Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 382, Issue 1, Pages 426-447Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2011.04.058
Keywords
Fractional diffusion equation; Initial value/boundary value problem; Well-posedness; Inverse problem
Categories
Funding
- 21st Century COE program
- GCOE program
- Doctoral Course Research Accomplishment Cooperation System at Graduate School of Mathematical Sciences of The University of Tokyo
- Japan Student Services Organization
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We consider initial value/boundary value problems for fractional diffusion-wave equation: partial derivative(alpha)(t) u(x, t) = Lu (x, t), where 0 < alpha <= 2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to infinity and the proofs are based on the eigenfunction expansions. Second for alpha is an element of (0,1). we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t -> infinity, (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0, T). (C) 2011 Elsevier Inc. All rights reserved.
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