一类带外力项的半线性抛物方程在抛物型Morrey空间中的整体适定性

Global Well-Posedness of a Class of Semilinear Parabolic Equations with External Force in Parabolic Morrey Spaces

摘要: 本文研究了一类带外力项的广义半线性抛物方程在抛物型Morrey空间$M_{\alpha }^{p_1,\frac{k\left(d+\alpha \right)}{\alpha -\beta }}$ 中的整体适定性问题。该方程包含分数阶拉普拉斯算子耗散项、非局部非线性项及一般外力项，其形式为$u_t+{\left(-\Delta \right)}^{\frac{\alpha }{2}}u={\left(-\Delta \right)}^{\frac{\beta }{2}}\left({\left|u\right|}^{k}u\right)+f$ ，其中$\beta <\alpha <\left(k+1\right)\beta$ (亚临界情形)。该模型具有高度的普适性，通过参数选取可退化为多个经典物理模型。本文通过Duhamel原理将初值问题转化为等价的积分方程，在线性项、非线性项及外力项满足充分小范数的条件下，系统建立了各项在所选空间中的关键先验估计。最终，应用Banach不动点定理，证明了该方程整体温和解在抛物型Morrey空间中的存在性、唯一性及对初值的连续依赖性。本研究推广了现有经典结果，彰显了抛物型Morrey空间为处理此类兼具非局部结构与临界增长的非线性发展方程提供了统一而有效的分析工具。

Abstract: This paper investigates the global well-posedness of a class of generalized semilinear parabolic equations with external force in parabolic Morrey spaces $M_{\alpha }^{p_{{}_1},\frac{k\left(d+\alpha \right)}{\alpha -\beta }}$ . The equation incorporates fractional Laplacian dissipation, nonlocal nonlinear terms, and general external force, given by $u_t+{\left(-\Delta \right)}^{\frac{\alpha }{2}}u={\left(-\Delta \right)}^{\frac{\beta }{2}}\left({\left|u\right|}^{k}u\right)+f$ , where $\beta <\alpha <\left(k+1\right)\beta$ (subcritical case). This model is highly universal and can degenerate into multiple classical physical models through parameter selection. By means of the Duhamel principle, the initial value problem is reformulated as an equivalent integral equation. Under the assumption that the linear, nonlinear, and external force terms possess sufficiently small norms, we systematically establish the crucial a priori estimates for each of these terms within the selected function space. Finally, the Banach fixed-point theorem is applied to prove the existence, uniqueness, and continuous dependence on initial data of the global mild solution in parabolic Morrey spaces. This study extends existing classical results and demonstrates that parabolic Morrey spaces provide a unified and effective analytical framework for handling such nonlinear evolution equations with nonlocal structures and critical growth.