一类广义半线性抛物方程在时间加权$L^{\infty }$空间中的适定性理论

Well-Posedness Theory for a Class of Generalized Semilinear Parabolic Equations in Time-Weighted $L^{\infty }$ Spaces

摘要: 在描述流体运动、热传导等演化过程时，半线性抛物方程是一类重要方程。现有研究多围绕特定形式的方程展开，对广义化模型的适定性分析有待完善。本文以Diego Chamorro与Maxence Mansais提出的分数阶Navier-Stokes方程为基础，保留非线性项核心结构并进行参数优化，建立了一类一般化的半线性抛物方程模型。利用Duhamel原理将方程转化为等价的积分形式，运用Banach不动点原理、Besov空间刻画、分数阶半群时间衰减估计及卷积Young不等式等关键工具，在临界空间$L_{\alpha ,\beta ,k}^{\infty }\left(\left[0,+\infty \right)\times R^d\right)$ 中，证明了：当初始数据与外力项的范数满足小性阈值约束时，方程存在唯一的全局温和解。该一般化模型拓展了传统半线性抛物方程的适用场景，为研究不同分数阶扩散效应与非线性强度下的动力学行为提供了统一分析框架，相关理论结果对分数阶抛物方程的理论延伸具有参考价值。

Abstract: Semilinear parabolic equations play a crucial role in describing evolutionary processes such as fluid motion and heat conduction. Existing research has predominantly focused on equations with specific forms, leaving the well-posedness analysis of generalized models insufficiently explored. Building upon the fractional Navier-Stokes equations proposed by Diego Chamorro and Maxence Mansais, this paper retains the core structure of the nonlinear term while optimizing parameters to establish a generalized model for a class of semilinear parabolic equations. By applying the Duhamel principle to transform the equation into an equivalent integral form, and utilizing key tools such as the Banach fixed-point theorem, characterization of Besov spaces, time-decay estimates for fractional semigroups, and the Young convolution inequality, we prove that in the critical space $L_{\alpha ,\beta ,k}^{\infty }\left(\left[0,+\infty \right)\times R^d\right)$ , when the norms of the initial data and external force term satisfy a smallness threshold constraint, the equation admits a unique global mild solution. This generalized model extends the applicability of traditional semilinear parabolic equations and provides a unified analytical framework for studying dynamical behaviors under different fractional diffusion effects and nonlinear intensities. The theoretical results offer valuable insights for extending the theory of fractional parabolic equations.