Existence and uniqueness of solutions to the Kuramoto-Sakaguchi nonlinear parabolic integrodifferential equation. (English) Zbl 0997.35029

This paper deals with an initial-boundary value problem for a nonlinear parabolic equation with periodic boundary conditions on a line segment where the nonlinear component has an integral form. The authors prove uniqueness and global existence in time of classical solutions for the equations under consideration. They derive smoothness, regularity, and time-independent estimates for all derivatives of classical solutions. In particular, they study this integro-differential equation as a model that describes the dynamical behaviour of a system of infinitely many nonlinearly coupled random oscillators.


35K55 Nonlinear parabolic equations
45K05 Integro-partial differential equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35R10 Partial functional-differential equations