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.  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.  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.  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, 23 (1968).   Google Scholar

[6]

G. Pincet Mailly, "Explosion des Solutions de Problèmes Paraboliques Sous Conditions au Bord Dynamiques,", Ph.D. Thesis, (2001).   Google Scholar

[7]

W. Walter, "Differential and Integral Inequalities,", Springer-Verlag, (1970).   Google Scholar

show all references

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.  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.  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.  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, 23 (1968).   Google Scholar

[6]

G. Pincet Mailly, "Explosion des Solutions de Problèmes Paraboliques Sous Conditions au Bord Dynamiques,", Ph.D. Thesis, (2001).   Google Scholar

[7]

W. Walter, "Differential and Integral Inequalities,", Springer-Verlag, (1970).   Google Scholar

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