For the splitting method of the nonlinear heat equation with initial datum in W^1, q.

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in Ω ⊂ R (d ≥ 1): { ∂tu = ∆u+ λ|u| u in Ω× (0,∞), u = 0 in ∂Ω× (0,∞), u(x, 0) = φ(x) in Ω, where λ ∈ {−1, 1} and φ ∈ W (Ω) ∩ L∞(Ω) with 2 ≤ p < ∞ and d(p − 1)/2 < q < ∞. We establish the well-posedness of the approximation of u in L-space (r ≥ q), and furthermore, we derive its convergence rate of order O(τ) for a time step τ > 0. Finally, we give some numerical examples to confirm the reliability of the analyzed result.