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We prove the existence of a fundamental solution of the Cauchy initialboundary value problem on the whole space for a parabolic partial differentialequation with discontinuous unbounded first-order coefficient at the origin. Weestablish also non-asymptotic, rapidly decreasing at infinity, upper and lowerestimates for the fundamental solution. We extend the classical parametrixmethod provided by E.E. Levi.

Fundamental solution for Cauchy initial value problem for parabolic PDEs  with discontinuous unbounded first-order coefficient at the origin. Extension  of the classical parametrix method