Harnack's inequality for a space-time fractional diffusion equation and applications to an inverse source problem

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• 作者：Junxiong Jia^a ; ^b ; ^{jjx323@mail.xjtu.edu.cn} ; Jigen Peng^a ; ^b ; ^{jgpen@mail.xjtu.edu.cn} ; Jiaqing Yang^a ; ^{jiaq.yang@mail.xjtu.edu.cn}
• 关键词：35R30 ; 35R11 ; 35B50
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：262
• 期：8
• 页码：4415-4450
• 全文大小：471 K
• 卷排序：262

文摘

In this paper, we focus on a space–time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.