### Abstract

We consider the non-autonomous system of nonlinear parabolic equations { u_{t}+ t^{r}Δ_{α}u = v^{q} v _{t} + t^{s}Δ_{β}v=u^{p} posed in Q := (0, ∞) × R^{N}, subject to the initial data (u(0,x) = u_{0}(x), v(0, x) = vo(x)), where p > 1 and q > 1 are positive real numbers, α, β €]0,2] and Δ_{γ} := (-Δ)^{γ/2} is the (-Δ)^{γ/2} fractional power of -Δ in the x variable defined via the Fourier transform and its inverse ^{-1} by (-Δ)^{γ/2}ω(x,t) = ^{-1} (v^{γ}(ω) (ξ)) (x,t), where r > -1 and s > -1. The Fujita critical exponent which separates the case of blowing-up solutions from the case of globally in time existing solutions is determined.

Original language | English |
---|---|

Pages (from-to) | 203-212 |

Number of pages | 10 |

Journal | Communications in Applied Analysis |

Volume | 14 |

Issue number | 2 |

Publication status | Published - Apr 2010 |

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### Keywords

- Critical exponent
- Non-autonomous reaction-diffusion systems

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Modelling and Simulation
- Numerical Analysis

### Cite this

**Non-existence of global solutions to systems of non-autonomous nonlinear parabolic equations.** / Kerbal, Sebti.

Research output: Contribution to journal › Article

*Communications in Applied Analysis*, vol. 14, no. 2, pp. 203-212.

}

TY - JOUR

T1 - Non-existence of global solutions to systems of non-autonomous nonlinear parabolic equations

AU - Kerbal, Sebti

PY - 2010/4

Y1 - 2010/4

N2 - We consider the non-autonomous system of nonlinear parabolic equations { ut+ trΔαu = vq v t + tsΔβv=up posed in Q := (0, ∞) × RN, subject to the initial data (u(0,x) = u0(x), v(0, x) = vo(x)), where p > 1 and q > 1 are positive real numbers, α, β €]0,2] and Δγ := (-Δ)γ/2 is the (-Δ)γ/2 fractional power of -Δ in the x variable defined via the Fourier transform and its inverse -1 by (-Δ)γ/2ω(x,t) = -1 (vγ(ω) (ξ)) (x,t), where r > -1 and s > -1. The Fujita critical exponent which separates the case of blowing-up solutions from the case of globally in time existing solutions is determined.

AB - We consider the non-autonomous system of nonlinear parabolic equations { ut+ trΔαu = vq v t + tsΔβv=up posed in Q := (0, ∞) × RN, subject to the initial data (u(0,x) = u0(x), v(0, x) = vo(x)), where p > 1 and q > 1 are positive real numbers, α, β €]0,2] and Δγ := (-Δ)γ/2 is the (-Δ)γ/2 fractional power of -Δ in the x variable defined via the Fourier transform and its inverse -1 by (-Δ)γ/2ω(x,t) = -1 (vγ(ω) (ξ)) (x,t), where r > -1 and s > -1. The Fujita critical exponent which separates the case of blowing-up solutions from the case of globally in time existing solutions is determined.

KW - Critical exponent

KW - Non-autonomous reaction-diffusion systems

UR - http://www.scopus.com/inward/record.url?scp=77954012251&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954012251&partnerID=8YFLogxK

M3 - Article

VL - 14

SP - 203

EP - 212

JO - Communications in Applied Analysis

JF - Communications in Applied Analysis

SN - 1083-2564

IS - 2

ER -