A Study on the Blow-up of Solutions for a Lam´E System of Inverse Problem
![](../images/pdf.png)
Authors: M. SHAHROUZI
DOI: 10.46793/KgJMat2501.081S
Abstract:
We consider the Lamé system of inverse problem in a bounded domain with nonlinear boundary condition. When 2 < m ≤
![p
4](4cc2fb76c07e42f87801dfc74711baba0x.png)
Keywords:
Blow-up, Lamé system, Inverse problem.
References:
[1] A. Bchatnia and M. Daolati, Behaviour of the energy for Lamé system in bounded domains with nonlinear damping and internal force, Electron. J. Differ. Equ. 1 (2013), 1–17.
[2] A. Bchatnia and A. Guesmia, Well-posedness and asymptotic stability for the Lamé system with infinite memories in a bounded domain, Math. Control Relat. Fields 4(4) (2014), 451–463.
[3] M. I. Belishev and I. Lasiecka, The dynamical Lamé system: regularity of solutions, boundary controllability and boundary data continuation, ESAIM Control Optim. Calc. Var. 8 (2002), 143–167.
[4] A. Beniani, N. Taouaf and A. Benaissa, Well-posedness and exponential stability for coupled Lamé system with viscoelastic term and strong damping, Comput. Math. Appl. 75(12) (2018), 4397–4404.
[5] L. E. Bocanegra-Rodríguez, M. A. J. Silva, T. F. Ma and P. N. Seminario-Huertas, Longtime dynamics of a semilinear Lamé system, Dynam. Differential Equations (2021), 1–22. https://doi.org/10.1007/s10884-021-09955-7
[6] S. Boulaaras, A well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source term, Appl. Anal. 100(7) (2021), 1514–1532.
[7] A. L. Bukhgeǐm, G. V. Dyatlov, V. B. Kardakov and E. V. Tantserev, Uniqueness in one inverse problem for the elasticity system, Sib. Math. J. 45(4) (2004), 618–627.
[8] A. Eden and V. K. Kalantarov, On global behavior of solutions to an inverse problem for nonlinear parabolic equations, J. Math. Anal. Appl. 228 (1998), 181–205.
[9] B. Feng, Z. Hajjej and M. Balegh, Existence and general decay rate estimates of a coupled Lamé system only with viscoelastic dampings, Math. Methods Appl. Sci. (2020), 1–18. https://doi.org/10.1002/mma.6586
[10] B. Feng and H. Li, Decay rates for a coupled viscoelastic Lamé system with strong damping, Math. Model. Anal. 25(2) (2020), 226–240.
[11] M. Grasselli, M. Ikehata and M. Yamamoto, Direct and inverse inequalities for the isotropic Lamé system with variable coefficients and applications to an inverse source problem, Appl. Anal. 84(4) (2005), 357–375.
[12] M. Ikehata, G. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo 5 (1998), 627–692.
[13] O. Yu. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Comm. Pure Appl. Math. 56 (2003), 1366–1382.
[14] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd Edition, Springer-Verlag, New York, 2006. https://10.1007/0-387-32183-7
[15] V. Isakov, G. Nakamura and J. N. Wang, Uniqueness and stability in the Cauchy problem for the elasticity system with residual stress, Contemp. Math. 333 (2003), 99–113.
[16] V. K. Kalantarov and O. A. Ladyzhenskaya, Formation of collapses in quasilinear equations of parabolic and hyperbolic types, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 69 (1977), 77–102.
[17] H. A. Levine and S. Ro. Park, Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation, J. Math. Anal. Appl. 228 (1998), 181–205.
[18] H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equation with dissipation, Arch. Ration. Mech. Anal. 137 (1997), 341–361.
[19] F. Li and G. Y. Bao, Uniform stability of the solution for a memory-type elasticity system with nonhomogeneous boundary control condition, J. Dyn. Control Syst. 23 (2017), 301–315.
[20] A. I. Prilepko, D. G. Orlovskii and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, Inc., New York, 2000.
[21] A. G. Ramm, Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering, Springer Science Publishing House, 2005. https://doi:10.1007/b100958
[22] M. Shahrouzi, Asymptotic stability and blow up of solutions for a class of viscoelastic inverse problem with boundary feedback, Math. Methods Appl. Sci. 39 (2016), 2368–2379.
[23] M. Shahrouzi, On behavior of solutions to a class of nonlinear hyperbolic inverse source problem, Acta Math. Sin. (Engl. Ser.) 32 (2016), 683–698.
[24] M. Shahrouzi, Blow up of solutions to a class of damped viscoelastic inverse source problem, Differ. Equ. Dyn. Syst. 28 (2020), 889–899.
[25] M. Shahrouzi, General decay and blow up of solutions for a class of inverse problem with elasticity term and variable-exponent nonlinearities, Math. Methods Appl. Sci. (2021), 1–15. https://doi.org/10.1002/mma.7891
[26] M. Shahrouzi and F. Tahamtani, Global nonexistence and stability of the solutions of inverse problems for a class of Petrovsky systems, Georgian Math. J. 19 (2012), 575–586.
[27] F. Tahamtani and M. Shahrouzi, Asymptotic stability and blow up of solutions for a Petrovsky inverse source problem with dissipative boundary condition, Math. Methods Appl. Sci. 36 (2013), 829–839.