Extremal case of parabolic differential equations having discontinuous unbounded coefficients. Existence of fundamental solution for an initial Cauchy problem. Parametrix method

Abstract

We prove in this short report the existence of a fundamental solution (F.S.) for the Cauchy initial boundary problem on the whole space for the parabolic differential equation having at origin the point of non-integrable unbounded discontinuity for coefficient before a first order derivative. We give also the non-asymptotic rapidly decreasing at infinity estimate for these function. We extend the classical parametrix method offered by E.E.Levi.