Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

Joanna Rencławowicz

Applicationes Mathematicae (1998)

  • Volume: 25, Issue: 3, page 313-326
  • ISSN: 1233-7234

Abstract

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We examine the parabolic system of three equations u t - Δu = v p , v t - Δv = w q , w t - Δw = u r , x ∈ ℝ N , t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.

How to cite

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Rencławowicz, Joanna. "Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations." Applicationes Mathematicae 25.3 (1998): 313-326. <http://eudml.org/doc/219206>.

@article{Rencławowicz1998,
abstract = {We examine the parabolic system of three equations $u_t$ - Δu = $v^p$, $v_t$ - Δv = $w^q$, $w_t$ - Δw = $u^r$, x ∈ $ℝ^N$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.},
author = {Rencławowicz, Joanna},
journal = {Applicationes Mathematicae},
keywords = {reaction-diffusion system; global existence; blow up; system of three equations; blow up unconditionally},
language = {eng},
number = {3},
pages = {313-326},
title = {Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations},
url = {http://eudml.org/doc/219206},
volume = {25},
year = {1998},
}

TY - JOUR
AU - Rencławowicz, Joanna
TI - Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations
JO - Applicationes Mathematicae
PY - 1998
VL - 25
IS - 3
SP - 313
EP - 326
AB - We examine the parabolic system of three equations $u_t$ - Δu = $v^p$, $v_t$ - Δv = $w^q$, $w_t$ - Δw = $u^r$, x ∈ $ℝ^N$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.
LA - eng
KW - reaction-diffusion system; global existence; blow up; system of three equations; blow up unconditionally
UR - http://eudml.org/doc/219206
ER -

References

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  1. [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. Zbl0877.35019
  2. [EH] M. Escobedo and M. A. Herrero, z Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations 89 (1991), 176-202. Zbl0735.35013
  3. [EL] M. Escobedo and H. A. Levine, z Critical blow up and global existence numbers for a weakly coupled system of reaction-diffusion equations, Arch. Rational Mech. Anal. 129 (1995), 47-100. Zbl0822.35068
  4. [F1] H. Fujita, z On the blowing up of solutions of the Cauchy problem for u t = ∇u + u 1 + α , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109-124. 
  5. [F2] H. Fujita, z On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, in: Proc. Sympos. Pure Math. 18, Amer. Math. Soc., 1970, 105-113. 

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