Parabolic equations with changing time direction

Abstract

We study the parabolic equations with a changing time direction using the theory of singular integral equations with the Cauchy kernel, as well as behavior of the Cauchy type integral at the ends of the integration contour and at the density discontinuity points in Holder spaces Hx tp,p/2n. Along with the smoothness of the problems data, the theory of singular equations makes it possible to find additional necessary and sufficient conditions ensuring that the solution belongs to the Holder spaces with p > 2n. Moreover, a unified approach applied under general matching conditions for such equations shows that noninteger values of p – [p] in can have a large effect on both the number of solvability conditions and the smoothness of the desired solution to the equation.We study the parabolic equations with a changing time direction using the theory of singular integral equations with the Cauchy kernel, as well as behavior of the Cauchy type integral at the ends of the integration contour and at the density discontinuity points in Holder spaces Hx tp,p/2n. Along with the smoothness of the problems data, the theory of singular equations makes it possible to find additional necessary and sufficient conditions ensuring that the solution belongs to the Holder spaces with p > 2n. Moreover, a unified approach applied under general matching conditions for such equations shows that noninteger values of p – [p] in can have a large effect on both the number of solvability conditions and the smoothness of the desired solution to the equation.