Well-posedness of the generalized Burgers equation on a finite interval

References

• Burgers JM. A mathematical model illustrating the theory of turbulence. New York: Academic Press; 1948. 171–199. (Advances in Applied Mechanics; vol. 1).
   Google Scholar
• Hopf E. The partial differential equation ut+uux=νuxx. Commun Pure Appl Math. 1950;3:201–230.
   Web of Science ®Google Scholar
• Cole JD. On a quasilinear parabolic equation occurring in aerodynamics. Quart Appl Math. 1951;9:225–236.
   Web of Science ®Google Scholar
• Dix DB. Nonuniqueness and uniqueness in the initial-value problem for Burgers equation. SIAM J Math Anal. 1996;27:708–724.
   Web of Science ®Google Scholar
• Bekiranov D. The initial-value problem for the generalized Burgers equation. Differ Integral Equ. 1996;9:1253–1265.
   Google Scholar
• Molinet L, Ribaud F, Youssfi A. Ill-posedness issues for a class of parabolic equations. Proc R Soc Edinb. 2002;132:1407–1416.
   Google Scholar
• Deng K, Kwong MK, Levine HA. The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers equation. Quart Appl Math. 1992;50:173–200.
   Web of Science ®Google Scholar
• Henkin GM, Shananin AA. Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations. J Math Pures Appl. 2004;83:1457–1500.
   Web of Science ®Google Scholar
• Kato M. Large time behavior of solutions to the generalized Burgers equations. Osaka J Math. 2007;44:923–943.
   Web of Science ®Google Scholar
• Kiselev A, Nazarov F, Shterenberg R. Blow up and regularity for fractal Burgers equation. Dyn Partial Differ Equ. 2008;5:211–240.
   Web of Science ®Google Scholar
• Joseph KT, Sachdev PL. Initial boundary value problems for scalar and vector Burgers equations. Stud Appl Math. 2001;106:481–505.
   Web of Science ®Google Scholar
• Tani A. On the first initial-boundary value problem of the generalized Burgers equation. Publ Res Inst Math Sci. 1974;10:209–233.
   Google Scholar
• Burns JA, Kang S. A control problem for Burgers equations with bounded input/output. Nonlinear Dynam. 1991;2:235–262.
   Google Scholar
• Krstic M. On global stabilization of Burgers' equation by boundary control. Syst Control Lett. 1999;37:123–141.
   Web of Science ®Google Scholar
• Bona JL, Sun SM, Zhang B-Y. Non-homogeneous boundary value problems for the Korteweg-de Vries and the Korteweg-de Vrie-Burgers equations in a quarter plane. Ann Inst H Poincar$\acute{e}$ Anal Non Lin$\acute{e}$aire. 2008;25:1145–1185.
   Web of Science ®Google Scholar
• Molinet L, Vento S. Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case. Ann Sc Norm Super Pisa Cl Sci. 2011;10:531–560.
   Google Scholar
• Molinet L, Vento S. Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case. Trans Amer Math Soc. 2013;365:123–141.
   Web of Science ®Google Scholar
• Bona J, Scott L. Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces. Duke Math J. 1976;43:87–99.
   Web of Science ®Google Scholar
• Tartar L. Interpolation non line´aire et re´gularite´. J Funct Anal. 1972;9:469–489.
   Google Scholar
• Bona JL, Sun S, Zhang B-Y. A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane. Trans Amer Math Soc. 2001;354:427–490.
   Web of Science ®Google Scholar
• Bona JL, Sun SM, Zhang B-Y. A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Comm Partial Differ Equ. 2003;28:1391–1436.
   Web of Science ®Google Scholar
• Bona JL, Sun SM, Zhang B-Y. A non-homogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain II. J Differ Equ. 2009;247:2558–2596.
   Web of Science ®Google Scholar
• Lions JL, Magenes E. Non-homogeneous boundary value problems and applications. Berlin: Springer-Verlag; 1972.
   Google Scholar
• Li J, Zhang B-Y, Zhang ZX. A nonhomogeneous boundary value problem for the Kuramoto-Sivashinsky equation in a quarter plane. Math Meth Appl Sci. 2017;40:5619–5641.
   Web of Science ®Google Scholar
• Li J, Zhang ZX. Nonhomogeneous boundary value problems for the complex Ginzburg-Landau equation posed on a finite interval. Appl Anal. DOI:10.1080/00036811.2017.1343468
   Web of Science ®Google Scholar
• Pazy A. Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag; 1983. (Applied Mathematical Sciences; vol. 44).
   Google Scholar
• Esfahani A. Sharp well-posedness of the Ostrovsky, Stepanyams and Tsimring equation. Math Commun. 2013;18:323–335.
   Web of Science ®Google Scholar
• Pilod D. Sharp well-posedness results for the Kuramoto-Velarde equation. Commun Pure Appl Anal. 2008;7:867–881.
   Web of Science ®Google Scholar