American Institute of Mathematical Sciences

Advanced
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, 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

[1]

Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627
[2]

Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244
[3]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[4]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[5]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[6]

Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403
[7]

Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $ x $-dependent coefficients. Evolution Equations & Control Theory, 2020, 9 (1) : 153-179. doi: 10.3934/eect.2020001
[8]

Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025
[9]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[10]

V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191
[11]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[12]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[13]

Juntang Ding, Xuhui Shen. Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4243-4254. doi: 10.3934/dcdsb.2018135
[14]

John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84
[15]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[16]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595
[17]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
[18]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205
[19]

Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105
[20]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (38)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences