# American Institute of Mathematical Sciences

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:
 [1] H. Amann, "Ordinary Differential Equations. An Introduction to Nonlinear Analysis,", Translated from the German by Gerhard Metzen, 13 (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.   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.  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.   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.  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.   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, 13 (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.   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.  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.   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.  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.   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
 [1] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [2] Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093 [3] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [4] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [5] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [6] Tetsuya Ishiwata, Shigetoshi Yazaki. A fast blow-up solution and degenerate pinching arising in an anisotropic crystalline motion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2069-2090. doi: 10.3934/dcds.2014.34.2069 [7] Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809 [8] Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225 [9] Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077 [10] Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 [11] Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117 [12] Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure & Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 [13] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [14] Monica Marras, Stella Vernier Piro. On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients. Conference Publications, 2013, 2013 (special) : 535-544. doi: 10.3934/proc.2013.2013.535 [15] Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333 [16] Mingzhu Wu, Zuodong Yang. Existence of boundary blow-up solutions for a class of quasiliner elliptic systems for the subcritical case. Communications on Pure & Applied Analysis, 2007, 6 (2) : 531-540. doi: 10.3934/cpaa.2007.6.531 [17] Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549 [18] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [19] Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051 [20] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058

2019 Impact Factor: 1.233