Stable and unstable initial configuration in the theory wave fronts

Previous Article
On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces
DCDS-S Home
This Issue
Next Article
Multiple solutions for a perturbed system on strip-like domains

August 2012, 5(4): 797-808. doi: 10.3934/dcdss.2012.5.797

Stable and unstable initial configuration in the theory wave fronts

Ruediger Landes ^1,

1.	University of Oklahoma, Noman, OK 73019, United States

Received March 2011 Revised May 2011 Published November 2011

Abstract
Related Papers

In this paper we study the wavefront like phase transition of solutions of a parabolic nonlinear boundary value problem used to model phase transitions in the theory of boiling liquids. Using weak supersolutions we provide bounds for the propagation speed of such a phase transition. Also we construct stable supersolutions to initial configurations which have locally supercritical values.

Keywords: Wavefronts, Parabolic equations, Nonlinear Boundary Conditions, Phastransitions, Supersolutions..

Mathematics Subject Classification: Primary: 35K99, Secondary: 35B35, 35B40, 35D3.

Citation: Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797

References:

[1]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.
[2]	H. Auracher, W. Marquardt, M. Buchholz, R. Hohl, T. Lüttich and J. Blum, New experimental results on steady-state and transient pool boiling heat transfer, Therm. Sci. Engng, 9 (2001), 29-39.
[3]	J. Blum, T. Lüttich and W. Marquardt, Temperature Wave Propagation as a Route from Nucleate to Film Boiling?, In "2nd International Symposium on Two-Phase Flow Modelling and Experimentation" (eds. G. P. Celata, P. DiMarco and R. K. Shah), 1, Rome, Edizioni ETS, Pisa, (1999), 137-144.
[4]	V. K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech., 30 (1998), 365-401. doi: 10.1146/annurev.fluid.30.1.365.
[5]	P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.
[6]	R. Landes, Wavefront solution in the theory of boiling liquids, Analysis (Munich), 29 (2009), 283-298.
[7]	T. Lüttich, W. Marquardt, M. Buchholz and H. Auracher, "Towards a Unifying Heat Transfer Correlation for the Entire Boiling Curve," 5th International Conference on Boiling Heat Transfer, Montego Bay, Jamaica, May 2003.
[8]	M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a nonlinear pool boiling model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1475-1494. doi: 10.1016/j.cnsns.2006.11.001.
[9]	M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1518-1537. doi: 10.1016/j.cnsns.2006.11.002.
[10]	J. R. Thome, Boiling, in "Handbook of Heat Transfer" (eds. A. Bejan and A. D. Krause), Wiley & Sons, New York, (2003), 635-717.

show all references

References:

[1]	D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.
[2]	H. Auracher, W. Marquardt, M. Buchholz, R. Hohl, T. Lüttich and J. Blum, New experimental results on steady-state and transient pool boiling heat transfer, Therm. Sci. Engng, 9 (2001), 29-39.
[3]	J. Blum, T. Lüttich and W. Marquardt, Temperature Wave Propagation as a Route from Nucleate to Film Boiling?, In "2nd International Symposium on Two-Phase Flow Modelling and Experimentation" (eds. G. P. Celata, P. DiMarco and R. K. Shah), 1, Rome, Edizioni ETS, Pisa, (1999), 137-144.
[4]	V. K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech., 30 (1998), 365-401. doi: 10.1146/annurev.fluid.30.1.365.
[5]	P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.
[6]	R. Landes, Wavefront solution in the theory of boiling liquids, Analysis (Munich), 29 (2009), 283-298.
[7]	T. Lüttich, W. Marquardt, M. Buchholz and H. Auracher, "Towards a Unifying Heat Transfer Correlation for the Entire Boiling Curve," 5th International Conference on Boiling Heat Transfer, Montego Bay, Jamaica, May 2003.
[8]	M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a nonlinear pool boiling model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1475-1494. doi: 10.1016/j.cnsns.2006.11.001.
[9]	M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1518-1537. doi: 10.1016/j.cnsns.2006.11.002.
[10]	J. R. Thome, Boiling, in "Handbook of Heat Transfer" (eds. A. Bejan and A. D. Krause), Wiley & Sons, New York, (2003), 635-717.

[1]	Dung Le. Strong positivity of continuous supersolutions to parabolic equations with rough boundary data. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1521-1530. doi: 10.3934/dcds.2015.35.1521
[2]	Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
[3]	Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637
[4]	Asadollah Aghajani, Craig Cowan. Explicit estimates on positive supersolutions of nonlinear elliptic equations and applications. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2731-2742. doi: 10.3934/dcds.2019114
[5]	Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations and Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[6]	Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763
[7]	Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432
[8]	Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[9]	Giuseppe Da Prato, Alessandra Lunardi. On a class of elliptic and parabolic equations in convex domains without boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 933-953. doi: 10.3934/dcds.2008.22.933
[10]	Ciprian G. Gal, Mahamadi Warma. Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evolution Equations and Control Theory, 2016, 5 (1) : 61-103. doi: 10.3934/eect.2016.5.61
[11]	B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure and Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29
[12]	Raluca Clendenen, Gisèle Ruiz Goldstein, Jerome A. Goldstein. Degenerate flux for dynamic boundary conditions in parabolic and hyperbolic equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 651-660. doi: 10.3934/dcdss.2016019
[13]	Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047
[14]	Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
[15]	Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control and Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[16]	Junichi Harada, Mitsuharu Ôtani. $H^2$-solutions for some elliptic equations with nonlinear boundary conditions. Conference Publications, 2009, 2009 (Special) : 333-339. doi: 10.3934/proc.2009.2009.333
[17]	Le Thi Phuong Ngoc, Khong Thi Thao Uyen, Nguyen Huu Nhan, Nguyen Thanh Long. On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions. Communications on Pure and Applied Analysis, 2022, 21 (2) : 585-623. doi: 10.3934/cpaa.2021190
[18]	Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure and Applied Analysis, 2007, 6 (2) : 487-503. doi: 10.3934/cpaa.2007.6.487
[19]	Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193
[20]	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

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Ruediger Landes
on Google Scholar
Ruediger Landes

[Back to Top]