论文标题
具有nagumo型源和扰动顺序的非线性分数扩散方程
The nonlinear fractional diffusion equations with Nagumo-type sources and perturbed orders
论文作者
论文摘要
我们考虑一类非线性分数方程，具有时间变量的caputo分数衍生物，这是希尔伯特空间中自动接合正确定的无界操作员的分数顺序和一个奇异的非线性源。这些方程是一些众所周知的分数方程的概括，例如分数Cahn-Allen方程，分数汉堡方程，分数Cahn-Hilliard方程，分数Kuramoto-Sivashinsky方程等。 我们研究初始值和最终值问题。 在某些合适的假设下，我们研究了问题相对于扰动的分数顺序的存在，最大解决方案的独特性以及解决方案的稳定性。 对于$ t = 0 $，我们证明最终值问题是不起眼的，并推断出该问题是错误的。提出了一种正则化方法，以从不精确的分数订单和最终数据中恢复初始数据。根据问题的确切解决方案的某些规律性假设，我们获得了Hölder类型的错误估计。
We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular nonlinear source. These equations are generalizations of some well-known fractional equation such as the fractional Cahn-Allen equation, the fractional Burger equation, the fractional Cahn-Hilliard equation, the fractional Kuramoto-Sivashinsky equation, etc. We study both the initial value and the final value problem. Under some suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. For $t=0$, we show that the final value problem is instable and deduce that the problem is ill-posed. A regularization method is proposed to recover the initial data from the inexact fractional orders and the final data. By some regularity assumptions of the exact solutions of the problems, we obtain an error estimate of Hölder type.
