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On the blow-up of solutions of the Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations. (English) Zbl 1238.35101
Summary: We study sufficient conditions of the blow-up of solutions of initial-boundary-value problems for the well-known Benjamin-Bona-Mahony-Burgers and Rosenau-Burgers equations on a segment. Note that this is the first result for these equations in the “blow-up” area.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Model equations for long waves in nonlinear dispersive systems, Phil. trans. R. soc. ser. A, 272, 47-78, (1972) · Zbl 0229.35013
[2] Rosenau, P., Extending hydrodynamics via the regularization of the chapman – enskog expansion, Phys. rev. A, 40, 12, 7193-7196, (1989)
[3] Albert, J.P., On the decay of solutions of the generalized benjamin – bona – mahony equation, J. math. anal. appl., 141, 2, 527-537, (1989) · Zbl 0697.35116
[4] Avrin, J.D.; Goldstein, J.A., Global existence for the benjamin – bona – mahony equation in arbitrary dimensions, Nonlinear anal., 9, 8, 861-865, (1985) · Zbl 0591.35012
[5] Bisognin, E.; Bisognin, V.; Charao, C.R.; Pazoto, A.F., Asymptotic expansion for a dissipative benjamin – bona – mahony equation with periodic coefficients, Port. math. (NS), 60, 4, 473-504, (2003) · Zbl 1102.35074
[6] Biler, P., Long-time behavior of the generalized benjamin – bona – mahony equation in two space dimensions, Differential integral equations, 5, 4, 891-901, (1992) · Zbl 0759.35012
[7] Chen, Yu., Remark on the global existence for the generalized benjamin – bona – mahony equations in arbitrary dimension, Appl. anal., 30, 1-3, 1-15, (1988) · Zbl 0631.35080
[8] Hayashi, N.; Kaikina, E.I.; Naumkin, P.I.; Shishmaryov, I.A., Asymptotics for dissipative nonlinear equations, (2006), Springer-Verlag New York
[9] Hagen, T.; Turi, J., On a class of nonlinear BBM-like equations, Comput. appl. math., 17, 2, 161-172, (1998) · Zbl 0914.35069
[10] Mei, M.; Liu, L.; Wong, Y.S., Asymptotic behavior of solutions to the rosenau – burgers equation with a periodic initial boundary, Nonlinear anal., 67, 8, 2527-2539, (2007) · Zbl 1123.35057
[11] Mei, M.; Liu, L., A better asymptotic profile of the rosenau – burgers equation, Appl. math. comput., 131, 1, 147-170, (2002) · Zbl 1020.35097
[12] Park, M.A., On the rosenau equation in multidimensional space, J. nonlinear anal., 21, 1, 77-85, (1993) · Zbl 0811.35142
[13] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 11, 1661-1664, (1993) · Zbl 0972.35521
[14] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta math., 181, 2, 229-243, (1998) · Zbl 0923.76025
[15] Galaktionov, V.A.; Pokhozhaev, S.I., Equations of nonlinear third-order dispersion: shock waves, rarefaction waves, and breakdown waves, Zh. vychisl. mat. mat. fiz., 48, 10, 1819-1846, (2008) · Zbl 1177.76183
[16] Mitidieri, E.L.; Pokhozhaev, S.I., A priori estimates and the absence of solutions of partial differential inequalities, Tr. mat. inst. steklova, 234, (2001) · Zbl 0987.35002
[17] Pokhozhaev, S.I., On the blow-up of solutions of the kuramoto – sivashinsky equation, Mat. sb., 199, 9, 97-106, (2008) · Zbl 1161.35491
[18] Samarsky, A.A.; Galaktionov, V.A.; Kurdyumov, S.P.; Mikhailov, A.P., Sharpening regimes in problems for quasilinear parabolic equations, (1987), Nauka Moscow, (in Russian)
[19] Mitidieri, E.L.; Galaktionov, V.A.; Pohozaev, S.I., On global solutions and blow-up for kuramoto – sivashinsky-type models and well-posed Burnett equations, Nonlinear. anal. TMA, 70, 8, 2930-2952, (2009) · Zbl 1176.35094
[20] Pokhozaev, S.I., Critical nonlinearities in partial differential equations, Milan J. math., 77, 1, 127-150, (2009) · Zbl 1205.35024
[21] Sveshnikov, A.G.; Al’shin, A.B.; Korpusov, M.O.; Pletner, Yu.D., Linear and nonlinear Sobolev-type equations, (2007), Fizmatlit Moscow, (in Russian) · Zbl 1179.35007
[22] Levine, H.A., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \(P u_t = - A u + F(u)\), Arch. ration. mech. anal., 51, 371-386, (1973) · Zbl 0278.35052
[23] Levine, H.A.; Pucci, P.; Serrin, J., Some remarks on the global nonexistence problem for nonautonomous abstract evolution equations, Contemp. math., 208, 253-263, (1997) · Zbl 0882.35081
[24] Evans, L.K., Partial differential equations, (1998), AMS New York
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