Gaussian estimates for fundamental solutions to certain parabolic systems

Abstract

Auscher proved Gaussian upper bound estimates for the fundamental solutions to parabolic equations with complex coefficients in the case when coefficients are time-independent and a small perturbation of real coefficients. We prove the equivalence between the local boundedness property of solutions to a parabolic system and a Gaussian upper bound for its fundamental matrix. As a consequence, we extend Auscher's result to the time dependent case.

On the Green's matrices of strongly parabolic systems of second order

“…Here, we consider the case R c = ∞ and prove (2.40). In [14], by following methods of Davies [4] and Fabes-Stroock [9], Hofmann and Kim derived the upper Gaussian bound of Aronson [1] under a further qualitative assumption that the coefficients of L are smooth. For the reader's convenience, we reproduce their argument here, dropping the technical assumption that the coefficients are smooth.…”

“…In [2], Auscher gave a new proof of Aronson's Gaussian upper bound for the fundamental solution of parabolic equations with time independent coefficients, which carries over to the case of a complex perturbation of real coefficients. Recently, it is noted in [14] that Aronson's upper bound is equivalent to the local boundedness property of weak solutions of strongly parabolic systems. Green's functions of elliptic equations of divergence form in bounded domains have been extensively studied by Littman, Stampacchia, and Weinberger [20] and Grüter and Widman [12], whereas the Green's matrices of the elliptic systems with continuous coefficients in bounded C 1 domains have been discussed by Fuchs [11] and Dolzmann and Müller [5].…”

Global Estimates for Green’s Matrix of Second Order Parabolic Systems with Application to Elliptic Systems in Two Dimensional Domains

“…We may assume that is a bounded Lipschitz domain with a sufficiently small Lipschitz constant, and ω ρ ( A αβ ) is also sufficiently small for some ρ > 0; see e.g., [11]. Therefore, to prove estimate (3.2), we only need to consider the case when t > s. To derive the estimate (3.2), we modify the proof in [8, §5.1], which was based on that in [24]. We mention that the method in [24] was in turn based on the ideas appeared in [9] and [15].…”

Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients

“…Theorem 2.8 (Theorem 1.1, [12]). Assume that the system (2.4) and its adjoint system (2.5) satisfy the local boundedness property for weak solutions.…”