Some remarks on the abstract Cauchy problem

Inhomogeneous Two-Parameter Abstract Cauchy Problem

Some remarks about Cauchy integrals

A basic theme in the wonderful books and surveys of Stein, Weiss, and Zygmund is that Hilbert transforms, Poisson kernels, heat kernels, and related objects are quite interesting and fundamental. I certainly like this point of view. There is a variety of ways in which things can be interesting or fundamental, of course. In the last several years there have been striking developments connected t...

Remarks on the Well-Posedness of the Nonlinear Cauchy Problem

We show that hyperbolicity is a necessary condition for the well posedness of the noncharacteristic Cauchy problem for nonlinear partial differential equations. We give conditions on the initial data which are necessary for the existence of solutions and we analyze Hadamard’s instabilities in Sobolev spaces. We also show that genuinely nonlinear equations raise new interesting problems.

inhomogeneous two-parameter abstract cauchy problem

A singular abstract cauchy problem.

In this note a singular abstract Cauchy problem is considered. A solution is obtained for this problem in terms of a regular abstract Cauchy problem. As an application we obtain a new solution of the initial value problem for a class of singular partial differential equations.