Abstract
We consider the initial boundary value problem for the well-known three-dimensional Rosenau-Burgers equation in the cylinder \((0,L) \otimes \mathbb{S}\) (where \(\mathbb{S} \subset \mathbb{R}^2\)) for some boundary conditions. Using the test-function method, we obtain the result on the blowup of solutions of this initial boundary value problem during a finite time. This is one of the first results in the “blowup” direction for this equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Rosenau, Phys. Rev. A, 40, 7193–7196 (1989).
R. Camassa and D. D. Holm, Phys. Rev. Lett., 71, 1661–1664 (1993); arXiv:patt-sol/9305002v1 (1993).
A. Constantin and J. Escher, Acta Math., 181, 229–243 (1998).
C. Escudero, Phys. Rev. E, 71, 047302 (2005).
H. Liu, J. Nonlinear Math. Phys., 13, 441–466 (2006).
V. A. Galaktionov and S. I. Pokhozhaev, Comput. Math. Math. Phys., 48, 1784–1810 (2008).
E. Mitidieri and S. I. Pokhozhaev, Proc. Steklov Inst. Math., 234, 1–362 (2001).
S. I. Pokhozhaev, Sb. Math., 199, 1355–1365 (2008).
A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Peaking Modes in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1987).
V. A. Galaktionov, E. Mitidieri, and S. I. Pohozaev, Nonlinear Anal., 70, 2930–2952 (2009).
S. I. Pohozaev, Milan J. Math., 77, 127–150 (2009).
N. Hayashi, E. I. Kaikina, P. I. Naumkin, and I. A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations (Lect. Notes Math., Vol. 1884), Springer, Berlin (2006).
L. Liu and M. Mei, Appl. Math. Comput., 131, 147–170 (2002).
M. A. Park, Nonlinear Anal., 21, 77–85 (1993).
A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner, Linear and Nonlinear Equations of Sobolev Type [in Russian], Fizmatlit, Moscow (2007).
H. A. Levine, Arch. Rational Mech. Anal., 51, 371–386 (1973).
H. A. Levine, Trans. Amer. Math. Soc., 192, 1–21 (1974).
V. K. Kalantarov and O. A. Ladyzhenskaya, J. Soviet Math., 10, 53–70 (1978).
L. C. Evans, Partial Differential Equations (Grad. Stud. Math., Vol. 19), Amer. Math. Soc., Providence, R. I. (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 170, No. 3, pp. 342–349, March, 2012.
Rights and permissions
About this article
Cite this article
Korpusov, M.O. Blowup of solutions of the three-dimensional Rosenau-Burgers equation. Theor Math Phys 170, 280–286 (2012). https://doi.org/10.1007/s11232-012-0030-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-012-0030-9