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Gaussian bounds for the fundamental solutions of \(\nabla(A\nabla u)+B\nabla u-u_ t=0\). (English) Zbl 0886.35004

Summary: We study the parabolic equation \(\nabla(A\nabla u)+B\nabla u-u_t=0\), where \(B=B(x,t)\) is in a class of singular functions more general than the \(L^{p,q}\) class in Aronson’s paper [D. G. Aronson, Ann. Sc. Norm. Super. Pisa 22, 607-694 (1968; Zbl 0182.13802)]. We prove the existence of Gaussian bounds for the fundamental solutions. As a corollary, we show that if \(|B|\in K_{n,1}\) and \(B=B(x)\) is compactly supported, then the heat kernel of \(\Delta+B\nabla\) has Gaussian upper and lower bound. B. Simon [Bull. Am. Math. Soc. 7, 447-526 (1982; Zbl 0524.35002)] obtained Gaussian bounds when the zero order term is singular. However, the case of singular drift terms (corresponding to magnetic fields) has been open. This question is settled by the corollary. In addition, we give a condition so that the upper bound is global in time.

MSC:

35A08 Fundamental solutions to PDEs
35K10 Second-order parabolic equations
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References:

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