Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball

The nonlinear heat equation with a fractional Laplacian ${\displaystyle [u_t+{(-\Delta )}^{\frac{\alpha }{2}}u=u^2,0<\alpha \le 2]}$, is considered in a unit ball ${\displaystyle B}$. Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space ${\displaystyle C⁰([0,\infty ),H{₀}^{\kappa }\left(B\right))}$ with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.