In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\Omega\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$ in $\Omega\times(0,\infty)$, $u=0$ in $\partial\Omega\times(0,\infty)$, $u ({\bf x},0) =\phi ({\bf x})$ in $\Omega$. where $\lambda\in\{-1,1\}$ and $\phi \in W^{1,q}(\Omega)\cap L^{\infty} (\Omega)$ with $2\leq p < \infty$ and $d(p-1)/2<q<\infty$. We establish the well-posedness of the approximation of $u$ in $L^r$-space ($r\geq q$), and furthermore, we derive its convergence rate of order $\mathcal{O}(\tau)$ for a time step $\tau>0$. Finally, we give some numerical examples to confirm the reliability of the analyzed result.

翻译：本文分析了$\Omega\subset\mathbb{R}^d$（$d\geq 1$）中非线性热方程的算子分裂格式：$\partial_t u = \Delta u + \lambda |u|^{p-1} u$ on $\Omega\times(0,\infty)$，$u=0$ on $\partial\Omega\times(0,\infty)$，$u ({\bf x},0) =\phi ({\bf x})$ in $\Omega$，其中$\lambda\in\{-1,1\}$，$\phi \in W^{1,q}(\Omega)\cap L^{\infty} (\Omega)$，且$2\leq p < \infty$，$d(p-1)/2<q<\infty$。我们建立了$u$在$L^r$空间（$r\geq q$）中近似解的适定性，并进一步推导了时间步长$\tau>0$下其收敛阶为$\mathcal{O}(\tau)$。最后，我们给出数值算例以验证分析结果的可靠性。