On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity

References

[1] M. M. Al-Gharabli and S. A. Messaoudi, Existence and a general decay result for a plate equation with non-linear damping and a logarithmic source term, J. Evol. Equ. 18 (2018), 105–125. 10.1007/s00028-017-0392-4Search in Google Scholar

[2] I. Białynicki Birula and J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Polon. Sci. 23 (1975), no. 4, 461–466. Search in Google Scholar

[3] J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys. 118 (1988), no. 1, 15–29. 10.1007/BF01218475Search in Google Scholar

[4] J. Boussinesq, Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris 72 (1871), 755–759. Search in Google Scholar

[5] T. Cazenave and A. Haraux, Équations d’évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5) 2 (1980), no. 1, 21–51. 10.5802/afst.543Search in Google Scholar

[6] G. Chen, Y. Wang and S. Wang, Initial boundary value problem of the generalized cubic double dispersion equation, J. Math. Anal. Appl. 299 (2004), no. 2, 563–577. 10.1016/j.jmaa.2004.05.044Search in Google Scholar

[7] C. I. Christov and M. G. Velarde, Evolution and interactions of solitary wave (solitons) in nonlinear dissipative systems, Phys. Scripta 1994 (1994), no. T55, 101–106. 10.1088/0031-8949/1994/T55/017Search in Google Scholar

[8] H. Ding and J. Zhou, Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearity, Nonlinear Anal. Real World Appl. 67 (2022), Paper No. 103587. 10.1016/j.nonrwa.2022.103587Search in Google Scholar

[9] G. Eilenberger, Solitons: Mathematical Methods for Physicists, Springer Ser. Solid-State Sci. 19, Springer, Berlin, 2012. Search in Google Scholar

[10] H. A. Erbay, S. Erbay and A. Erkip, Instability and stability properties of traveling waves for the double dispersion equation, Nonlinear Anal. 133 (2016), 1–14. 10.1016/j.na.2015.11.019Search in Google Scholar

[11] L. G. Farah, Local solutions in Sobolev spaces with negative indices for the “good” Boussinesq equation, Comm. Partial Differential Equations 34 (2009), no. 1–3, 52–73. 10.1080/03605300802682283Search in Google Scholar

[12] P. Górka, Logarithmic Klein–Gordon equation, Acta Phys. Polon. B 40 (2009), no. 1, 59–66. Search in Google Scholar

[13] A. Hasegawa, Plasma Instabilities and Nonlinear Effects, Springer, Berlin, 2012. Search in Google Scholar

[14] Q. Hu and H. Zhang, Initial boundary value problem for generalized logarithmic improved Boussinesq equation, Math. Methods Appl. Sci. 40 (2017), no. 10, 3687–3697. 10.1002/mma.4255Search in Google Scholar

[15] Q. Hu, H. Zhang and G. Liu, Global existence and exponential growth of solution for the logarithmic Boussinesq-type equation, J. Math. Anal. Appl. 436 (2016), no. 2, 990–1001. 10.1016/j.jmaa.2015.11.082Search in Google Scholar

[16] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University, Cambridge, 2000. 10.1017/CBO9781139171281Search in Google Scholar

[17] S. Jihong, Z. Mingyou, W. Xingchang, L. Bowei and X. Runzhang, Global well-posedness for strongly damped multidimensional generalized Boussinesq equations, Math. Methods Appl. Sci. 39 (2016), no. 15, 4437–4450. 10.1002/mma.3873Search in Google Scholar

[18] N. Kutev, N. Kolkovska and M. Dimova, Global existence of Cauchy problem for Boussinesq paradigm equation, Comput. Math. Appl. 65 (2013), no. 3, 500–511. 10.1016/j.camwa.2012.05.024Search in Google Scholar

[19] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form P ⁢ u t ⁢ t = - A ⁢ u + ℱ ⁢ ( u ) , Trans. Amer. Math. Soc. 192 (1974), 1–21. 10.2307/1996814Search in Google Scholar

[20] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal. 5 (1974), 138–146. 10.1137/0505015Search in Google Scholar

[21] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal. 9 (2020), no. 1, 613–632. 10.1515/anona-2020-0016Search in Google Scholar

[22] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations 106 (1993), no. 2, 257–293. 10.1006/jdeq.1993.1108Search in Google Scholar

[23] F. Linares and M. Scialom, Asymptotic behavior of solutions of a generalized Boussinesq type equation, Nonlinear Anal. 25 (1995), no. 11, 1147–1158. 10.1016/0362-546X(94)00236-BSearch in Google Scholar

[24] Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal. 26 (1995), no. 6, 1527–1546. 10.1137/S0036141093258094Search in Google Scholar

[25] Y. Liu and R. Xu, Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation, Phys. D 237 (2008), no. 6, 721–731. 10.1016/j.physd.2007.09.028Search in Google Scholar

[26] Y. Liu and R. Xu, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl. 338 (2008), no. 2, 1169–1187. 10.1016/j.jmaa.2007.05.076Search in Google Scholar

[27] E. Pişkin and N. Irkıl, Well-posedness results for a sixth-order logarithmic Boussinesq equation, Filomat 33 (2019), no. 13, 3985–4000. 10.2298/FIL1913985PSearch in Google Scholar

[28] A. V. Porubov, Amplification of Nonlinear Strain Waves in Solids, World Scientific, River Edge, 2003. 10.1142/5238Search in Google Scholar

[29] X. Runzhang, Y. Yanbing, L. Bowei, S. Jihong and H. Shaobin, Global existence and blowup of solutions for the multidimensional sixth-order “good” Boussinesq equation, Z. Angew. Math. Phys. 66 (2015), 955–976. 10.1007/s00033-014-0459-9Search in Google Scholar

[30] J. S. Russell, The Wave of Translation in the Oceans of Water, Air, and Ether, Trübner, London, 1885. Search in Google Scholar

[31] A. M. Samsonov, On the existence of solitons of longitudinal deformation in an infinite nonlinearly elastic rod, Sov. Phys. Tch. Phys. 33 (1988), 989–991. Search in Google Scholar

[32] A. M. Samsonov, Nonlinear strain waves in elastic waveguides, Nonlinear Waves in Solids (Udine 1993), CISM Courses and Lect. 341, Springer, Vienna (1994), 349–382. 10.1007/978-3-7091-2444-4_6Search in Google Scholar

[33] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968), 148–172. 10.1007/BF00250942Search in Google Scholar

[34] M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon. 36 (1991), no. 2, 371–379. Search in Google Scholar

[35] V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations 9 (1996), no. 3, 619–634. 10.57262/die/1367969976Search in Google Scholar

[36] S. Wang and G. Chen, The Cauchy problem for the generalized IMBq equation in W s , p ⁢ ( 𝐑 n ) , J. Math. Anal. Appl. 266 (2002), no. 1, 38–54. 10.1006/jmaa.2001.7670Search in Google Scholar

[37] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal. 64 (2006), no. 1, 159–173. 10.1016/j.na.2005.06.017Search in Google Scholar

[38] S. Wang and X. Hongxia, Global solution for a generalized Boussinesq equation, Appl. Math. Comput. 204 (2008), 130–136. 10.1016/j.amc.2008.06.059Search in Google Scholar

[39] S. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl. 45 (2019), 116–141. 10.1016/j.nonrwa.2018.06.012Search in Google Scholar

[40] A.-M. Wazwaz, Gaussian solitary waves for the logarithmic Boussinesq equation and the logarithmic regularized Boussinesq equation, Ocean Eng. 94 (2015), 111–115. 10.1016/j.oceaneng.2014.11.024Search in Google Scholar

[41] R.-Z. Xu, Y.-B. Luo, J.-H. Shen and S.-B. Huang, Global existence and blow up for damped generalized Boussinesq equation, Acta Math. Appl. Sin. (Engl. Ser.) 33 (2017), no. 1, 251–262. 10.1007/s10255-017-0655-4Search in Google Scholar

[42] R. Xue, Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation, J. Math. Anal. Appl. 316 (2006), no. 1, 307–327. 10.1016/j.jmaa.2005.04.041Search in Google Scholar