### Impacto

Díaz Díaz, Jesús Ildefonso and Vrabie, Ioan I.
(1994)
*Existence for reaction-diffusion systems - a compactness method approach.*
Journal of Mathematical Analysis and Applications, 188
(2).
pp. 521-540.
ISSN 0022-247X

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Official URL: http://www.sciencedirect.com/science/article/pii/S0022247X84714430

## Abstract

The authors study the existence of weak solutions for the following system: ut−Δφ(u)∈F(u,v), vt−Δψ(v)∈G(u,v) in (0,T)×Ω, φ(u)=ψ(v)=0 on (0,T)×∂Ω, u(0,x)=u0(x), v(0,x)=v0(x) in the region Ω⊂⊂Rn with smooth boundary ∂Ω. The functions ψ,φ:R→R are assumed to be continuous and nondecreasing with ψ(0)=φ(0)=0, u0,v0∈L∞(Ω), F,G:R2→2R with F an upper semicontinuous mapping (u.s.c.). The following local existence results are shown: (1) for the diffusive case, i.e. when both ψ and φ are strictly increasing with u.s.c. G; (2) for the semi-diffusive case (only one function φ is strictly increasing) with G being either with separated variables (i.e. having the form of the product or of the sum of two functions g(u) and H(v)) or globally Lipschitz with respect to its second variable (i.e. |G(u,v)−G(u,v′)|≤L|v−v′| for each u∈B⊂⊂R, v,v′∈R and some L=L(B)). Additional conditions (of linear form) on the growth of F and G are indicated to guarantee global existence results.

Item Type: | Article |
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Uncontrolled Keywords: | reaction diffusion systems; local and global existence of weak solutions |

Subjects: | Sciences > Mathematics > Numerical analysis |

ID Code: | 15949 |

References: | O. ARINO, S. GAUTHIER, AND J. PENOT. A fixed point theorem for secquentially conlinuolls mapping with applications to ordinary differential equations. Funkcial Ekvac. 27 (1984), 273-279. R. ARIS, "The Mathematical Theory of Diffusion ami Reaction in Permeable Catalysts,"Oxford Univ, Press (Clarendon). London/New YorK, 1975. C. BANDLE AND I. STAKGOLD. Reaction-diffusion with dead cores, in "Free BoundaryProblems: Theory and Applications," Vol.IV, pp. 436-448, Pitman, New York 1985. J. BEAR, "Dynamics of Fluids in Porous Media," Elsevier, Amsterdam/New York, 1972, PH. BENILAN, "Equations d'Evolution dans un Espace de Banach Quelconque et Applicalions,"Thése. Orsay, 1972. D. BLANCHARD. A. DAMLAMIAN, AND H. GHID0UCHE. A nonlinear system for phase change with dissipation, Differential Integral Equations 2 (1989),344-362. H. BREZIS. "Operateurs Maximaux Monotoncs et Semi-Groupes de Contractions dans les espaces de Hilbert," North-Holland, Amsterdam, 1973. H. BREZIS AND M. G. CRANDALL, Uniqueness of solutions of the initial value problem for u - (u) = 0, j, Math. Pures Appl. 58 (1979),153-163. S. CARL, The monotone interative techniquc for a parabolic boundary value problem with discountinuous nonlinearity, Nonlnear Anal. 13 (1989).1399-1407. K.C.CHANG, Free boundary problems amd the set-valued mapping. J. Differential Equations 49 (1983), 1-28. J. I. DíAZ, Mathemalical analysis of some diffusive energy balance models in Climalology,in "Mathematics, Climate and Environment" (J.I. Díaz and J, L, Lions, Eds.),pp. 28-56, Masson, Paris, 1993, J. I. DíAZ AND J. HERNÁNDEZ, Some results on the existence of free boundaries for parabolic reaction-diffusion systems. in "Trends in Theory amI Practice of Nonlinear Differential Equations" (Y. Laksmikantham, Ed,), pp. 149-156, Dekker, New York, 1984. J. I. DíAZ AND I. STAKGOLD. Mathematical analysis of the conversion of a porous solid by a dislributed gas reaction, in "Actas del XI CEDYA," pp. 217-223, Univ. of Malaga, Malaga, Spain, 1989, J. I. DíAZ AND I. STAKGOLD. Mathematicat apects of the combustion of a solid by a distributed isothermal gas reaction, SIAM Math. Anal. to appear. J. I. DíAZ AND I. I. VRABIE, Existence for reaction diffusion systems, unpublished manuscripl, 1987. J. I. DíAZ AND I. I. VRABIE, Propriétés de compacité de l'operateur de Geen géneralisé pour l'équation des milieux poreux, C.R. Acad. Sci, Paris 309 (1989),221-223, J. I. DíAZ AND I. I. VRABIE, Compactness of the green operator of nonlinear diffusion equations: Application to boussinesq type systems in fluid dynamics, Topological Methods in Nonlinear Anal. To appear. E. DIBENEDETTO AND R. E. SHOWALTER, A free-boundary problem for a degenerate parabolic system, J. Differential Equations 50 (1983), 1-19. A. DILIDDO AND I. STAKGOLD. Isothermal combustion Whit two moving fronts. J. Math. Anal. Appl. 152 (1990),584-599. C. J. VAN DUIJN AND P. KNABNER. Solute transport through porous media whith slow adsorption, in "Free Boundary Problems: Theory ami Applicalions" (K. H, HofTman and J. Sprekels, Eds), pp. 375--388, Pitman, New York, 1988. E. FEIREISL AND J. NORHURY. Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discuntinuous nonlinearities. Proc. Roy. Soc. Edinburg. Sect. A 1l9A (1991),1-17. V.GALAKTIONOV,S. P.KURDYUMOV. AND A.A.SAMARSKH. A parabolic system of quasiilinear equations, Differential'nye Uravneniya 19 (1983). 2123-2140. R. GIANNI AND J. HULSHOF, The semi linear heat equation with a Heaviside source term, European J. Appl. Math. 3 (1992). 367-379. G. HETZER AN D P.G.SCMIDT. A global attractor and stationary solutions for a reaction-diffusion systems arising from climate modeling, Nonlinear Anal. 14 (1990),915-926. S.L.HOLLIS, R.H.MARTIN, AND M. PIERRE. Global existence and boundedncss in reaction-diffusion systems, SIAM.J. Math. Anal. 18(1987),744-761. A.S.KALASHNIKOV, A certain class of systems of reaction-diffusion type. Trudy Sem. Petrovsk. 14 (1980). 78-88. L. MADDALENA, Exislence of global solutions for reaction-diffusin systems with density dependent diffusion. Nonliear Anal. 8 (1984). 1383-1394. L. MADDALENA, Existence, uniqueness, and qualitative properties of the solution of a degenerate nonlinear parabolic system, J. Math. Anal. Appl. 127 (1987), 443-458. C. V. PAO, "Nonlinear Parabolic Equatiolls," Pkenum, New York, 1992. I.STAKGOLD, Diffusion with strong absorption, in "Shape Optimization and Free Boundaries"(M. C. Delfour, Ed.), pp. 341-347, Boston, 1992. I. VRABIE, "Compactness Methods for Nonlinear Evolutions," Longman Scientific and Technical. London, 1987. |

Deposited On: | 16 Jul 2012 11:07 |

Last Modified: | 06 Feb 2014 10:35 |

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