AI 中文总结
该研究针对时间分数阶Amari神经场方程，通过Volterra方程方法证明了全局前沿解的存在唯一性，确定了渐近速度与行波剖面，并刻画了其尾部渐近行为。
AI 中文摘要
我们研究了带有Caputo导数和Heaviside发放率函数的时间分数阶Amari神经场方程中的前沿传播。对于一类非负偶对称连接核和严格递减的$C^1$类前沿型初始数据，我们将Cauchy问题归结为关于阈值位置的非线性Volterra方程。我们证明了全局前沿解的存在唯一性，确定了其阈值位置的短时行为，并识别了其渐近速度。我们还构造了相关的渐进行波剖面，并在非零速度情形下建立了速度和剖面（在平移意义下）的唯一性。利用刚性论证，我们证明了阈值位置局部增量的渐近线性性，并在阈值中心坐标系中获得了对剖面的均匀收敛。最后，我们刻画了波剖面的依赖于核的尾部渐近行为。
英文摘要
We study front propagation in a time-fractional Amari neural field equation with a Caputo derivative and a Heaviside firing-rate function. For a class of nonnegative even connectivity kernels and strictly decreasing $C^1$ front-like initial data, we reduce the Cauchy problem to a nonlinear Volterra equation for the threshold location. We prove the existence and uniqueness of a global front solution, determine the short-time behavior of its threshold location, and identify its asymptotic speed. We also construct the associated asymptotic traveling-wave profile and establish uniqueness of the speed and profile, up to translation, in the nonzero-speed case. Using a rigidity argument, we prove the asymptotic linearity of the local increments of the threshold location and obtain uniform convergence to the profile in the threshold-centered frame. Finally, we characterize the kernel-dependent tail asymptotics of the wave profile.
