# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

### Tom 142 / 2000

Studia Mathematica 142 (2000), 71-99 DOI: 10.4064/sm-142-1-71-99

#### Streszczenie

The nonlinear heat equation with a fractional Laplacian $[u_t+(-Δ)^{α/2} u = u^2, 0 < α ≤ 2]$, is considered in a unit ball $B$. 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 $C⁰([0,∞), H₀^{κ}(B))$ 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.