### Impacto

Escobedo, M. and Herrero, Miguel A.
(1991)
*A uniqueness result for a semilinear reaction-diffusion system.*
Proceedings of the American Mathematical Society, 112
(1).
pp. 175-185.
ISSN 0002-9939

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Official URL: http://www.ams.org/journals/proc/1991-112-01/S0002-9939-1991-1043410-9/S0002-9939-1991-1043410-9.pdf

## Abstract

Let (u(t, x), v(t, x)) and (uBAR(t, x), vBAR(t, x)) be two nonnegative classical solutions of (S)[GRAPHICS:{ut=Δu+vp, p>0 ; vt=Δv+uq, q>0] in some strip S(T) = (0, T) x R(N), where 0 < T ≤ ∞, and suppose that u(0, x) = uBAR(0, x), v(0, x) = vBAR(0, x), where u(0, x) and v(0, x) are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one has u(t, x) = uBAR(t, x), v(t, x) = vBAR(t, x) in S(T). If pq ≥ 1, the result is also true if u(0, x) = v(0, x) = 0. On the other hand, when 0 < pq < 1, the set of solutions of (S) with zero initial values is given by u(t; s) = c1(t - s)+(p+1)/(1-pq), v(t; s) = c2(t - s)+(q+1)/(1-qp), where 0 ≤ s ≤ t, c1 and c2 are two positive constants depending only on p and q, and (ξ)+ = max{ξ,0}.

Item Type: | Article |
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Uncontrolled Keywords: | Reaction diffusion systems; uniqueness |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 17116 |

References: | J. Aguirre and M. Escobedo, A Cauchy problem for ut - Δu = uP with 0 < p < 1 : Asymptotic behaviour of solutions, Ann. Fac. Sei. Toulouse 8 (1986-87), 175-203. D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), 33-76. M. Escobedo and M. A. Herrero, Boundedness and blow-up for a semilinear reaction diffusion system, J. Differential Equations (to appear). M. Floater and J. B. McLeod, in preparation. A. Friedman and Y. Giga, A single point blow up for solutions of semilinear parabolic systems, J. Fac. Sei. Univ. of Tokyo, Sect. I 34 (1987), 65-79. H. Fujita, On the blowing up of solutions of the Cauchy problem for ut-Δu = u(1+α) , J. Fac. Sei. Univ. of Tokyo, Sect. I 13 (1960), 109-124. V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskii, A parabolic system of quasilinear equations I, Differential Equations 19 (1983), 2123-2143. ___, A parabolic system of quasilinear equations II, Differential Equations 21 (1985), 1544-1559. F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. of Math. 38 (1981), 29-40. |

Deposited On: | 16 Nov 2012 09:03 |

Last Modified: | 07 Feb 2014 09:42 |

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