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

## A bifurcation for a generalized Burgers' equation in dimension one

 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.
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:
  H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis,", Translated from the German by Gerhard Metzen, 13 (1990). Google Scholar  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. Google Scholar  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.  doi: 10.1155/S1025583400000266.  Google Scholar  J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion,, Comm. Partial Differential Equations, 21 (1996), 255. Google Scholar  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.  doi: 10.1081/PDE-120019380.  Google Scholar  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  J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. Google Scholar  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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##### References:
  H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis,", Translated from the German by Gerhard Metzen, 13 (1990). Google Scholar  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. Google Scholar  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.  doi: 10.1155/S1025583400000266.  Google Scholar  J. von Below and S. Nicaise, Dynamical interface transition in ramified media with diffusion,, Comm. Partial Differential Equations, 21 (1996), 255. Google Scholar  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.  doi: 10.1081/PDE-120019380.  Google Scholar  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  J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. Google Scholar  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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