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Blow up of Solutions to a Class of Damped Viscoelastic Inverse Source Problem

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This article is devoted to the study blow up of solutions to a quasilinear inverse source problem with memory and damping terms. We obtain sufficient conditions on initial functions for which the solutions blow up in a finite time. Estimates of the lifespan of solutions are also given.

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References

  1. Belgin, 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)

    Article  MathSciNet  Google Scholar 

  2. Belov, Yu.Ya., Shipina, T.N.: The problem of determining the source function for a system of composite type. J. Inverse Ill Posed Probl. 6, 287–308 (1988)

  3. Bui, A.T.: An inverse problem for a nonlinear Schrodinger equation. Abstr. Appl. Anal. 7(7), 385–399 (2002)

    Article  MathSciNet  Google Scholar 

  4. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. J. Differ. Equ. 2002, 1–14 (2002)

    MATH  Google Scholar 

  5. Cavalcanti, M.M., Oquendo, H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 4, 1310–1324 (2003)

    Article  MathSciNet  Google Scholar 

  6. Eden, A., Kalantarov, V.K.: Global behavior of solutions to an inverse problem for semilinear hyperbolic equations. J. Math. Sci. 2, 3718–3727 (2006)

    Article  Google Scholar 

  7. Eden, A., Kalantarov, V.K.: On global behavior of solutions to an inverse problem for nonlinear parabolic equations. J. Math. Anal. Appl. 307(1), 120–133 (2005)

    Article  MathSciNet  Google Scholar 

  8. Gbur, G.: Uniqueness of the solution to the inverse source problem for quasi-homogeneous sources. Opt. Commun. 187, 301–309 (2001)

    Article  Google Scholar 

  9. Georgiev, V., Todorova, G.: Existence of a solution of the wave equation with a nonlinear damping term. J. Differ. Equ. 109, 295–308 (1994)

    Article  MathSciNet  Google Scholar 

  10. Gozukizil, O.F., Yaman, M.: A note on the unique solvability of an inverse problem with integral over determination. Appl. Math. E-Notes 184, 223–230 (2008)

    MATH  Google Scholar 

  11. Gozukizil, O.F., Yaman, M.: Long-time behavior of the solutions to inverse problems for parabolic equations. Appl. Math. Comput. 184, 669–673 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Guvenilir, A.F., Kalantarov, V.K.: The asymptotic behavior of solutions to an inverse problem for diferential operator equations. Math. Comput. Model. 37, 907–914 (2003)

    Article  Google Scholar 

  13. Kalantarov, V.K., Ladyzhenskaya, O.A.: Formation of collapses in quasilinear equations of parabolic and hyperbolic types. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). 69, 77–102 (1977)

  14. Levine, H.A., Park, S.R.: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation. J. Math. Anal. Appl. 228, 181–205 (1998)

    Article  MathSciNet  Google Scholar 

  15. Levine, H.A., Serrin, J.: Global nonexistence theorems for quasilinear evolution equation with dissipation. Arch. Ration. Mech. Anal. 137, 341–361 (1997)

    Article  MathSciNet  Google Scholar 

  16. Messaoudi, S.A.: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260, 58–66 (2003)

    Article  MathSciNet  Google Scholar 

  17. Prilepko, A.I., Orlovskii, D.G., Vasin, I.A.: Methods for Solving Inverse Problems in Mathematical Physics. Marcel Dekker Inc, New York (2000)

    Google Scholar 

  18. Shahrouzi, M.: On the Petrovsky inverse problem with memory term and nonlinear boundary feedback. Iran. J. Sci. Technol. 39(A1), 45–50 (2015)

    MathSciNet  Google Scholar 

  19. Shahrouzi, M.: Blow-up of solutions for a class of fourth-order equation involving dissipative boundary condition and positive initial energy. J. Partial Differ. Equ. 27(4), 347–356 (2014)

    Article  MathSciNet  Google Scholar 

  20. Shahrouzi, M., Tahamtani, F.: Global nonexistence and stability of the solutions of inverse problems for a class of Petrovsky systems. Georgian Math. J. 19, 575–586 (2012)

    Article  MathSciNet  Google Scholar 

  21. 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)

    Article  MathSciNet  Google Scholar 

  22. Tahamtani, F., Shahrouzi, M.: Asymptotic stability and blow up of solutions for a Petrovsky inverse source problem with dissipative boundary condition. Math. Methods Appl. Sci. 36, 829–839 (2013)

    Article  MathSciNet  Google Scholar 

  23. Tahamtani, F., Shahrouzi, M.: Existence and blow up of solutions to a Petrovsky equation with memory and nonlinear source term. Bound Value Probl. 2012(50), 1–15 (2012)

  24. Wang, Y.: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy. Appl. Math. Lett. 22, 1394–1400 (2009)

    Article  MathSciNet  Google Scholar 

  25. Wu, S.T.: Blow-up of solutions for an integro-differential equation with a nonlinear source. Electron. J. Differ. Equ. 45, 1–9 (2006)

    MathSciNet  Google Scholar 

  26. 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. Sci. 33B(1), 84–106 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Jahrom University, P.O. Box: 74137-66171, Jahrom, Iran

    Mohammad Shahrouzi

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  1. Mohammad Shahrouzi
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Correspondence to Mohammad Shahrouzi.

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Shahrouzi, M. Blow up of Solutions to a Class of Damped Viscoelastic Inverse Source Problem. Differ Equ Dyn Syst 28, 889–899 (2020). https://doi.org/10.1007/s12591-017-0356-8

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  • DOI: https://doi.org/10.1007/s12591-017-0356-8

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