# American Institute of Mathematical Sciences

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

## Stable and unstable initial configuration in the theory wave fronts

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

Received  March 2011 Revised  May 2011 Published  November 2011

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.
Citation: Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete & 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.  Google Scholar [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. Google Scholar [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. Google Scholar [4] V. K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech., 30 (1998), 365-401. doi: 10.1146/annurev.fluid.30.1.365.  Google Scholar [5] P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.  Google Scholar [6] R. Landes, Wavefront solution in the theory of boiling liquids, Analysis (Munich), 29 (2009), 283-298.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. R. Thome, Boiling, in "Handbook of Heat Transfer" (eds. A. Bejan and A. D. Krause), Wiley & Sons, New York, (2003), 635-717. Google Scholar

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.  Google Scholar [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. Google Scholar [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. Google Scholar [4] V. K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech., 30 (1998), 365-401. doi: 10.1146/annurev.fluid.30.1.365.  Google Scholar [5] P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.  Google Scholar [6] R. Landes, Wavefront solution in the theory of boiling liquids, Analysis (Munich), 29 (2009), 283-298.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. R. Thome, Boiling, in "Handbook of Heat Transfer" (eds. A. Bejan and A. D. Krause), Wiley & Sons, New York, (2003), 635-717. Google Scholar
 [1] Dung Le. Strong positivity of continuous supersolutions to parabolic equations with rough boundary data. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021190 [18] Chunlai Mu, Zhaoyin Xiang. Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux. Communications on Pure & 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 & 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

2020 Impact Factor: 2.425