AI 中文总结
研究一类超扩散积分 - 微分方程柯西问题，通过建立乘子估计确定局部适定性临界阈值\(q_c\)，证明超临界区域不适定性，追踪结构交叉解析全局渐近动力学，确定临界指数\(\rho_F\)，为小初始数据建立全局存在性和代数衰减，适用于规范物理模型。
AI 中文摘要
本文研究了一类在\(\mathbb{R}^N\)中控制反常超扩散输运的非线性积分 - 微分方程的柯西问题。线性动力学由双尺度记忆核驱动，其拉普拉斯变换是扇形的，在高低频呈现不同幂律渐近性。通过建立严格的\(L^q - L^p\)乘子估计确定了局部适定性的临界勒贝格阈值\(q_c\)，证明了超临界区域\(1 < q < q_c\)存在瞬时范数膨胀和不适定性，追踪到长时间松弛参数的结构交叉解析了全局渐近动力学，确定了非局部藤田型临界指数\(\rho_F\)，并为小初始数据建立了全局时间存在性和代数衰减。该框架直接适用于规范物理模型。
英文摘要
This manuscript investigates the Cauchy problem for a class of nonlinear integro-differential equations governing anomalous super-diffusive transport in $\mathbb{R}^N$. The linear dynamics are driven by a dual-scale memory kernel whose Laplace transform is sectorial and exhibits distinct power-law asymptotics at high and low frequencies. This super-diffusive structure precludes the infinite regularizing capacity characteristic of classical parabolic theory; consequently, the associated resolvent operator possesses a heavy algebraic tail in Fourier space, acting as a pseudo-differential operator in the Hörmander class $S^{-2}_{1,0}$ and restricting spatial smoothing. By establishing rigorous $L^q-L^p$ multiplier estimates, the critical Lebesgue threshold $q_c$ for local well-posedness is determined. To demonstrate the sharpness of this threshold, instantaneous norm inflation -- and consequent ill-posedness -- is proven in the supercritical regime $1 < q < q_c$. Furthermore, tracking the structural crossover to the long-time relaxation parameter resolves the global asymptotic dynamics. The nonlocal Fujita-type critical exponent $ρ_F$ is identified, and global-in-time existence along with algebraic decay is established for small initial data in intersection spaces, provided the nonlinearity remains supercritical and overcomes the structural algebraic barrier connecting the dual scales. This general framework applies directly to canonical physical models, including Cole-Cole fractional retardation and multi-scale Prabhakar memory.
Comments34 pages
