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Auscher, McIntosh and Tchamitchian studied the heat kernels of second orderelliptic operators in divergence form with complex bounded measurablecoefficients on $\mathbb{R}^n$. In particular, in the case when $n=2$ theyobtained Gaussian upper bound estimates for the heat kernel without imposingfurther assumption on the coefficients. We study the fundamental solutions ofthe systems of second order parabolic equations in the divergence form withbounded, measurable, time-independent coefficients, and extend their results tothe systems of parabolic equations.

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