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June  2012, 5(3): 683-706. doi: 10.3934/dcdss.2012.5.683

A bifurcation for a generalized Burgers' equation in dimension one

Jean-François Rault 1,

1. 

LMPA Joseph Liouville (ULCO) FR 2956 CNRS, Université Lille Nord de France, 50 rue F. Buisson, B.P. 699, F-62228 Calais Cedex, France

Received  August 2010 Revised  April 2011 Published  October 2011

We consider the generalized Burgers' equation \begin{eqnarray*} \left\{ \begin{array}{ll} \partial_t u = \partial_x^2u - u \partial_x u + u^p - \lambda u &\textrm{ in } \overline{\Omega} \textrm{ for } t>0, \\ \mathcal{B}(u)=0 & \textrm{ on } \partial \Omega \textrm{ for } t>0, \\ u(\cdot,0) = \varphi \geq 0 & \textrm{ in } \overline{\Omega}, \end{array} \right. \end{eqnarray*} with $p>1$, $\lambda \in \mathbb{R}$, $\Omega$ a subdomain of $\mathbb{R}$, and where $\mathcal{B}(u)=0$ denotes some boundary conditions. First, using some phase plane arguments, we study the existence of stationary solutions under the Dirichlet or the Neumann boundary conditions and prove a bifurcation depending on the parameter $\lambda$. Then, we compare positive solutions of the parabolic equation with appropriate stationary solutions to prove that global existence can occur when $\mathcal{B}(u)=0$ stands for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions $\sigma \partial_t u + \partial _\nu u=0$. Finally, for many boundary conditions, global existence and blow up phenomena for solutions of the nonlinear parabolic problem in an unbounded domain $\Omega$ are investigated by using some standard super-solutions and some weighted $L^1-$norms.
Keywords: Bifurcation, Blow-up, Existence of solution, Phase plane..
Mathematics Subject Classification: Primary: 35A01, 35B32, 35K55; Secondary: 35B4.
Citation: Jean-François Rault. A bifurcation for a generalized Burgers' equation in dimension one. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 683-706. doi: 10.3934/dcdss.2012.5.683
References:
[1]

H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis," Translated from the German by Gerhard Metzen, de Gruyter Studies in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1990.  Google Scholar

[2]

C. Bandle, J. von Below and W. Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Math. Appl., 17 (2006), 35-37.  Google Scholar

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J. von Below and C. De Coster, A qualitative theory for parabolic problems under dynamical boundary conditions, Journal of Inequalities and Applications, 5 (2000), 467-486. doi: 10.1155/S1025583400000266.  Google Scholar

[4]

J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21 (1996), 255-279.  Google Scholar

[5]

J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Communications in Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.  Google Scholar

[6]

J. von Below and G. Pincet Mailly, Blow Up for some nonlinear parabolic problems with convection under dynamical boundary conditions,, in, 2007 (): 1031.   Google Scholar

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J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364.  Google Scholar

[8]

J.-F. Rault, Phénomène d'Explosion et Existence Globale pour Quelques Problèmes Paraboliques sous les Conditions au Bord Dynamiques, Thèse Doctorale à l'Université du Littoral Côte d'Opale, 2010. Google Scholar

show all references

References:
[1]

H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis," Translated from the German by Gerhard Metzen, de Gruyter Studies in Mathematics, 13, Walter de Gruyter & Co., Berlin, 1990.  Google Scholar

[2]

C. Bandle, J. von Below and W. Reichel, Parabolic problems with dynamical boundary conditions: Eigenvalue expansions and blow up, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Math. Appl., 17 (2006), 35-37.  Google Scholar

[3]

J. von Below and C. De Coster, A qualitative theory for parabolic problems under dynamical boundary conditions, Journal of Inequalities and Applications, 5 (2000), 467-486. doi: 10.1155/S1025583400000266.  Google Scholar

[4]

J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion, Comm. Partial Differential Equations, 21 (1996), 255-279.  Google Scholar

[5]

J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Communications in Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.  Google Scholar

[6]

J. von Below and G. Pincet Mailly, Blow Up for some nonlinear parabolic problems with convection under dynamical boundary conditions,, in, 2007 (): 1031.   Google Scholar

[7]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364.  Google Scholar

[8]

J.-F. Rault, Phénomène d'Explosion et Existence Globale pour Quelques Problèmes Paraboliques sous les Conditions au Bord Dynamiques, Thèse Doctorale à l'Université du Littoral Côte d'Opale, 2010. Google Scholar

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