Yosida逼近下时间分数阶演化方程的适定性

中文摘要 AI 辅助

我们考虑希尔伯特空间$X$中时间分数阶演化方程的初值问题：$$ \partial_t^{\alpha} (u(t)-a) = Au(t) \qquad \mbox{对于$0<t<T$}, $$ 其中$\partial_t^{\alpha}$表示阶数$0<\alpha<1$的Caputo型分数阶微分算子，$u\colon (0,T) \to X,$ $a$描述初值，而定义域为$\mathcal{D}(A)$的$A$是$X$中压缩C$_0$半群的生成元。我们证明了一个分数阶Hille-Yosida定理，刻画了对于每个$a\in X$弱解的唯一存在性。我们还讨论了对于$a \in \mathcal{D}(A)$强解的唯一存在性，以及弱解的连续性。我们的结果可应用于例如时间分数阶输运方程和Boltzmann方程。证明主要基于利用Yosida逼近构造解的逼近序列。

英文摘要

We consider an initial value problem for a time-fractional evolution equation in a Hilbert space $X$: $$ \partial_t^α (u(t)-a) = Au(t) \qquad \mbox{for $0<t<T$}, $$ where $\partial_t^α$ denotes a fractional differential operator of Caputo type with order $0<α<1,$ $u\colon (0,T) \to X,$ $a$ describes an initial value, and $A$, with domain $\mathcal{D}(A),$ is the generator of a contraction C$_0$ semigroup in $X$. We prove a fractional Hille-Yosida theorem characterizing the unique existence of a weak solution for every $a\in X$. We also discuss the unique existence of a strong solution for $a \in \mathcal{D}(A),$ as well as the continuity of weak solutions. Our results are applicable, for example, to time-fractional transport equations and Boltzmann equations. The proofs are mainly based on constructing an approximating sequence of solutions using the Yosida approximation.

发表机构

• Cadi Ayyad University, UCA, Faculty of Sciences Semlalia（卡迪·阿雅德大学，塞姆拉利亚理学院）
• Department of Mathematics, Faculty of Science, Zonguldak Bülent Ecevit University（宗古尔达克比尔詹杰奇大学，理学院数学系）
• Graduate School of Mathematical Sciences, The University of Tokyo（东京大学，数理科学研究科）

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