Abstract
This article is concerned with a class of hyperbolic inverse source problem with memory term and nonlinear boundary damping. Under appropriate assumptions on the initial data and parameters in the equation, we establish two results on behavior of solutions. At first we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity and finally a blow-up result is established for certain solution with positive initial energy.
Similar content being viewed by others
References
Belov, Ya., Shipina, T. N.: The problem of determining the source function for a system of composite type. J. Inv. Ill-Posed Problems, 6, 287–308 (1988)
Berrimi, S., Messaoudi, S. A.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal., 64, 2314–2331 (2006)
Bilgin, B. A., Kalantarov, V. K.: Blow up of solutions to the initial boundary value problem for quasilinear strongly damped wave equations. J. Math. Anal. Appl., 403, 89–94 (2013)
Bosello, C. A., Fabrizio, M.: Stability and well posedness for a dissipative boundary condition with memory in electromagnetism. Appl. Math. Lett., 40, 59–64 (2015)
Boukhatem, Y., Benabderrahmane, B.: Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. Nonlinear Anal., 97, 191–209 (2014)
Bui, A. T.: An inverse problem for a nonlinear Schrödinger equation. Abstract Appl. Anal., 7 7, 385–399 (2002)
Cavalcanti, M. M., Domingos Cavalcanti, V. N., Soriano, J. A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differential Equaions, 2002, 1–4 (2002)
Cavalcanti, M. M., Oquendo, H. P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim., 4, 1310–1324 (2003)
Chen, W., Xiong, Y.: Blow-up and general decay of solutions for a nonlinear viscoelastic equation. Electron. J. Differential Equaions, 12, 1–11 (2013)
Dong, W., Xu, J.: Existence of weak solutions for a p-Laplacian problem involving Dirichlet boundary condition. Appl. Math. Comput., 248, 511–518 (2014)
Eden, A., Kalantarov, V. K.: Global behavior of solutions to an inverse problem for semilinear hyperbolic equations. J. Math. Sci., 2, 3718–3727 (2006)
Gbur, G.: Uniqueness of the solution to the inverse source problem for quasi-homogeneous sources. Optics Commun., 187, 301–309 (2001)
Georgiev, V., Todorova, G.: Existence of a solution of the wave equation with a nonlinear damping term. J. Differential Equaions, 109, 295–308 (1994)
Gerbi, S., Said-Houari, B.: Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term. Appl. Math. Comput., 218, 11900–11910 (2012)
Guvenilir, A. F., Kalantarov, V. K.: The asymptotic behavior of solutions to an inverse problem for diferential operator equations. Math. Comp. Modeling, 37, 907–914 (2003)
Kalantarov, V. K., Ladyzhenskaya, O. A.: Formation of collapses in quasilinear equations of parabolic and hyperbolic types. Zap. Nauchn. Semin. LOMI, 69, 77–102 (1977)
Liu, W., Yu, J.: On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Anal., 74, 2175–2190 (2011)
Messaoudi, S. A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr., 260, 58–66 (2003)
Prilepko, A. I., Orlovskii, D. G., Vasin, I. G.: Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, Inc, New York, Basel, 2000
Shahrouzi, M.: Blow-up of solutions for a class of fourth-order equation involving dissipative boundary condition and positive initial energy. J. Partial Differential Equations, 27 4, 347–356 (2014)
Shahrouzi, M.: On the Petrovsky inverse problem with memory term and nonlinear boundary feedback. Iranian J. Sci. Tech., 39(A1), 45–50 (2015)
Tahamtani, F., Shahrouzi, M.: Global nonexistence and stability of the solutions of inverse problems for a class of Petrovsky systems. Georgian Math. J., 19, 575–586 (2012)
Shidfar, A., Babaei, A., Molabahrami, A.: Solving the inverse problem of identifying an unknown source term in a parabolic equation. Comput. Math. Appl., 60, 1209–1213 (2010)
Tahamtani, F., Shahrouzi, M.: Asymptotic stability and blow up of solutions for a Petrovsky inverse source problem with dissipative boundary condition. Math. Meth. Appl. Sci., 36, 829–839 (2013)
Wang, Y.: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett., 22, 1394–1400 (2009)
Wu, S. T.: Blow-up of solutions for an integro-differential equation with a nonlinear source. Electron. J. Differential Equaions, 45, 1–9 (2006)
Zarai, A., Tatar, N. A., Abdelmalek, S.: Elastic membrane equation with memory term and nonlinear boundary damping: global existence, decay and blowup of the solution. Acta Math. Scientia, 33B(1), 84–106 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shahrouzi, M. On behavior of solutions to a class of nonlinear hyperbolic inverse source problem. Acta. Math. Sin.-English Ser. 32, 683–698 (2016). https://doi.org/10.1007/s10114-016-5081-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-016-5081-7