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July  2020, 13(7): 2047-2055. doi: 10.3934/dcdss.2020157

A note on a class of 4th order hyperbolic problems with weak and strong damping and superlinear source term

Monica Marras , and  Stella Vernier-Piro

Università di Cagliari, Dipartimento di Matematica e Informatica, Viale Merello 92, 09123 Cagliari, Italy

* Corresponding author: Monica Marras

To our friend Patrizia on the occasion of her sixty-fifth birthday.

Received  September 2018 Revised  September 2018 Published  November 2019

In this paper we study a initial-boundary value problem for 4th order hyperbolic equations with weak and strong damping terms and superlinear source term. For blow-up solutions a lower bound of the blow-up time is derived. Then we extend the results to a class of equations where a positive power of gradient term is introduced.

Keywords: Initial-boundary value problems for higher-order hyperbolic equations, non linear hyperbolic problems, blow-up, lower bound.
Mathematics Subject Classification: Primary: 35L35; Secondary: 35B44.
Citation: Monica Marras, Stella Vernier-Piro. A note on a class of 4th order hyperbolic problems with weak and strong damping and superlinear source term. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 2047-2055. doi: 10.3934/dcdss.2020157
References:
[1] R. A. Adams and J. J. F. Fourier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

L. J. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar

[3]

G. Autuori, F. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Variables and Elliptic Equations, 57 (2012), 379-395.  doi: 10.1080/17476933.2011.592584.  Google Scholar

[4]

G. Autuori and P. Pucci, Local asymptotic stability for polyharmonic Kirchhoff systems, Applicable Analysis, 90 (2011), 493-514.  doi: 10.1080/00036811.2010.483433.  Google Scholar

[5]

W. Y. Chen and Y. Zhou, Global nonexistence of a semilinear Petrovsky equation, Nonlinear Anal., 70 (2009), 3203-3208.  doi: 10.1016/j.na.2008.04.024.  Google Scholar

[6] V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-Up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.   Google Scholar
[7]

G. Li, Y. N. Sun and W. J. Liu, Global existence, and blow-up of solutions for a strongly damped Petrosky system with nonlinear damping, Applicable Analysis, 91 (2012), 575-586.  doi: 10.1080/00036811.2010.550576.  Google Scholar

[8]

G. Li, Y. N. Sun and W. J. Liu, Global existence, uniform decay and blow-up solutions for a system of Petrosky equations, Nonlinear Anal., 74 (2011), 1523-1538.  doi: 10.1016/j.na.2010.10.025.  Google Scholar

[9]

Y. C. Liu and R. Z. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.  Google Scholar

[10]

M. Marras and S. Vernier-Piro, Lifespan for solutions to 4-th order nonlinear hyperbolic systems with time dependent coefficients, J. Math. Anal. Appl., 480 (2019), 123387, 14 pp. doi: 10.1016/j.jmaa.2019.123387.  Google Scholar

[11]

S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.  Google Scholar

[12]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[13]

A. Peyravi, Lower bounds of blow-up time for a system of semi-linear hyperbolic Petrovsky equations, Acta Math. Sci. Ser. B, 36 (2016), 683-688.  doi: 10.1016/S0252-9602(16)30031-5.  Google Scholar

[14]

G. A. Philippin, Blow-up phenomena for a class of 4th order parabolic problems, Proc. Amer. Math. Soc., 143 (2015), 2507-2513.  doi: 10.1090/S0002-9939-2015-12446-X.  Google Scholar

[15]

G. A. Philippin and S. Vernier-Piro, Lower bounds for the lifespan of solutions for a class of fourth order wave equations, Applied Mathemathics Letters, 50 (2015), 141-145.  doi: 10.1016/j.aml.2015.06.016.  Google Scholar

[16]

G. A. Philippin and S. Vernier-Piro, Behaviour in time of solutions to a class of fourth order evolution equations, J. Math. Anal. Appl., 436 (2016), 718-728.  doi: 10.1016/j.jmaa.2015.11.066.  Google Scholar

[17]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372. doi: 10.1007/BF02418013.  Google Scholar

[18]

S.-T. Wu, Blowup of solutions for a system of nonlinear wave equations with nonlinear damping, Elec. J. Diff. Equ., 2009 (2009), 11 pp.  Google Scholar

[19]

S.-T. Wu and L.-Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Tawainese J. Math., 13 (2009), 545-558.  doi: 10.11650/twjm/1500405355.  Google Scholar

[20]

J. Zhou, Lower bounds for blow-up time of two nonlinear wave equations, Applied Mathemathics Letter, 45 (2015), 64-68.  doi: 10.1016/j.aml.2015.01.010.  Google Scholar

show all references

References:
[1] R. A. Adams and J. J. F. Fourier, Sobolev Spaces, Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

L. J. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math., 55 (1995), 136-155.  doi: 10.1137/S0036139993255327.  Google Scholar

[3]

G. Autuori, F. Colasuonno and P. Pucci, Blow up at infinity of solutions of polyharmonic Kirchhoff systems, Complex Variables and Elliptic Equations, 57 (2012), 379-395.  doi: 10.1080/17476933.2011.592584.  Google Scholar

[4]

G. Autuori and P. Pucci, Local asymptotic stability for polyharmonic Kirchhoff systems, Applicable Analysis, 90 (2011), 493-514.  doi: 10.1080/00036811.2010.483433.  Google Scholar

[5]

W. Y. Chen and Y. Zhou, Global nonexistence of a semilinear Petrovsky equation, Nonlinear Anal., 70 (2009), 3203-3208.  doi: 10.1016/j.na.2008.04.024.  Google Scholar

[6] V. A. Galaktionov, E. L. Mitidieri and S. I. Pohozaev, Blow-Up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.   Google Scholar
[7]

G. Li, Y. N. Sun and W. J. Liu, Global existence, and blow-up of solutions for a strongly damped Petrosky system with nonlinear damping, Applicable Analysis, 91 (2012), 575-586.  doi: 10.1080/00036811.2010.550576.  Google Scholar

[8]

G. Li, Y. N. Sun and W. J. Liu, Global existence, uniform decay and blow-up solutions for a system of Petrosky equations, Nonlinear Anal., 74 (2011), 1523-1538.  doi: 10.1016/j.na.2010.10.025.  Google Scholar

[9]

Y. C. Liu and R. Z. Xu, Fourth order wave equations with nonlinear strain and source terms, J. Math. Anal. Appl., 331 (2007), 585-607.  doi: 10.1016/j.jmaa.2006.09.010.  Google Scholar

[10]

M. Marras and S. Vernier-Piro, Lifespan for solutions to 4-th order nonlinear hyperbolic systems with time dependent coefficients, J. Math. Anal. Appl., 480 (2019), 123387, 14 pp. doi: 10.1016/j.jmaa.2019.123387.  Google Scholar

[11]

S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.  Google Scholar

[12]

L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.  doi: 10.1007/BF02761595.  Google Scholar

[13]

A. Peyravi, Lower bounds of blow-up time for a system of semi-linear hyperbolic Petrovsky equations, Acta Math. Sci. Ser. B, 36 (2016), 683-688.  doi: 10.1016/S0252-9602(16)30031-5.  Google Scholar

[14]

G. A. Philippin, Blow-up phenomena for a class of 4th order parabolic problems, Proc. Amer. Math. Soc., 143 (2015), 2507-2513.  doi: 10.1090/S0002-9939-2015-12446-X.  Google Scholar

[15]

G. A. Philippin and S. Vernier-Piro, Lower bounds for the lifespan of solutions for a class of fourth order wave equations, Applied Mathemathics Letters, 50 (2015), 141-145.  doi: 10.1016/j.aml.2015.06.016.  Google Scholar

[16]

G. A. Philippin and S. Vernier-Piro, Behaviour in time of solutions to a class of fourth order evolution equations, J. Math. Anal. Appl., 436 (2016), 718-728.  doi: 10.1016/j.jmaa.2015.11.066.  Google Scholar

[17]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372. doi: 10.1007/BF02418013.  Google Scholar

[18]

S.-T. Wu, Blowup of solutions for a system of nonlinear wave equations with nonlinear damping, Elec. J. Diff. Equ., 2009 (2009), 11 pp.  Google Scholar

[19]

S.-T. Wu and L.-Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Tawainese J. Math., 13 (2009), 545-558.  doi: 10.11650/twjm/1500405355.  Google Scholar

[20]

J. Zhou, Lower bounds for blow-up time of two nonlinear wave equations, Applied Mathemathics Letter, 45 (2015), 64-68.  doi: 10.1016/j.aml.2015.01.010.  Google Scholar

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