Publication: A uniqueness result for a semilinear reaction-diffusion system
Loading...
Full text at PDC
Publication Date
1991-05
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
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}.
Description
UCM subjects
Unesco subjects
Keywords
Citation
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.