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

By Kenichi Sakamoto, Masahiro Yamamoto. 2011

1,240 citations recorded by the source corpus.

Abstract 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 ) .

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/5e650624bb293f6d6abaac2414614c3b32ea848d

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Citations in the corpus

Existence and regularity results of a backward problem for fractional diffusion equations (2019, background): “In order to check that ∂tu belongs to the space β,θ((0,T];L2(Ω)) and satisfies the estimate (23), for the definition of operators i, i = 1, 2, 3, 4, we need to show that 1g, if ∈ β,θ((0,T];L2(Ω)).”
Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative (2016, methodology): “His research has been continued in a work by Sakamoto and Yamamoto [37], which is a pioneer work in the theory of weak solutions for (1.8).”
The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation (2015, methodology): “In [24], [17], the existence and properties of solutions for a time fractional equation in a bounded domain were considered by applying eigenfunction expansions.”
Too much regularity may force too much uniqueness (2016, background): “Time-dependent problems with a fractional temporal derivative of order α ∈ (0, 1) ∪ (1, 2) have attracted much attention in recent years, since these problems arise in several models of physical processes—see the references in [5, 8, 11, 12, 14].”
An inverse time-dependent source problem for a time-fractional diffusion equation (2016, neutral): “Wei and Zhang in [39] proposed a numerical method to solve the inverse time-dependent source problem issued in [28].”
A UNIQUENESS RESULT FOR AN INVERSE PROBLEM IN A SPACE-TIME FRACTIONAL DIFFUSION EQUATION (2013, background): “Important references documenting the recent interest on inverse problems with fractional derivatives are [4, 6, 9, 10, 13, 14, 16, 17, 18].”
A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation (2019, background): “It is worth to mention that the solution of many time-fractional differential equations typically displays a weak singularity near the initial time [8, 9, 10, 19, 20, 21, 26, 29, 30], which leads to the loss of time accuracy for many related high-order numerical methods.”