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

Colloquium Mathematicae (2000)

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

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topRencławowicz, 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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- [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
- [R2] J. Rencławowicz, Global existence and blow up of solutions for a class of reaction-diffusion systems, J. Appl. Anal., to appear.
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- [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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