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Lower bounds for blow-up time in a class of nonlinear wave equations

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Abstract

This paper deals with the blow-up of solutions u(x, t) to a class of nonlinear hyperbolic problems. Under certain conditions on the data, we construct a lower bound for the blow-up time t* when blow-up occurs. A Sobolev-type inequality to be used in our investigation will also be established.

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References

  1. Ball J.M. : Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. Oxf. 28(2), 473–486 (1977)

    Article  MATH  Google Scholar 

  2. Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source terms. J. Differ. Eqs. 109(2), 295–308 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gazzola F, Squassina M: Global solutions and finite time blow up for damped semilinear wave equations. Ann. I. H. Poincaré 23(2), 185–207 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Glassey R.T. : Blow-up theorems for nonlinear wave equations. Math. Z. 132, 183–203 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  5. Haraux A, Zuazua E: Decay estimates for some semilinear damped hyperbolic problems. Arch. Rat. Mech. Anal. 100(2), 191–206 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hardy G.H., Littlewood J.E., Pólya G: Inequalities. Cambridge Univ. Press, Cambridge (1988)

    MATH  Google Scholar 

  7. Kalantarov, V. K., Ladyzenskaja, O. A.: Formation of collapses in quasilinear equations of parabolic and hyperbolic types (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions 10. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69, (1977), 77-102, 274

  8. Kopackova M: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation. Comment. Math. Univ. Carolin. 30(4), 713–719 (1989)

    MATH  MathSciNet  Google Scholar 

  9. Levine H.A. : Instability and nonexistence of global solutions of nonlinear wave equations of the form Pu tt =  −Au +  F(u). Trans. Am. Math. Soc. 192, 1–21 (1974)

    MATH  Google Scholar 

  10. Levine H.A. : Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  11. Levine H.A., Payne L.E. : Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations. J. Math. Anal. Appl. 55(2), 329–334 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  12. Levine H.A., Serrin J: Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch. Rat. Mech. Anal. 137(4), 341–361 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Messaoudi S.A. : Blow up in a nonlinearly damped wave equation. Math. Nachr. 231, 105–111 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Payne L.E., Sattinger D.H. : Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22(3–4), 273–303 (1975)

    Article  MathSciNet  Google Scholar 

  15. Ohta M: Remarks on blowup of solutions for nonlinear evolution equations of second order. Adv. Math. Sci. Appl. 8(2), 901–910 (1998)

    MATH  MathSciNet  Google Scholar 

  16. Vitillaro E: Global nonexistence theorems for a class of evolution equations with dissipation. Arch. Rat. Mech. Anal. 149(2), 155–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  17. Xu R. : Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data. Q. Appl. Math. 68(3), 459–468 (2010)

    MATH  Google Scholar 

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  1. Département de mathématiques et de statistique, Université Laval, Québec, G1V 0A6, Canada

    G. A. Philippin

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  1. G. A. Philippin
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Philippin, G.A. Lower bounds for blow-up time in a class of nonlinear wave equations. Z. Angew. Math. Phys. 66, 129–134 (2015). https://doi.org/10.1007/s00033-014-0400-2

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  • DOI: https://doi.org/10.1007/s00033-014-0400-2

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