## Nonlinear Heat Equation with a Fractional Laplacian in a Disk

### Volume 81 / 1999

Colloquium Mathematicum 81 (1999), 101-122
DOI: 10.4064/cm-81-1-101-122

#### Abstract

For the nonlinear heat equation with a fractional Laplacian $u_t + (-Δ)^{α/2} u = u^2$, 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained.