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

摘要

We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish also non-asymptotic, rapidly decreasing at infinity, upper and lower estimates for the fundamental solution. We extend the classical parametrix method of E.E. Levi.