# Inverse Problems of Finding the Absorption Parameter in the Diffusion Equation

Article

First Online:

- 9 Downloads

## Abstract

The paper is devoted to the study of inverse problems of finding, together with a solution the parameter

*u(x, t)*of the diffusion equation$${u_t} - \Delta u + [c(x,t) + a{q_0}(x,t)]u = f(x,t),$$

*a*characterizing absorption (*c(x,t)*and*q*_{0}*(x,t)*are given functions). It is assumed that, on the function*u(x,t)*, nonpercolation conditions and some special overdetermination conditions of integral form are imposed. We prove existence theorems for solutions (*u(x,t),a*) such that the function*u(x, t)*has all Sobolev generalized derivatives appearing in the equation and*a*is a nonnegative number.## Keywords

diffusion equation nonpercolation condition unknown parameter inverse problems final integral overdetermination conditions existence## Preview

Unable to display preview. Download preview PDF.

## Notes

### Funding

This work was supported by the Russian Foundation for Basic Research under grant 18-01-00620.

## References

- 1.V. S. Vladimirov,
*Equations of Mathematical Physics*(Nauka, Moscow, 1988) [in Russian].zbMATHGoogle Scholar - 2.A. Lorenzi, “Recovering two constants in a linear parabolic equation,” J. Phys.: Conf. Ser.
**73**(012014) (2007).Google Scholar - 3.A. Lorenzi and G. Mola, “Identification a real constant in linear evolution equation in a Hilbert spaces,” Inverse Probl. Imaging
**5**(3), 695–714 (2011).MathSciNetCrossRefGoogle Scholar - 4.G. Mola, “Identification of the diffusion coefficient in linear evolution equation in a Hilbert spaces,” J. Abstr. Differ. Equ. Appl.
**2**(1), 14–28 (2011).MathSciNetzbMATHGoogle Scholar - 5.A. Lorenzi and G. Mola, “Recovering the reaction and the diffusion coefficients in a linear parabolic equations,” Inverse Problems
**28**(075006) (2012).MathSciNetCrossRefGoogle Scholar - 6.A. Lorenzi and E. Paparoni, “Identifications of two unknown coefficients in an integro-differential hyperbolic equation,” J. Inverse Ill-Posed Probl.
**1**(4), 331–348 (1993).MathSciNetCrossRefGoogle Scholar - 7.A. S. Lyubanova, “Identification a constant coefficient in an elliptic equation,” Appl. Anal.
**87**(10-11), 1121–1128 (2008).MathSciNetCrossRefGoogle Scholar - 8.A. I. Kozhanov and R. R. Safiullova, “Determination of parameters in the telegraph equation,” Ufimsk. Mat. Zh.
**9 (1)**, 63–74 (2017) [Ufa Math. J.**9**(1), 62–74 (2017)].MathSciNetCrossRefGoogle Scholar - 9.A. I. Kozhanov, “Nonlinear loaded equations and inverse problems,” Zh. Vychisl. Mat. Mat. Fiz.
**44 (4)**, 694–716 (2004) [Comput. Math. Math. Phys.**44**(4), 657–675 (2004)].MathSciNetzbMATHGoogle Scholar - 10.A. I. Kozhanov, “Parabolic equations with unknown time-dependent coefficients,” Zh. Vychisl. Mat. Mat. Fiz.
**57 (6)**, 961–972 (2017) [Comput. Math. Math. Phys.**57**(6), 956–966 (2017)].MathSciNetzbMATHGoogle Scholar - 11.M. T. Dzhenaliev,
*On the Theory of Linear Boundary-Value Problems for Loaded Differential Equations*(Inst. Teor. Prikl. Mat., Almaty, 1995) [in Russian].Google Scholar - 12.A. M. Nakhushev,
*Loaded Equations and Their Application*(Nauka, Moscow, 2012) [in Russian].Google Scholar - 13.O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva,
*Linear and Quasilinear Equations of Parabolic Type*(Nauka, Moscow, 1967) [in Russian].zbMATHGoogle Scholar - 14.O. A. Ladyzhenskaya and N. N. Ural'tseva,
*Linear and Quasilinear Equations of Elliptic type*(Nauka, Moscow, 1973) [in Russian].zbMATHGoogle Scholar - 15.V. A. Trenogin,
*Functional Analysis*(Nauka, Moscow, 1980) [in Russian].zbMATHGoogle Scholar - 16.S. L. Sobolev,
*Some Applications of Functional Analysis in Mathematical Physics*(Nauka, Moscow, 1988) [in Russian].zbMATHGoogle Scholar

## Copyright information

© Pleiades Publishing, Ltd. 2019