Complutense University Library

On the initial growth of interfaces in reaction-diffusion equations with strong absorption


Díaz Díaz, Jesús Ildefonso and Álvarez León, Luis (1993) On the initial growth of interfaces in reaction-diffusion equations with strong absorption. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 123 (5). pp. 803-817. ISSN 0308-2105

[img] PDF
Restringido a Repository staff only hasta 31 December 2020.


Official URL:


We study the initial growth of the interfaces of non-negative local solutions of the equation u(t) = (u(m))xx - lambdau(q) when m greater-than-or-equal-to 1 and 0 < q < 1. We show that if u(x, 0) greater-than-or-equal-to C(-x)+2/(m-q) with C > C0, for some explicit C0 = C0(lambda, m, q), then the free boundary zeta(t) = sup {x: u(x, t) > 0} is a ''heating front''. More precisely zeta(t) greater-than-or-equal-to at(m-q)/2(1-q) for any t small enough and for some a > 0. If on the contrary, u(x, 0) less-than-or-equal-to C(-x)+2/(m-q) with C < C0, then zeta(t) is a ''cooling front'' and in fact zeta(t) less-than-or-equal-to -at(m-q)/2(1-q) for any t small enough and for some a > 0. Applications to solutions of the associated Cauchy and Dirichlet problems are also given.

Item Type:Article
Uncontrolled Keywords:heat-equation; thermal waves; media
Subjects:Sciences > Mathematics > Differential equations
ID Code:16257

L. Alvarez. On the behavior or the free boundary or some nonhomogeneous elliptic problems.Applicable Anal. 36 (1990). 131-144.

L. Alvarez and J. I. Díaz. On the behavior near the free boundary of solutions of some nonhomogeneous elliptie problems. In Actas IX C.E.D.YA. (Ed. Univ. de Valladolid, 1986), pp. 55-59.

L. Alvarez and .J. I. Díaz. Sufficient and necessary initial mass conditions for the existence of a waiting time in nonlinear-convection processes. J. Math. Anal. Appl. 155 (1991), 378-392.

L. Alvarez, J. I. Díaz and R. Kersner. On the initial growth of the interfaces in non linear diffusion-convection processes. In Nonlillear Diffusion Equations and Their Equilibrium States,eds W.-M. Ni, L. A. Peletier and.J. Serrin, pp. 1-20 (Berlin: Springer, 1988).

S. N. Antontsev and J. I. Díaz. New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by the energy methods. Dokl. Acad. Nauk 303 (1988), 524-529 (in Russian); English translation: Soviet. Mat. Dokl. 38 (1988), 535-539.

C. Bandle and I. Stakgold. The formation or the dead core in parabolic reaction-diffusion problems. Trans. Amer. Math. Soc. 286 (1984),275-293.

Ph. Benilan, M. G. Crandall and M. Pierre. Solutions of the porous medium equation in RN under optimal conditions on initial value. Indiana Univ. Math. J. 33 (1984), 51-87.

F. Bernis. Finite speed of propagation and asymptotic rates for sorne nonlinear higher order parabolic equations with absorption. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 1-19.

M. Bertsch. A class of degenerate diffusion equations with a singular non linear term. Nonlinear Anal. 7 (1983),117-127.

J. Crank and R. S. Gupta. A moving boundary problem arising from the diffusion of oxygen in absorbing tissue. J. Inst. Math. Appl. 10 (1972),19-33.

G. Díaz. On the positivity set of solutions or semilinear equations by stochastic methods. In Free Boundary Problems: Theory and Apptications, Vol. II, eds K. H. Haffman and J. Sprekels pp.833-840 (Harlow: Longman, 1990).

J. I. Díaz. Nonlinear Partia! Differential Equations and Free Boundaries: Vol. I Elliptic Equations, Research Notes in Mathematics 106 (London: Pitman, 1985.

J. I. Díaz and .J. Hernández. Some results on the existence of free boundaries for parabolic reaction-diffusion systems. In Trends in Theory and Practice of Nonlinear Differential Equations, ed. V. Lakshmikanthan, Procceding of a meeting held at Lexington,Texas,June,1982,pp.149-156 (New York: Marcel Dekker. 1984).

J. I. Díaz and J. Hernández. Qualitative properties of free boundaries for some nonlinear degenerate parabolic equations. In Nonlinear Parabolic equations: qualitative properties of Solutions, pp. 85-93 (Harlow: Longman, 1987).

A. Friedman. Partial Differential Equations of the Parabolic Type (Englewood Cliffs N.J.:Prentice-Hall 1969).

A. Friedman and M. A. Herrero. Extinction properties of semilinear heat equations with strong absorption.J. Math. Anal. Appl. 124 (1987). 530-546.

R. E. Grundy. Asymptotic solutions of a model diffusion-reaction equation. IMA J. Appl. Math.40 (1988),53-72.

R. E. Grundy and L. A. Peletier. Short time behavior of a singular solution to the heat equation with absorption. Proc. Roy. Soc. Edinburgh Sect A 107 (1987), 271-288.

R. E. Grundy and L. A. Peletier. The initial interface development for a reaction-diffusion equation with power law initial data. Quart. J. Mech. Appl. Matiz. (to appear).

S. Gutman and R. H. Martin Jr. The porous medium equation with nonlinear absorption and moving boundaries. Israel J. Math. 54 (1986), 81-109.

M. A. Herrero and J. L. Vázquez. The one-dimensional nonlinear heat equation with absorption:Regularity of solutions and interfaces. SIAM J. Math. Anal. 18 (1987), 149-167.

M. A. Herrero and J. L. Vázquez. Thermal waves in absorbing media. J. Differential Equations 74(1988),218-233.

M. A. Herrero and J. J. L. Velazquez. On the dynamics of a semilinear heat equation with strong absorption. Comm. Partial Differential Equations 14 (1989),1653-1715.

A. S. Kalashnikov. The propagation of disturbances in problems of non-linear heat conduction with absorption. USSR. Comput. Math. and Math. Phys. 14 (1974),70-85.

A. S. Kalashnikov. The effect of absorption on heat propagation in a medium in which the thermal conductivity depends on temperature. U.S.S.R. Comput. Math. and Math. Phys. 16(1976),141-149.

S. Kamin, L. A. Peletier and J. L. Vázquez. A nonlinear diffusion-absorption equation with unbounded data. In Nonlinear diffusion equations and their equilibrium states. III,eds N.G.Lloyd et al.,pp.243-263 (Boston: Birkhäuser, 1992).

R. Kersner. The behavior of temperature fronts in media with nonlinear thermal conductivity under absorption. Moscow Univ. Math. Bull. 31 (1976), 90-95.

R. Kersner. Degenerate parabolic equations with general nonlinearities. Nonlinear Anal. 4(1984),1043-1062.

B. F. Knerr. The behavior of the support of solutions of the equation of non linear heat conduction with absorption in one dimension. Trans.Amer.Math.Soc.249(1979),409-424.

M. Langlais and D. Phillips. Stabilization of solutions of nonlinear and degenerate evolution problems. Nonlinear Anal. 9 (1985), 321-333.

I. B. Palymskii. Some qualitative properties of solutions of nonlinear heat equations for nonlinear heat conductivity with absorption. In Chislenye Met. Mekhz. Sploshnoi Sredy (Novosibirsk) 16(1985),136-145.

P. Rosenau and S. Kamin. Thermal waves in an absorbing and convecting medium. Phys. D 8(1983),273-283.

J. L. Vázquez. The interfaces of one-dimensional flows in porous media. Trans. Amer. Matiz. Soc.285 (1984), 717-737.

Deposited On:10 Sep 2012 09:37
Last Modified:07 Feb 2014 09:26

Repository Staff Only: item control page