Abstract
The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u t + g(t, u)u x + f(t, u) = εu xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max |u x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well.
Article PDF
Similar content being viewed by others
References
Bardos C., Leroux A.Y., Nedelec J.C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979)
Berestycki H., Kamin S., Sivashinsky G.: Nonlinear dynamics and metastability in a Burgers type equation (for upward propagating flames). C. R. Acad. Sci. Paris Ser. I Math. 321(2), 185–190 (1995)
Kruzhkov, S.N.: First order quasilinear equations with several independent variables. Math. Sbornik, 81(123), n.2, 228–255 (1970) (Russian). English trans. in: Math. USSR Sbornik, 10, 217–243 (1970)
Kruzhkov, S.N.: Quasilinear parabolic equations and systems with two independent variables. Trudy Sem. Petrovsk. 5, 217–272 (1979) (Russian). English transl. in: Topics in Modern Math., Consultant Bureau, New York (1985)
Ladyzhenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. (in Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23 American Math. Society, Providence, R.I. 1967, 648 pp.
Lardner R.W., Arya J.C.: Two generalizations of Burger’s equation. Acta Mech. 37, 179–190 (1980)
Murray J.D.: Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations. SIAM J. Appl. Math. 19, 273–298 (1970)
Murray J.D.: On the Gunn effect and other physical examples of perturbed conservation equations. J. Fluid Mech. 44, 315–346 (1970)
Murray J.D.: On Burgers’ model equations for turbulence. J. Fluid Mech. 59, 263–279 (1973)
Rao C.S., Sachdev P.L., Ramaswamy M.: Self-similar solutions of a generalized Burgers equation with nonlinear damping. Nonlinear Anal. Real World Appl. 4, 723–741 (2003)
Sadchev P.L., Nair K.R.C., Tikekar V.G.: Generalized Burgers equations and Euler-Painleve transcendents I. J. Math. Phys. 27(6), 1506–1522 (1986)
Sun X., Ward M.J.: Metastability for a generalized Burgers equation with applications to propagating flame fronts. Eur. J. Appl. Math. 10(1), 27–53 (1999)
Sadchev P.L., Nair K.R.C.: Generalized Burgers equations and Euler-Painleve transcendents II. J. Math. Phys. 28(5), 997–1004 (1987)
Tersenov Al.S., Tersenov Ar.S.: On the Bernstein-Nagumo’s condition in the theory of nonlinear parabolic equations. J. Reine Angew. Math. 572, 197–217 (2004)
Tersenov Al.S., Tersenov Ar.S.: The Cauchy problem for a class of quasilinear parabolic equations. Ann. Mat. Pura Appl. (4) 182(3), 325–336 (2003)
Vaganan M., Kumaran S.: Similarity solutions of the Burgers equation with linear damping. Appl. Math. Lett. 17(10), 1191–1196 (2004)
Yu J., Kevorkian J.: Nonlinear evolution of small disturbances into roll-waves in an inclined open channel. J. Fluid. Mech. 243, 575–594 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tersenov, A.S. On the generalized Burgers equation. Nonlinear Differ. Equ. Appl. 17, 437–452 (2010). https://doi.org/10.1007/s00030-010-0061-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-010-0061-6