Parabolic Differential Equations with Unbounded Coefficients – A Generalization of the Parametrix Method

We consider uniformly parabolic differential equations with unbounded first- and zero-order coefficients. A fundamental solution is constructed based on the classical parametrix method of E. Levi. From this the existence and uniqueness of the corresponding Cauchy problem is derived. Our approach does not require differentiable coefficients, as is usually assumed in the unbounded case. It only requires Hölder continuous coefficients. In this respect, our new proof also extends known results. We briefly discuss applications which make essential use of this extension.