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A bifurcation for a generalized Burgers' equation in dimension one

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  • 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.
    Mathematics Subject Classification: Primary: 35A01, 35B32, 35K55; Secondary: 35B44.


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  • [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.


    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.


    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.


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


    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.


    J. von Below and G. Pincet MaillyBlow Up for some nonlinear parabolic problems with convection under dynamical boundary conditions, in "Discrete Continuous Dynam. Systems," 2007, Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, suppl., 1031-1041.


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


    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.

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