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 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 ) .