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## Colloquium Mathematicum

2000 | 86 | 1 | 43-66
Tytuł artykułu

### Blow up, global existence and growth rate estimates in nonlinear parabolic systems

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. $u_{it} - d_{i} Δu_{i} = \prod_{k=1}^m u_{k}^{p_k^i}, i=1,...,m, x ∈ ℝ^{N}, t > 0,$ with nonnegative, bounded, continuous initial values and $p_{k}^{i} ≥ 0$, $i,k = 1,...,m$, $d_i > 0$, $i = 1,...,m$. For solutions which blow up at $t = T <≤ ∞$, we derive the following bounds on the blow up rate: $u_i(x,t) ≤ C(T - t)^{-α_{i}}$ with C > 0 and $α_i$ defined in terms of $p_k^i$.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
43-66
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-12-10
poprawiono
1999-07-19
Twórcy
autor
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
• [AHV] D. Andreucci, M. A. Herrero and J. J. L. Velázquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems, Ann. Inst. H. Poincaré 14 (1997), 1-53.
• [CM] G. Caristi and E. Mitidieri, Blow up estimates of positive solutions of a parabolic system, J. Differential Equations 113 (1994), 265-271.
• [EH] M. Escobedo and M. A. Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, ibid. 89 (1991), 176-202.
• [EL] M. Escobedo and H. A. Levine, Critical blow up and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal. 129 (1995), 47-100.
• [Fu1] H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, in: Proc. Sympos. Pure Math. 18, Amer. Math. Soc., 1970, 105-113.
• [Fu2] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = Δ u + u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109-124.
• [GK] Y. Giga and R. V. Kohn, Characterizing blow-up using similarity variables, Indiana Univ. Math. J. 36 (1987), 1-40.
• [H] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Heidelberg, 1981.
• [L] G. Lu, Global existence and blow-up for a class of semilinear parabolic systems: a Cauchy problem, Nonlinear Anal. 24 (1995), 1193-1206.
• [LS] G. Lu and B. D. Sleeman, Subsolutions and supersolutions to systems of parabolic equations with applications to generalized Fujita type systems, Math. Methods Appl. Sci. 17 (1994), 1005-1016.
• [R1] J. Rencławowicz, Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations, Appl. Math. (Warsaw) 25 (1998), 313-326.
• [R2] J. Rencławowicz, Global existence and blow up of solutions for a class of reaction-diffusion systems, J. Appl. Anal., to appear.
• [R3] J. Rencławowicz, Global existence and blow up of solutions for a weakly coupled Fujita type system of reaction-diffusion equations, ibid., to appear.
• [R4] J. Rencławowicz, Global existence and blow-up for a completely coupled Fujita type system, Appl. Math. (Warsaw) 27 (2000), 203-218.
Typ dokumentu
Bibliografia
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