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Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients. (English) Zbl 1156.35005

In 1967 Aronson proved Gaussian estimates for the fundamental solution of parabolic equations in divergence form by using the Harnack inequality established by Moser. In 1985 Fabes and Stroock, by using Nash’s ideas, proved Gaussian estimates and from them they derived Moser’s estimates. In 1996 Auscher gave an alternative proof that can be applied also in the case of parabolic equation with complex coefficients (provided that they are a small perturbation of real coefficients). In 1995 Auscher, McIntosh and Tchamitichian proved that the heat kernel of second order elliptic operators in divergence form with complex bounded measurable coefficients in two dimensions satisfies a Gaussian upper bound. Starting from such results, in this paper the author considers parabolic systems in divergence form in two dimensions with time independent coefficients and shows that its fundamental solutions enjoy a Gaussian upper bound.

MSC:

35A08 Fundamental solutions to PDEs
35B45 A priori estimates in context of PDEs
35K40 Second-order parabolic systems
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