Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian
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Hajime Koba
und
Hideki Matsuoka
Abstract
This paper considers a backward problem on a heat equation with a fractional Laplacian. It is not easy to solve a backward heat equation directly. This problem is a well-known ill-posed problem. In order to consider a backward heat equation with a fractional Laplacian, we apply the N-th power of the Dirichlet-Laplacian and small parameters to regularize the equation. This method is called a quasi-reversibility method. We use the generalized quasi-reversibility method to change the backward heat system into another system. This paper shows the existence of a strong solution of the modified backward heat system, and derives L2-estimates of the difference between a solution of the heat equation with the fractional Laplacian and a solution of our system.
The authors gratefully acknowledge the precious comments of Professor Yoshikazu Giga. The authors would like to thank Professor Masahiro Yamamoto for helpful suggestions and for introducing them the references [Stable Solution of Inverse Problems, Friedr. Vieweg & Sohn, Braunschweig, 1987], [Inverse Problems 25 (2009), no. 12, Article ID 123013]. This paper was written when the second author was a graduate student at the University of Tokyo.
© 2015 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Partial boundary regularity of non-linear parabolic systems in low dimensions
- A symmetry result for strictly convex domains
- Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces
- Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian
- Certain identities, connection and explicit formulas for the Bernoulli and Euler numbers and the Riemann zeta-values
Artikel in diesem Heft
- Frontmatter
- Partial boundary regularity of non-linear parabolic systems in low dimensions
- A symmetry result for strictly convex domains
- Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces
- Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian
- Certain identities, connection and explicit formulas for the Bernoulli and Euler numbers and the Riemann zeta-values
