Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems

We consider initial value/boundary value problems for fractional diffusion-wave equation: ∂ t α u ( x , t ) = L u ( x , t ) , where 0 < α ⩽ 2 , where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution a...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 382; no. 1; pp. 426 - 447
Main Authors: Sakamoto, Kenichi, Yamamoto, Masahiro
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Inc 01.10.2011
Elsevier
Subjects:
Exact sciences and technology
Fractional diffusion equation
Initial value/boundary value problem
Inverse problem
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Well-posedness
Fractional diffusion equation
Initial value/boundary value problem
Well-posedness
Inverse problem
Wave equation
Solution uniqueness
Existence of solution
Eigenfunction
Initial value problem
Asymptotic behavior
Well posed problem
Boundary value problem
Mathematical analysis
Diffusion equation
Elliptic operator
Numerical stability
ISSN:0022-247X, 1096-0813
Online Access:Get full text
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Summary:We consider initial value/boundary value problems for fractional diffusion-wave equation: ∂ t α u ( x , t ) = L u ( x , t ) , where 0 < α ⩽ 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 ∞ and the proofs are based on the eigenfunction expansions. Second for α ∈ ( 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 → ∞ , (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over ( 0 , T ) .
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2011.04.058