具有奇异初始数据的半线性次扩散方程在\(L^\infty\)框架之外的正则性和高阶时间步长
Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the $L^\infty$ framework
浏览论文内容
中文总结 AI 辅助
研究\(L^\infty\)框架外有奇异初始数据的半线性次扩散问题，利用次扩散解算子平滑性质及合适非线性假设，建立温和解相关结果并推导误差估计，数值实验验证收敛速率。
中文摘要 AI 辅助
本文旨在分析\(L^\infty\)框架之外具有奇异初始数据的半线性次扩散问题的数值格式。主要困难在于非线性项的奇异行为比之前的分析更强。由于奇异初始数据过于粗糙，无法保证解的一致\(L^\infty\)界，基础空间中常用的Lipschitz框架不再适用。分析必须在较弱的分数Sobolev型空间中进行，其中非线性复合更微妙，\(f(u(t))\)项相对于\(u(t)\)可能表现出放大的奇异性。为克服此困难，利用次扩散解算子的平滑性质，在分数算子空间中制定合适的非线性假设。这些平滑估计使部分奇异性从非线性项转移到解算子，在那里可以控制。在此假设下，建立了温和解的适定性和正则性结果，并推导了指数卷积求积法的逐点时间误差估计。数值实验证实了预测的收敛速率。
英文摘要
This paper aims to analyze a numerical scheme for semilinear subdiffusion problems with singular initial data beyond the $L^\infty$ framework. The main difficulty lies in the stronger singular behavior of the nonlinear term compared with previous analyses. Since the singular initial datum is too rough to guarantee a uniform $L^\infty$ bound for the solution, the usual Lipschitz framework in the base space is no longer sufficient. The analysis must instead be carried out in weaker fractional Sobolev-type spaces, where nonlinear composition is more delicate and the term $f(u(t))$ may exhibit an amplified singularity relative to that of $u(t)$. To overcome this difficulty, we exploit the smoothing properties of the subdiffusion solution operators and formulate suitable nonlinear assumptions in fractional operator spaces. These smoothing estimates allow part of the singularity to be transferred from the nonlinear term to the solution operators, where it can be controlled. Under these assumptions, we establish well-posedness and regularity results for the mild solution and derive a pointwise-in-time error estimate for the exponential convolution quadrature method. Numerical experiments confirm the predicted convergence rates.
