Global solutions and smoothing effects for semi-linear evolution equations in circular domains

Abstract

The paper deals with global-in-time solutions of semi-linear evolution equations in circular domains. These solutions are constructed by means of the series of eigenfunctions of the Laplace operator in the corresponding region. A forced Cahn–Hilliard-type equation in a unit disc Ω is considered as an example. The current work focuses on revealing the mechanism of nonlinear smoothing, i.e., on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem. To this end convolutions of Rayleigh functions with respect to the Bessel index are employed. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424; V. Varlamov, Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains, J. Math. Anal. Appl. 327 (2007) 1461–1478]. In order to reveal the effect of nonlinear smoothing, anisotropic Sobolev spaces H s , α ( Ω ) are introduced. They are based on the Sobolev spaces H s ( Ω ) “weighted” by tangential derivatives, so that the index α is responsible for the smoothness in θ and s is the usual Sobolev index. Global-in-time solutions of the mixed problem in question are constructed, and additional smoothness with respect to the angular coordinate is established in the anisotropic Sobolev spaces. The above mentioned special functions are essentially used for this purpose.