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

[1]

Joachim von Below, Gaëlle Pincet Mailly. Blow up for some nonlinear parabolic problems with convection under dynamical boundary conditions. Conference Publications, 2007, 2007 (Special) : 1031-1041. doi: 10.3934/proc.2007.2007.1031

[2]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671

[3]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267

[4]

Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015

[5]

Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193

[6]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[7]

Monica Marras, Stella Vernier Piro. Blow up and decay bounds in guasi linear parabolic problems. Conference Publications, 2007, 2007 (Special) : 704-712. doi: 10.3934/proc.2007.2007.704

[8]

Julián López-Gómez, Pavol Quittner. Complete and energy blow-up in indefinite superlinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 169-186. doi: 10.3934/dcds.2006.14.169

[9]

Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637

[10]

Igor Chueshov, Björn Schmalfuss. Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 315-338. doi: 10.3934/dcds.2007.18.315

[11]

Davide Guidetti. Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1401-1417. doi: 10.3934/cpaa.2016.15.1401

[12]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63

[13]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[14]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523

[15]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

[16]

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

[17]

Davide Guidetti. Classical solutions to quasilinear parabolic problems with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 717-736. doi: 10.3934/dcdss.2016024

[18]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181

[19]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737

[20]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

[Back to Top]