Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
The aim of the paper is to study the nonlinear heat equation with a fractional Laplacian NEWLINE\[NEWLINEu_t+ (-\Delta)^{\alpha/2}n= u^2,\quad 0< \alpha\leq 2,NEWLINE\]NEWLINE in a unit ball. Homogeneous boundary conditions and small initial conditions are considered. The paper concentrates on the higher-order long time asymptotics. The first initial-boundary value problem is discussed in a unit ball and its solutions are constructed in the form of an eigenfunction expansion series, proving the well-posedness. The higher-order long time asymptotics are determined by means of nonlinear iterations. It is pointed out that the success of constructing solutions depends greatly on the convergence of the spatial eigenfunctions series and this convergence is rather poor.
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