Summary
In this paper we examine the first initial boundary value problem for ut=uxx + ε(1 − u)−β,ɛ > 0,β > 0,on (0, 1) × (0, ∞) from the point of view of dynamical systems. We construct the set of stationary solutions, determine those which are stable, those which are not and show that there are solutions with initial data arbitrarily close to unstable stationary solutions which quench (reach one in finite time). We also examine the related problem ut=uxx, 0 <x < 1,t > 0;u(0,t)=0, ε(1 − u(1, t))−β. The set of stationary solutions for this problem, and the dynamical behavior of solutions of the time dependent problem are somewhat different.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.Acker - B.Kawohl,Remarks on quenching, Nonlinear Analysis - TMA (to appear).
A. Acker -W. Walter,On the global existence of solutions of parabolic equations with a singular nonlinear term, Nonlinear Analysis, TMA,2 (1978), pp. 499–504.
A. Acker -W. Walter,The quenching problem for nonlinear parabolic equations, Lecture Notes in Mathematics,564, Springer-Verlag, New York, 1976.
C. Bandle -C.-M. Brauner,Singular perturbation method in a parabolic problem with free boundary, inProc. BAIL IVth Conference, Novosibirsk, Boole Press, Dublin 1987.
B.Gidas -Wei-Ming Ni - L.Nirenberg,Symmetry and related properties via the maximum principle, Comm. Math. Phys.,68 (1979).
M. W. Hirsch,Differential equations and convergence almost everywhere strongly monotone flows, Contemp. Math.,17, A.M.S., Providence, R.I., 1983, pp. 267–285.
D. D. Joseph -T. S. Lundgren,Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal.,49 (1973), pp. 241–269.
H. Kawarada,On solutions of initial boundary value problem for u t =u xx + 1/(1 −u), RIMS Kyoto Univ.,10 (1975), pp. 729–736.
H. A. Levine,The phenomenon of quenching; a survey, inTrends in the Theory and Practice of Nonlinear Analysis,V. Lakshmikantham (ed.), Elsevier Science Publ., North Holland, 1985, pp. 275–286.
H. A. Levine,The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, SIAM J. Math. Anal.,14 (1983), pp. 1139–1153.
H. A. Levine -G. M. Lieberman,Quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions, J. Reine Ang. Math.,345 (1983), pp. 23–38.
H. A. Levine -J. T. Montgomery,The quenching of solutions of some nonlinear parabolic problems, SIAM J. Math. Anal.,11 (1980), pp. 842–847.
G. M. Lieberman,Quenching of solutions of evolution equations, Proc. Centre Math. Anal. Aust. Nat. Univ.,8 (1984), pp. 151–157.
H. Matano,Asymptotic behavior and stability of solutions of semilinear diffusion equations, Pub. Res Inst. Mat. Sci.,15 (1979), pp. 401–454.
H. Matano,Existence of nontrivial unstable sets for equilibrium of strongly order preserving systems, J. Fac. Sci. U. Tokyo Se. IA,30 (1984), pp. 645–673.
D. Phillips,Existence of solutions to a quenching problem, Appl. Anal.,24 (1987), pp. 353–364.
R. A.Smith,On a hyperbolic quenching problem in several dimensions, SIAM J. Math. Anal. (in press).
Keng Deng - H. A.Levine,On the blow up of ut at quenching, Proc. A.M.S. (in press).
Guo Jong-Sheng,On the quenching behavior of a semilinear parabolic equation, JMAA (in press).
Author information
Authors and Affiliations
Additional information
This research was sponsored by the U.S. Air Force Office of Scientific Research, Air Forse Systems Command Grants 84-0252 and 88-0031. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation therein.
Rights and permissions
About this article
Cite this article
Levine, H.A. Quenching, nonquenching, and beyond quenching for solution of some parabolic equations. Annali di Matematica pura ed applicata 155, 243–260 (1989). https://doi.org/10.1007/BF01765943
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01765943