobolev extension domain with finite Lebesgue measure, or an unbounded domain with compact Lipschitz boundary. Further, the authors establish global Gaussian bounds for the Neumann Green's function under the assumption that the weak solution with zero boundary data satisfies the local boundedness estimate.