On initial-boundary value problem on semiaxis for generalized Kawahara equation
Contemporary Mathematics. Fundamental Directions, Proceedings of the S.M. Nikolskii Mathematical Institute of RUDN University, Tome 65 (2019) no. 4, pp. 683-699.

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In this paper, we consider initial-boundary value problem on semiaxis for generalized Kawahara equation with higher-order nonlinearity. We obtain the result on existence and uniqueness of the global solution. Also, if the equation contains the absorbing term vanishing at infinity, we prove that the solution decays at large time values.
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A. V. Faminskii; E. V. Martynov. On initial-boundary value problem on semiaxis for generalized Kawahara equation. Contemporary Mathematics. Fundamental Directions, Proceedings of the S.M. Nikolskii Mathematical Institute of RUDN University, Tome 65 (2019) no. 4, pp. 683-699. http://geodesic.mathdoc.fr/item/CMFD_2019_65_4_a9/

[1] A. T. Il'ichev, “On the properties of one fifth-order nonlinear evolution equation describing wave processes in media with weak dispersion”, Proc. Math. Inst. Russ. Acad. Sci., 186, 1989, 222–226 (in Russian)

[2] R. V. Kuvshinov, A. V. Faminskiy, “Mixed problem in a half-strip for the Kawahara equation”, Differ. Equ., 45:3 (2009), 391–402 (in Russian) | MR | Zbl

[3] A. V. Marchenko, “On long waves in shallow water under the ice cover”, Appl. Math. Mech., 52:2 (1988), 230–234 (in Russian) | Zbl

[4] P. I. Naumkin, “Estimates of decreasing for solutions of the Cauchy problem for the modified Kawahara equation”, Math. Digest, 210:5 (2019), 72–108 (in Russian) | DOI | MR | Zbl

[5] M. A. Opritova, A. V. Faminskiy, “On the initial-boundary value problem in a half-strip for generalized Kawahara equation”, Ukr. Math. Bull., 11:3 (2014), 312–339 (in Russian)

[6] M. A. Opritova, A. V. Faminskiy, “On decreasing for large time values of solutions of the initial-boundary value problem for the generalized Kawahara equation in a half-strip”, Bull. Tambov State Univ., 20:5 (2015), 1331–1337 (in Russian)

[7] K. Sangare, “Mixed problem in a half-strip for the generalized Kawahara equation in the space of infinitely differentiable exponentially decreasing functions”, Bull. Peoples' Friendship Univ. Russ. Ser. Math., 10:1 (2003), 91–107 (in Russian)

[8] K. Sangare, A. V. Faminskiy, “Weak solutions of mixed problem for the generalized Kawahara equation in a half-strip”, Math. Notes, 85:1 (2009), 98–109 (in Russian) | DOI | MR | Zbl

[9] N. A. Shananin, “On partial quasianalyticity of generalized solutions of weakly nonlinear differential equations with weighted derivatives”, Math. Notes, 68:4 (2000), 608–619 (in Russian) | DOI | Zbl

[10] F. D. Araruna, R. A. Capisrano-Filho, G. G. Doronin, “Energy decay for the modified Kawahara equation posed in a bounded domain”, J. Math. Anal. Appl., 385:2 (2012), 743–756 | DOI | MR | Zbl

[11] M. M. Cavalcanti, V. N. Domingos Cavalcanti, A. Faminskii, F. Natali, “Decay of solutions to damped Korteweg—de Vries equation”, Appl. Math. Optim., 65 (2012), 221–251 | DOI | MR | Zbl

[12] M. Cavalcanti, Ch. Kwak, The initial-boundary value problem for the Kawahara equation on the half-line, 2018, arXiv: 1805.05229 [math.AP]

[13] W. Chen, Z. Guo, “Global well-posedness and I-method for the fifth-order Korteweg—de Vries equation”, J. Anal. Math., 114:1 (2011), 121–156 | DOI | MR | Zbl

[14] G. G. Doronin, N. A. Larkin, “Quarter-plane problem for the Kawahara equation”, Pac. J. Appl. Math., 1:3 (2008), 151–176

[15] G. G. Doronin, F. Natali, “Exponential decay for a locally damped fifth-order equation posed on a line”, Nonlinear Anal., Real World Appl., 30 (2016), 59–72 | DOI | MR | Zbl

[16] A. V. Faminskii, N. A. Larkin, “Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval”, Electron. J. Differ. Equ., 2010, no. 1, 1–20 | MR

[17] A. V. Faminskii, E. V. Martynov, “Large-time decay of solutions to the damped Kawahara equation posed on a half-line”, Differential Equations on Manifolds and Mathematical Physics (to appear)

[18] T. Kawahara, “Oscillatory solitary waves in dispersive media”, J. Phys. Soc. Jpn., 33:1 (1972), 260–264 | DOI

[19] S. Kichenassamy, P. J. Olver, “Existence and nonexistence of solitary wave solutions to higher-order model evolution equations”, SIAM J. Math. Anal., 23:5 (1992), 1141–1166 | DOI | MR | Zbl

[20] N. A. Larkin, M. Simoes, “The Kawahara equation on bounded intervals and on a half-line”, Nonlinear Anal., 127 (2015), 397–412 | DOI | MR | Zbl

[21] F. Linares, A. F. Pazoto, “Asymptotic behavior of the Korteweg—de Vries equation posed in a quarter plane”, J. Differ. Equ., 246 (2009), 1342–1353 | DOI | MR | Zbl

[22] A. F. Pazoto, R. Rosier, “Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line”, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1511–1535 | MR | Zbl

[23] S. P. Tao, S. B. Cui, “Local and global existence of solutions to initial value problems of modified nonlinear Kawahara equation”, Acta Math. Sin. (Engl. Ser.), 21:5 (2005), 1035–1044 | DOI | MR | Zbl

[24] C. F. Vasconcellos, P. N. da Silva, “Stabilization of the Kawahara equation with localized damping”, ESAIM Control. Optim. Calc. Var., 17 (2011), 102–116 | DOI | MR | Zbl