We construct Green’s functions for second order parabolic operators of the form Pu = ∂tu−div(A∇u+bu)+c·∇u+du in (−∞, ∞)×Ω, where Ω is an open connected set in Rn. It is not necessary that Ω to be bounded and Ω = Rn is not excluded. We assume that the leading coefficients A are bounded and measurable and the lower order coefficients b, c, and d belong to critical mixed norm Lebesgue spaces and satisfy the conditions d − divb ≥ 0 and div(b−c) ≥ 0. We show that the Green’s function has the Gaussian bound in the entire (−∞, ∞) × Ω.
Bibliographical noteFunding Information:
S. Kim was partially supported by the National Research Foundation of Korea under agreements NRF-2019R1A2C2002724 and NRF-20151009350. ∗ Corresponding author.
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All Science Journal Classification (ASJC) codes
- Applied Mathematics