Recovering the initial distribution for strongly damped wave equation

We study for the first time the inverse backward problem for the strongly damped wave equation. First, we show that the problem is severely ill-posed in the sense of Hadamard. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established

A final value problem with a non-local and a source term: regularization by truncation

This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Gr{\"o}nwalls' inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.Comment: Comments are welcome

Some inverse source problems of determining a space dependent source in fractional-dual-phase-lag type equations

The dual-phase-lag heat transfer models attract a lot of interest of researchers in the last few decades. These are used in problems arising from non-classical thermal models, which are based on a non-Fourier type law. We study uniqueness of solutions to some inverse source problems for fractional partial differential equations of the Dual-Phase-Lag type. The source term is supposed to be of the formh(t)f(x)with a known functionh(t). The unknown space dependent sourcef(x)is determined from the final time observation. New uniqueness results are formulated in Theorem 1 (for a general fractional Jeffrey-type model). Here, the variational approach was used. Theorem 2 derives uniqueness results under weaker assumptions onh(t)(monotonically increasing character ofh(t)was removed) in a case ofdominant parabolicbehavior. The proof technique was based on spectral analysis. Section Modified Model for tau q>tau Tshows that an analogy of Theorem 2 fordominant hyperbolicbehavior (fractional Cattaneo-Vernotte equation) is not possible