Existence and nonexistence of global solutions for logarithmic hyperbolic equation

Yaojun Ye; Qianqian Zhu; Yaojun Ye; Qianqian Zhu

doi:10.3934/era.2022054

Electronic Research Archive

2022, Volume 30, Issue 3: 1035-1051. doi: 10.3934/era.2022054

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Research article

Existence and nonexistence of global solutions for logarithmic hyperbolic equation

Yaojun Ye ^,,
Qianqian Zhu

Department of Mathematics and Statistics, Zhejiang University of Science and Technology, Hangzhou 310023, China

Received: 18 January 2022 Revised: 15 February 2022 Accepted: 20 February 2022 Published: 08 March 2022

This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.
- logarithmic hyperbolic equation,
- stable and unstable sets,
- existence and nonexistence,
- global solutions,
- initial-boundary value problem
Citation: Yaojun Ye, Qianqian Zhu. Existence and nonexistence of global solutions for logarithmic hyperbolic equation[J]. Electronic Research Archive, 2022, 30(3): 1035-1051. doi: 10.3934/era.2022054

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Abstract

This article is concerned with the initial-boundary value problem for a equation of quasi-hyperbolic type with logarithmic nonlinearity. By applying the Galerkin method and logarithmic Sobolev inequality, we prove the existence of global weak solutions for this problem. In addition, by means of the concavity analysis, we discuss the nonexistence of global solutions in the unstable set and give the lifespan estimation of solutions.

References

[1]	H. Buljan, A. Siber, M. Soljacic, T. Schwartz, M. Segev, D. N. Christodoulides, Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media, Phys. Rev. E, 68 (2003), 036607. https://doi.org/10.1103/PhysRevE.68.036607 doi: 10.1103/PhysRevE.68.036607
[2]	S. De Martino, M. Falanga, C. Godano, G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport, Europhys. Lett., 63 (2003), 472–475.
[3]	W. Krolikowski, D. Edmundson, O. Bang, Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, 61 (2000), 3122–3126. https://doi.org/10.1103/PhysRevE.61.3122
[4]	P. Gorka, Logarithmic Klein-Gordon equation, Acta Phys. Pol. B, 40 (2009), 59–66.
[5]	I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Pol. Sci. Ser. Sci. Phys. Astron., 23 (1975), 461–466.
[6]	I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann. Phys., 100 (1976), 62–93. https://doi.org/10.1016/0003-4916(76)90057-9
[7]	K. Bartkowski, P. Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, 41 (2008), 355201, 11 pp. https://doi.org/10.1088/1751-8113/41/35/355201 doi: 10.1088/1751-8113/41/35/355201
[8]	T. Cazenave, A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 2 (1980), 21–51.
[9]	J. Ball, Remarks on blow up and nonexistence theorems for nonlinear evolutions equations, Q. J. Math., 28 (1977), 473–486. https://doi.org/10.1093/qmath/28.4.473 doi: 10.1093/qmath/28.4.473
[10]	M. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17 (1972), 173–193.
[11]	H. A. Levine, L. E. Payne, Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations, J. Math. Anal. Appl., 55 (1976), 329–334. https://doi.org/10.1016/0022-247X(76)90163-3 doi: 10.1016/0022-247X(76)90163-3
[12]	Y. J. Ye, Existence and nonexistence of solutions of the initial-boundary value problem for some degenerate hyperbolic equation, Acta Math. Sci., 25B (2005), 703–709. https://doi.org/10.1016/S0252-9602(17)30210-2 doi: 10.1016/S0252-9602(17)30210-2
[13]	S. Ibrahim, A. Lyaghfouri, Blow-up solutions of quasilinear hyperbolic equations with critical Sobolev exponent, Math. Modell. Nat. Phenom., 7 (2012), 66–76. https://doi.org/10.1051/mmnp/20127206 doi: 10.1051/mmnp/20127206
[14]	V. A. Galaktionov, S. I. Pohozaev, Blow-up and critical exponents for nonlinear hyperbolic equations, Nonlinear Anal. TMA, 53 (2003), 453–466. https://doi.org/10.1016/S0362-546X(02)00311-5 doi: 10.1016/S0362-546X(02)00311-5
[15]	Y. Ye, Exponential decay of energy for some nonlinear hyperbolic equations with strong dissipation, Adv. Differ. Equations, (2010), 1–12. https://doi.org/10.1186/1687-1847-2010-357404 https://doi.org/10.1155/2010/357404
[16]	Y. J. Ye, Global existence and asymptotic behavior of solutions for some nonlinear hyperbolic equation, J. Inequal. Appl., 2010 (2010), 1–10. https://doi.org/10.1155/2010/895121 doi: 10.1155/2010/895121
[17]	V. Komornik, Exact Controllability and Stabilization: the Multiplier Method, Paris, 1994.
[18]	S. A. Messaoudi, B. S. Houari, Global non-existence of solutions of a class of wave equations with non-linear damping and source terms, Math. Methods Appl. Sci., 27 (2004), 1687–1696. https://doi.org/10.1002/mma.522 doi: 10.1002/mma.522
[19]	C. Chen, H. Yao, L. Shao, Global existence, uniqueness, and asymptotic behavior of solution for p-Laplacian type wave equation, J. Inequal. Appl., 2010 (2010), 1–13.
[20]	L. C. Nhan, T. X. Le, Global solution and blow-up for a class of $p$-Laplacian evolution equations with logarithmic nonlinearity, Acta Appl. Math., 151 (2017), 149–169. https://doi.org/10.1016/j.camwa.2017.02.030 doi: 10.1016/j.camwa.2017.02.030
[21]	Y. Z. Han, C. L. Cao, P. Sun, A $p$-Laplace equation with logarithmic nonlinearity at high initial energy level, Acta Appl. Math., 164 (2019), 155–164. https://doi.org/10.1007/s10440-018-00230-4 doi: 10.1007/s10440-018-00230-4
[22]	H. Ding, J. Zhou, Global existence and blow-up for a mixed pseudo-parabolic $p$-Laplacian type equation with logarithmic nonlinearity, J. Math. Anal. Appl., 478 (2019), 393–420. https://doi.org/10.1080/00036811.2019.1695784 https://doi.org/10.1080/00036811.2019.1695784 doi: 10.1080/00036811.2019.1695784
[23]	T. Boudjeriou, Global existence and blow-Up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 162 (2020), 1–24. https://doi.org/10.1007/s00009-020-01584-6 doi: 10.1007/s00009-020-01584-6
[24]	W. Lian, M. S. Ahmed, R. Z. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239–257. https://doi.org/10.1016/j.na.2019.02.015 doi: 10.1016/j.na.2019.02.015
[25]	Y. C. Liu, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differ. Equations, 192 (2003), 155–169.
[26]	L. E. Payne, D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22 (1975), 273–303. https://doi.org/10.1007/BF02761595 doi: 10.1007/BF02761595
[27]	D. H. Sattinger, On global solutions for nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148–172. https://doi.org/10.1007/BF00250942
[28]	Y. C. Liu, J. S. Zhao, On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal. TMA, 64 (2006) 2665–2687. https://doi.org/10.1016/j.na.2005.09.011
[29]	L. Wang, H. Garg, Algorithm for multiple attribute decision-making with interactive archimedean norm operations under pythagorean fuzzy uncertainty, Int. J. Comput. Intell. Syst., 14 (2021), 503–527. https://doi.org/10.2991/ijcis.d.201215.002 doi: 10.2991/ijcis.d.201215.002
[30]	J. L. Lions, Quelques Mthodes de Rsolution des Problmes aux Limites Nonlinaires, Paris, 1969.
[31]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uralyseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, (1967), 23.
[32]	S. M. Zheng, Nonlinear Evolution Equations, Chapman and Hall/CRC, 2004.
[33]	H. Chen, P. Luo, G. W. Liu, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422 (2015), 84–98. https://doi.org/10.1016/j.jmaa.2014.08.030 doi: 10.1016/j.jmaa.2014.08.030
[34]	H. Chen, S. Y. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differ. Equations, 258 (2015), 4424–4442. https://doi.org/10.1016/j.jde.2015.01.038 doi: 10.1016/j.jde.2015.01.038
[35]	L. Gross, Logarithmic Sobolev inequalities, Am. J. Math., 97 (1975), 1061–1083. https://doi.org/10.2307/2373688 doi: 10.2307/2373688
[36]	M. D. Pino, J. Dolbeault, I. Gentil, Nonlinear diffusions, hypercontractivity and the optimal $L^p-$ Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl., 293 (2004), 375–388. https://doi.org/10.1016/j.jmaa.2003.10.009 doi: 10.1016/j.jmaa.2003.10.009
[37]	F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincaré-AN, 23 (2006), 185–207. https://doi.org/10.1016/j.anihpc.2005.02.007 doi: 10.1016/j.anihpc.2005.02.007
[38]	H. A. Levine, Some nonexistence and instability theorems for formally parabolic equations of the form $Pu_t = -Au + F(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371–386.
[39]	V. K. Kalantarov, O. A. Ladyzhenskaya, The occurrence of collapse for quasi-linear equation of parabolic and hyperbolic typers, J. Sov. Math., 10 (1978), 53–70.

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