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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Vydáno v:	Journal of mathematical analysis and applications Ročník 382; číslo 1; s. 426 - 447
Hlavní autoři:	Sakamoto, Kenichi, Yamamoto, Masahiro
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier Inc 01.10.2011 Elsevier
Témata:	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
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Popis
Shrnutí:	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