Solution-giving formula to Cauchy problem for multidimensional parabolic equation with variable coefficients

TL;DR

This paper introduces a general method for solving multidimensional parabolic equations with variable coefficients, providing explicit solution representations and conditions for existence and smoothness of solutions.

Contribution

It develops a solution formula for the Cauchy problem using $C_0$-semigroups and Feynman formulas, extending the theoretical framework for variable coefficient parabolic equations.

Findings

Existence of a contraction $C_0$-semigroup under smoothness and boundedness conditions.

Representation of solutions via Feynman formulas as limits of multiple integrals.

Conditions ensuring the solution's smoothness and existence.

Abstract

We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the $C_0$-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and then find the representation (based on the family of translation operators) of the solution in terms of coefficients of the equation and initial condition. It is proved that if the coefficients of the equation are bounded, infinitely smooth and satisfy some other conditions then there exists a solution-giving $C_0$-semigroup of contraction operators. We also represent the solution as a Feynman formula (i.e. as a limit of a multiple integral with multiplicity tending to infinity) with generalized functions appearing in the integral kernel.