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various times. Despite their fundamental importance and practical application, the literature on inverse degenerate problems for parabolic PDEs is rather recent and scarce; see [ 5 , 4 , 9 , 8 , 11 , 12 , 15 , 16 , 17 ]. Therefore, this paper is aimed at investigating both forward and inverse source problems associated to degenerate parabolic PDEs in which the degeneracy occurs in the leading diffusivity coefficient which is allowed to vanish on a zero measure subset of the space and time solution domain. In comparison with the non-degenerate parabolic PDEs, the

DOI 10.1515/jiip-2014-0026 | J. Inverse Ill-Posed Probl. 2016; 24 (2):111–122 Research Article Lukáš Šeliga and Marián Slodička An inverse source problem for a damped wave equation with memory Abstract: Inverse problem of identifying the unknown spacewise dependent source f(x) in a nonlinear damped wave equation with a memory term is investigated. The missing coecient f(x) is reconstructed from the final time observation u(x, T) = ÷T(x). Uniqueness of a solution to the inverse source problem is proved. We also propose a Landweber-type algorithm for reconstruction

following inverse source problems. Inverse source problems Let ω ⊂ ⊂ ( 0 , L ) {\omega\subset\subset(0,L)} be a given nonempty sub-domain and 0 < t 0 < T {0<t_{0}<T} . Assume the source term F ⁢ ( x , t ) = f ⁢ ( x ) ⁢ R ⁢ ( x , t ) {F(x,t)=f(x)R(x,t)} in ( 1.1 ), where R is fixed and known, and let y satisfy ( 1.1 ). Problem 1.1. Determine f ⁢ ( x ) {f(x)} from interior observation data { y , y t } | ω × ( 0 , T ) {\{y,y_{t}\}|_{\omega\times(0,T)}} or from boundary measurements { y , y t } | { 0 , L } × ( 0 , T ) {\{y,y_{t}\}|_{\{0,L\}\times(0,T)}} . Here

function. However, they only considered homogenous equations in their study. To the authors’ best knowledge, except [ 4 ], there is no other work on the mathematical analysis of the space fractional advection dispersion equation. The goal of this paper is to mathematically analyze direct and inverse source problems for a space fractional advection dispersion equation. The inverse problem consists in recovering the source term using final observations by assuming that the source is time independent. This paper is organized as follows: In Section 2 , the problem and some

1 Introduction The inverse source problem arises from many scientific and industrial areas such as optical molecular imaging [ 15 ], viscous fluid flows in porous materials [ 20 ] and gas tomography [ 28 ]. As specific examples in medical imaging such as electroencephalography and magnetoencephalography, these imaging modalities are non-invasive neurophysiological techniques that measure the electric or magnetic fields generated by neuronal activity of the brain [ 26 , 5 ]. The spatial distributions of the measured fields around the skull are analyzed to

function (for distributed-order fractional model), diffusion and potential coefficients (when using a second-order elliptic operator in space), initial condition, source term, boundary conditions and domain geometry. This gives rise to a large variety of inverse problems for FDEs, which have attracted much attention in recent years. We can refer to Jin and Rundell [ 14 ] for a topical review and a comprehensive list of bibliographies. In this paper, we consider an inverse source problem for a DTFDE. To avoid unnecessary complications for the main theme, we will make the

1 Introduction While dealing with inverse source problems it is well known that in general one of the main encountered difficulties is the non-identifiability (uniqueness) of a source in its abstract form, see [ 9 ] for a counterexample. In the literature, to overcome this difficulty authors generally assume available some a priori information on the form of the sought source: For example, time-independent sources are treated by Cannon in [ 4 ] using spectral theory, then by Engl, Scherzer and Yamamoto in [ 7 ] using the approximated controllability of the heat

if $\mathcal E_x=\emptyset\forall\,x\in\Omega$ E x = ∅ ∀ x ∈ Ω although it can be conjectured. Nevertheless, we can prove the following result. Corollary 1.1 Let a ∈ L (Ω) satisfy a > 0 a.e. in Ω, F = 0, and u be the solution to (1.1) . Then u > 0 in Ω × (0,∞). Theorem 1.1 is a weaker result than our expected strong maximum principle, but is sufficient for some applications. Next we study an inverse source problem for (1.1) under the assumption that the inhomogeneous term F takes the form of separation of variables. Problem 1.1 Let x 0 ∈ Ω and

1 Introduction The area of inverse source problems has many applications and it, therefore, attracts the attention of the scientific community, see, e.g., [ 14 , 13 , 16 , 17 , 18 , 30 , 32 , 34 , 35 ]. The solutions of inverse source problems can be used to directly detect the source even when the source is inactive after a certain time. Here, we name some examples. In the case of the parabolic equation, the problem plays an important role in identifying the pollution sources in a river or a lake [ 14 ]. In the case of elliptic equations, the inverse

fractional diffusion equation is derived by the CTRW model with an argument similar to that used to obtain the classical diffusion equation from the random walk model (see [ 29 ]). Our main results are composed of two parts. One is the Carleman estimates, the other is the stability estimates in inverse source problems. We establish Carleman estimates for the one-dimensional half-order time fractional diffusion equation ( 1.1 ). Here we derive two kinds of Carleman estimates: with interior data (Theorem 2.5 ) and with boundary data (Theorem 2.6 ). Then we consider the