On the global existence and blow-up for the double dispersion equation with exponential term

Xiao Su; Hongwei Zhang; Xiao Su; Hongwei Zhang

doi:10.3934/era.2023023

Electronic Research Archive

2023, Volume 31, Issue 1: 467-491. doi: 10.3934/era.2023023

Previous Article Next Article

Research article

On the global existence and blow-up for the double dispersion equation with exponential term

Xiao Su ^,,
Hongwei Zhang

College of Science, Henan University of Technology, Zhengzhou 450001, China

Academic Editor: Michael Winkler

Received: 17 September 2022 Revised: 03 November 2022 Accepted: 06 November 2022 Published: 11 November 2022

This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively.
- double dispersion equation,
- nonlinear damped,
- exponential nonlinearity,
- global existence,
- blow-up
Citation: Xiao Su, Hongwei Zhang. On the global existence and blow-up for the double dispersion equation with exponential term[J]. Electronic Research Archive, 2023, 31(1): 467-491. doi: 10.3934/era.2023023

Related Papers:

Abstract

This paper deals with the initial boundary value problem for the double dispersion equation with nonlinear damped term and exponential growth nonlinearity in two space dimensions. We first establish the local well-posedness in the natural energy space by the standard Galërkin method and contraction mapping principle. Then, we prove the solution is global in time by taking the initial data inside the potential well and the solution blows up in finite time as the initial data in the unstable set. Moreover, finite time blow-up results are provided for negative initial energy and for arbitrary positive initial energy respectively.

References

[1]	A. M. Samsonov, E. V. Sokurinskaya, Energy exchange between nonlinear waves in elastic waveguides and external media, in Nonlinear Waves in Active Media, Springer, Berlin, (1989), 99–104. https://doi.org/10.1007/978-3-642-74789-2_13
[2]	A. M. Samsonov, Nonlinear strain waves in elastic waveguide, in Nonlinear Waves in Solids, Springer, Wien, 1994. https://doi.org/10.1007/978-3-7091-2444-4_6
[3]	G. Chen, Y. Wang, S. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl., 299 (2004), 563–577. https://doi.org/10.1016/j.jmaa.2004.05.044 doi: 10.1016/j.jmaa.2004.05.044
[4]	G. Chen, H. Xue, Periodic boundary value problem and Cauchy problem of the generalized cubic double dispersion equation, Acta Math. Sci., 28 (2008), 573–587. https://doi.org/10.1016/S0252-9602(08)60060-0 doi: 10.1016/S0252-9602(08)60060-0
[5]	S. Wang, G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64 (2006), 159–173. https://doi.org/10.1016/j.na.2005.06.017 doi: 10.1016/j.na.2005.06.017
[6]	Y. Liu, R. Xu, Potential well method for initial boundary value problem of the generalized double dispersion equations, Commun. Pur. Appl. Anal., 7 (2008), 63–81. https://doi.org/10.3934/cpaa.2008.7.63 doi: 10.3934/cpaa.2008.7.63
[7]	N. Polat, A. Ertaş, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10–20. https://doi.org/10.1016/j.jmaa.2008.08.025 doi: 10.1016/j.jmaa.2008.08.025
[8]	S. Wang, D. Fang, On the asymptotic behaviour of solution for the generalized double dispersion equation, Appl. Anal., 92 (2013), 1179–1193. https://doi.org/10.1080/00036811.2012.661044 doi: 10.1080/00036811.2012.661044
[9]	Y. Wang, S. Chen, Asymptotic profile of solutions to the double dispersion equation, Nonlinear Anal., 134 (2016), 236–254. https://doi.org/10.1016/j.na.2016.01.009 doi: 10.1016/j.na.2016.01.009
[10]	Y. Wang, C. Wei, Asymptotic profile of global solutions to the generalized double dispersion equation via the nonlinear term, Z. Angew. Math. Phys., 69 (2018), 34. https://doi.org/10.1007/s00033-018-0930-0 doi: 10.1007/s00033-018-0930-0
[11]	X. Su, S. Wang, The initial-boundary value problem for the generalized double dispersion equation, Z. Angew. Math. Phys., 68 (2017), 53. https://doi.org/10.1007/s00033-017-0798-4 doi: 10.1007/s00033-017-0798-4
[12]	S. Adachi, K. Tanaka, Trudinger type inequalities in $\mathbb{R}^N$ and their best exponent, Proc. Am. Math. Soc., 128 (2000), 2051–2057.
[13]	D. R. Adams, A Sharp inequality of J. Moser for higher order derivatives, Ann. Math., 128 (1988), 385–398. https://doi.org/10.2307/1971445 doi: 10.2307/1971445
[14]	J. Moser, A sharp form of an inequality by N. Trudinger, Indiana U. Math. J., 20 (1971), 1077–1092. https://www.jstor.org/stable/24890183
[15]	N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473–483. http://www.jstor.org/stable/24901677
[16]	Q. Hu, C. Zhang, H. Zhang, Global existence of solution for Cauchy problem of two-dimensional Boussinesq-type equation, ISRN Math. Anal., 2014 (2014). https://doi.org/10.1007/978-3-7091-2444-4_6
[17]	H. Zhang, Q. Hu, Global existence and nonexistence of solution for Cauchy problem of two-dimensional generalized Boussinesq equations, J. Math. Anal. Appl., 422 (2015), 1116–1130. https://doi.org/10.1016/j.jmaa.2014.09.036 doi: 10.1016/j.jmaa.2014.09.036
[18]	S. Guo, Y. Yang, High energy blow up for two-dimensional generalized exponential-type Boussinesq equation, Nonlinear Anal., 197 (2020), 111864. https://doi.org/10.1016/j.na.2020.111864 doi: 10.1016/j.na.2020.111864
[19]	M. Nakamura, T. Ozawa, Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth, Math. Z., 231 (1999), 479–487. https://doi.org/10.1007/PL00004737 doi: 10.1007/PL00004737
[20]	A. A. Baraket, Local existence and estimations for a semilinear wave equation in two dimension space, Boll. Unione Mat. Ital., 7-A (2004), 1–21. http://www.bdim.eu/item?id=BUMI_2004_8_7B_1_1_0
[21]	S. Ibrahim, M. Majdoub, N. Masmoudi, Global solutions for a semilinear, two-dimensional Klein-Gordon equation with exponential-type nonlinearity, Commun. Pur. Appl. Math., 59 (2006), 1639–1658. https://doi.org/10.1002/cpa.20127 doi: 10.1002/cpa.20127
[22]	M. Struwe, The critical nonlinear wave equation in two space dimensions, J. Eur. Math. Soc., 15 (2013), 1805–1823. https://doi.org/10.4171/JEMS/404 doi: 10.4171/JEMS/404
[23]	M. Struwe, Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in two space dimensions, Math. Ann., 350 (2011), 707–719. https://doi.org/10.1007/s00208-010-0567-6 doi: 10.1007/s00208-010-0567-6
[24]	T. F. Ma, J. A. Soriano, On weak solutions for an evolution equation with exponential nonlinearities, Nonlinear Anal., 37 (1999), 1029–1038. https://doi.org/10.1016/S0362-546X(97)00714-1 doi: 10.1016/S0362-546X(97)00714-1
[25]	O. Mahouachi, T. Saanouni, Global well-posedness and linearization of a semilinear 2D wave equation with exponential type nonlinearity, Georgian Math. J., 17 (2010), 543–562. https://doi.org/10.1515/gmj.2010.026 doi: 10.1515/gmj.2010.026
[26]	O. Mahouachi, T. Saanouni, Well and ill posedness issues for a 2D wave equation with exponential nonlinearity, J. Partial Differ. Equations, 24 (2011), 361–384. https://doi.org/10.4208/jpde.v24.n4.7 doi: 10.4208/jpde.v24.n4.7
[27]	C. O. Alves, M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var., 34 (2009), 377–411. https://doi.org/10.1007/s00526-008-0188-z doi: 10.1007/s00526-008-0188-z
[28]	S. Tarek, Fourth-order damped wave equation with exponential growth nonlinearity, Ann. Henri Poincaré, 18 (2017), 345–374. https://doi.org/10.1007/s00023-016-0512-7 doi: 10.1007/s00023-016-0512-7
[29]	T. Saanouni, Blowing-up semilinear wave equation with exponential nonlinearity in two space dimensions, Proc. Math. Sci., 123 (2013), 365–372. https://doi.org/10.1007/s12044-013-0132-9 doi: 10.1007/s12044-013-0132-9
[30]	E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. An., 149 (1999), 155–182. https://doi.org/10.1007/s002050050171 doi: 10.1007/s002050050171
[31]	S. A. Messaoudi, Blow up in a nonlinearly damped wave equation, Math. Nachr., 231 (2001), 105–111. https://doi.org/10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I
[32]	V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differ. Equations, 109 (1994), 295–308. https://doi.org/10.1006/jdeq.1994.1051 doi: 10.1006/jdeq.1994.1051
[33]	X. Su, S. Wang, A blow-up result with arbitrary positive initial energy for nonlinear wave equations with degenerate damping terms, J. Partial Differ. Equations, 32 (2019), 181–190. https://doi.org/10.4208/jpde.v32.n2.7 doi: 10.4208/jpde.v32.n2.7
[34]	V. Komornik, Exact Control Lability and Stabilization the Multiplier Method, John Wiley-Masson, Paris, 1994.
[35]	D. Fujiwara, Concrete characterization of the domains of fractional powers of some differential operators of second order, Proc. Japan. Acad., 43 (1967), 82–86. https://doi.org/10.3792/pja/1195521686 doi: 10.3792/pja/1195521686
[36]	J. L. Lions, Queleques Methods de résolution des problémes aux Limits Nonlineares, Dunod/Gautier-Villars, Paris, 1969.
[37]	W. A. Starauss, On continuity of functions with values in various Banach space, Pac. J. Math., 19 (1966), 543–551. https://doi.org/10.2140/pjm.1966.19.543 doi: 10.2140/pjm.1966.19.543
[38]	K. Agre, M. A. Rammaha, Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions, Differ. Integr. Equations, 14 (2001), 1315–1331.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)