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On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions
A note on a class of 4th order hyperbolic problems with weak and strong damping and superlinear source term
Università di Cagliari, Dipartimento di Matematica e Informatica, Viale Merello 92, 09123 Cagliari, Italy |
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.
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372.
doi: 10.1007/BF02418013. |
| [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. |
| [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. |
| [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. |
show all references
To our friend Patrizia on the occasion of her sixty-fifth birthday.
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372.
doi: 10.1007/BF02418013. |
| [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. |
| [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. |
| [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. |
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