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

✍ Scribed by Vladimir Varlamov

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
410 KB
Volume
69
Category
Article
ISSN
0362-546X

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✦ Synopsis

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-Hilliardtype 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,

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## Abstract As a basic example, we consider the porous medium equation (__m__ > 1) equation image where Ξ© βŠ‚ ℝ^__N__^ is a bounded domain with the smooth boundary βˆ‚Ξ©, and initial data $u\_0 \thinspace \varepsilon L^{\infty} \cap L^{1}$. It is well‐known from the 1970s that the PME admits separable