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

Joanna Rencławowicz

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 1, page 43-66
  • ISSN: 0010-1354

Abstract

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We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. u i t - d i Δ u i = ∏ 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 .

How to cite

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Rencławowicz, Joanna. "Blow up, global existence and growth rate estimates in nonlinear parabolic systems." Colloquium Mathematicae 86.1 (2000): 43-66. <http://eudml.org/doc/210841>.

@article{Rencławowicz2000,
abstract = {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 $p_k^i$.},
author = {Rencławowicz, Joanna},
journal = {Colloquium Mathematicae},
keywords = {invariant manifold; reaction-diffusion system; invariant region; global existence; blow up; bounds on the blow up rate},
language = {eng},
number = {1},
pages = {43-66},
title = {Blow up, global existence and growth rate estimates in nonlinear parabolic systems},
url = {http://eudml.org/doc/210841},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Rencławowicz, Joanna
TI - Blow up, global existence and growth rate estimates in nonlinear parabolic systems
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 43
EP - 66
AB - 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 $p_k^i$.
LA - eng
KW - invariant manifold; reaction-diffusion system; invariant region; global existence; blow up; bounds on the blow up rate
UR - http://eudml.org/doc/210841
ER -

References

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  10. [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. Zbl0807.35061
  11. [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. Zbl1002.35070
  12. [R2] J. Rencławowicz, Global existence and blow up of solutions for a class of reaction-diffusion systems, J. Appl. Anal., to appear. 
  13. [R3] J. Rencławowicz, Global existence and blow up of solutions for a weakly coupled Fujita type system of reaction-diffusion equations, ibid., to appear. 
  14. [R4] J. Rencławowicz, Global existence and blow-up for a completely coupled Fujita type system, Appl. Math. (Warsaw) 27 (2000), 203-218. Zbl0994.35055

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