Complutense University Library

Global existence for reaction-diffusion systems modelling ignition

Herrero, Miguel A. and Lacey, Andrew A. and Velázquez, J.J. L. (1998) Global existence for reaction-diffusion systems modelling ignition. Archive for Rational Mechanics and Analysis , 142 (3). pp. 219-251. ISSN 0003-9527

[img] PDF
Restricted to Repository staff only until 31 December 2020.

285kB

Official URL: http://www.springerlink.com/content/mq3y2ugwfqk5qc3l/fulltext.pdf

View download statistics for this eprint

==>>> Export to other formats

Abstract

The pair of parabolic equations u(t) = a Δ u + f(u,v), (1) v(t) = b Δ b - f(u, v), (2) with a > 0 and b > 0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f. Of particular interest is the case where f(u, v)= ve(u), (3) which appears in the Frank-Kamenetskii approximation to Arrhenius-type reactions, We show here that for a large choice of the nonlinearity f(u,v) in (1), (2) (including the model case (3)), the corresponding initial-value problem for(1), (2) in the whole space with bounded initial data has a solution which exists for all times. Finite-time blow-up might occur, though, for other choices of function f(ld, v), and we discuss here a linear example which strongly hints at such behaviour.

Item Type:Article
Uncontrolled Keywords:Semilinear heat-equation; blow-up; boundedness
Subjects:Sciences > Mathematics > Differential equations
ID Code:16949
References:

D. Andreucci, M. A. Herrero & J. J. L. Velazquez: Liouville theorems and blow-up behaviour in semilinear reaction-diffusion systems. Ann. Inst. Henri Poincaré 14 (1997), 1–53.

C. Alvarez-Pereira & J.M. Vega: Global stability of a premixed reaction zone (time-dependent Liñan’s problem), SIAM J. Math. Anal. 21 (1990), 884–904.

D. G. Aronson: Non-negative solutions of linear parabolic equations. Annali Sc. Norm. Sup. Pisa 22 (1968), 607–694.

A. Barabarova: On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity. Proc. Amer. Math. Soc. 122 (1994), 827–831.

P. Baras & L.Cohen: Complete blowup after Tmax for the solution of a semilinear heat equation. J. Func. Anal. 71 (1987), 142–174.

J. Bebernes & D. Eberly: Mathematical problems from combustion theory, Springer (1989).

J. Bebernes & A.A. Lacey: Finite-time blow-up for a particular parabolic system. SIAM J. Math. Anal. 21 (1990), 1415–1425.

J. Bebernes & A.A. Lacey: Finite-time blow-up for semilinear reactive-diffusive systems. J. Diff. Eqs. 96 (1992), 105–129.

J. Duoandikoetxea: Análisis de Fourier. Ediciones de la Universidad Autónoma de Madrid (1991).

M. Escobedo & M.A. Herrero: Boundedness and blow-up for a semilinear reaction-diffusion system. J. Diff. Eq. 89 (1991), 176–202.

S. Filippas & R.V.Kohn: Refined asymptotics for the blow up of ut−Δu = uP . Comm. Pure Appl. Math. 45 (1992), 821–869.

S. Filippas & F. Merle: Modulation theory for the blow-up of vector valued nonlinear heat equations. J. Diff. Eq. 116 (1995), 119–148.

Y. Giga & R.V.Kohn: Asymptotically self-similar blow-up of semilinear parabolic equations. Comm. Pure Appl. Math. 38 (1985), 297–319.

M. De Guzman: Differentiation of integrals in RN. Springer Lecture Notes in Mathematics nº 481 (1975).

S. L. Hollis, R.H.Martin & M.Pierre: Global existence and boundedness in reaction-diffusion systems. SIAM J. Math. Anal. 18 (1987), 744–761.

M. A. Herrero & J. J. L. Velazquez: Blow up behaviour of one-dimensional semilinear heat equations. Ann. Inst. Henri Poincaré 10 (1993), 131–189.

M. A. Herrero & J. J. L. Velazquez: Blow-up profiles in one-dimensional, semilinear parabolic problems. Comm. in PDE 17 (1992), 205–219.

A. Haraux & A. Youkana: On a result of K. Masuda concerning reaction- diffusion equations. Tohoku Math. J. 40 (1988), 159–163.

A. A. Lacey: Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J. Appl. Math. 43 (1983), 1350–1366.

A. A. Lacey & D. Tzanetis: Global existence and convergence to a singular steady state for a semilinear heat equation. Proc. Royal Soc. Edinburgh A105 (1987), 289–305.

A. A. Lacey & D. Tzanetis: Global, unbounded solutions to a parabolic equations. J. Diff. Eq. 101 (1993), 80–102.

K. Masuda: On the global existence and asymptotic behaviour of reaction–diffusion equations. Hokkaido Math. J. 12 (1983), 360–370.

R. H. Martin & M.Pierre: Nonlinear reaction-diffusion systems. In the book Nonlinear Equations in the Applied Sciences, W. F. Ames & C.Rogers, eds. Academic Press (1991).

J. J. L. Velazquez: Classification of singularities for blowing up solutions in higher dimensions. Trans. Amer. Math. Soc. 338 (1993), 441–464.

J. J. L. Velazquez: Estimates on the (n−1)-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation. Indiana Univ. Math. J. 42 (1993), 445–476.

Deposited On:31 Oct 2012 09:09
Last Modified:07 Feb 2014 09:38

Repository Staff Only: item control page