# American Institute of Mathematical Sciences

• Previous Article
Controllability problems for the 1-d wave equations on a half-axis with Neumann boundary control
• MCRF Home
• This Issue
• Next Article
Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws
June  2013, 3(2): 143-160. doi: 10.3934/mcrf.2013.3.143

## Stability of the determination of a time-dependent coefficient in parabolic equations

 1 LMAM, UMR 7122, Université de Lorraine, Ile du Saulcy, 57045 Metz, cedex 1, France 2 UMR-7332, Aix Marseille Université, Centre de Physique Théorique, Campus de Luminy, Case 907, 13288 Marseille, cedex 9, France

Received  February 2012 Revised  May 2012 Published  March 2013

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $\sigma(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-\Delta_x u+\sigma(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation is changed to the semi-linear parabolic equation $\partial_tu-\Delta_x u=F(x,t,\sigma(t),u(x,t))$.
Citation: Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control & Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143
##### References:

show all references

##### References:
 [1] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [2] Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661 [3] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [4] Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021060 [5] Mourad Choulli, El Maati Ouhabaz, Masahiro Yamamoto. Stable determination of a semilinear term in a parabolic equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 447-462. doi: 10.3934/cpaa.2006.5.447 [6] Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011 [7] Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052 [8] Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141 [9] Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005 [10] Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761 [11] Anne Mund, Christina Kuttler, Judith Pérez-Velázquez. Existence and uniqueness of solutions to a family of semi-linear parabolic systems using coupled upper-lower solutions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5695-5707. doi: 10.3934/dcdsb.2019102 [12] Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307 [13] J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176 [14] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [15] Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure & Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237 [16] Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018 [17] Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029 [18] Qiang Du, Jiang Yang, Zhi Zhou. Analysis of a nonlocal-in-time parabolic equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 339-368. doi: 10.3934/dcdsb.2017016 [19] Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083 [20] Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020

2020 Impact Factor: 1.284

## Metrics

• HTML views (0)
• Cited by (12)

• on AIMS