# American Institute of Mathematical Sciences

June  2013, 6(3): 825-836. doi: 10.3934/dcdss.2013.6.825

## Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions

 1 LMPA Joseph Liouville, FR 2596 CNRS, Université Lille Nord de France ULCO, 50, rue F. Buisson, B.P. 699, F-62228 Calais Cedex, France 2 LMPA Joseph Liouville, FR 2596 CNRS, Université Lille Nord de France ULCO, 50, rue F. Buisson, B.P. 699, F-62228 Calais Cedex,, France 3 LMPA Joseph Liouville (ULCO) FR 2956 CNRS, Université Lille Nord de France, 50 rue F. Buisson, B.P. 699, F-62228 Calais Cedex

Received  June 2010 Revised  April 2012 Published  December 2012

We investigate the blow up points of the one--dimensional parabolic Burgers' equation $$\partial_t u=\partial_x^2 u-u\partial_xu+u^p$$ under a dissipative dynamical boundary condition $\sigma \partial_t u+\partial_\nu u=0$ for one bump initial data. A numerical example of a solution pertaining exactly two bumps stemming from its initial data is presented. Moreover, we discuss the growth order of the $L^\infty$--norm of the solutions when approaching the blow up time.
Citation: Joachim von Below, Gaëlle Pincet Mailly, Jean-François Rault. Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 825-836. doi: 10.3934/dcdss.2013.6.825
##### References:
 [1] 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 [2] J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Comm. Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.  Google Scholar [3] J. von Below and G. Pincet Mailly, "Blow up for Some Nonlinear Parabolic Problems with Convection under Dynamical Boundary Conditions,", Discr. Cont. Dyn. Systems, 2007 (): 1031.   Google Scholar [4] A. Friedman and B. McLeod, Blow up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar [5] O. A. Ladyženskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Trans. of Math. Monographs 23, A.M.S., Providence, R.I., 1968.  Google Scholar [6] G. Pincet Mailly, "Explosion des Solutions de Problèmes Paraboliques Sous Conditions au Bord Dynamiques," Ph.D. Thesis, Université du Littoral Côte d'Opale, 2001. Google Scholar [7] W. Walter, "Differential and Integral Inequalities," Springer-Verlag, Berlin, 1970.  Google Scholar

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##### References:
 [1] 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 [2] J. von Below and G. Pincet Mailly, Blow up for reaction diffusion equations under dynamical boundary conditions, Comm. Partial Differential Equations, 28 (2003), 223-247. doi: 10.1081/PDE-120019380.  Google Scholar [3] J. von Below and G. Pincet Mailly, "Blow up for Some Nonlinear Parabolic Problems with Convection under Dynamical Boundary Conditions,", Discr. Cont. Dyn. Systems, 2007 (): 1031.   Google Scholar [4] A. Friedman and B. McLeod, Blow up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar [5] O. A. Ladyženskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type," Trans. of Math. Monographs 23, A.M.S., Providence, R.I., 1968.  Google Scholar [6] G. Pincet Mailly, "Explosion des Solutions de Problèmes Paraboliques Sous Conditions au Bord Dynamiques," Ph.D. Thesis, Université du Littoral Côte d'Opale, 2001. Google Scholar [7] W. Walter, "Differential and Integral Inequalities," Springer-Verlag, Berlin, 1970.  Google Scholar
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