### Abstract

We consider the following parabolic system whose nonlinearity has no gradient structure: {∂_{t}u=Δu+e^{pv},∂_{t}v=μΔv+e^{qu},u(⋅,0)=u_{0},v(⋅,0)=v_{0},p,q,μ>0, in the whole space R^{N}. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

Original language | English (US) |
---|---|

Pages (from-to) | 7523-7579 |

Number of pages | 57 |

Journal | Journal of Differential Equations |

Volume | 264 |

Issue number | 12 |

DOIs | |

State | Published - Jun 15 2018 |

### Fingerprint

### Keywords

- Blowup profile
- Blowup solution
- Semilinear parabolic system
- Stability

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*264*(12), 7523-7579. https://doi.org/10.1016/j.jde.2018.02.022

**Blowup solutions for a reaction–diffusion system with exponential nonlinearities.** / Ghoul, Tej-eddine; Nguyen, Van Tien; Zaag, Hatem.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 264, no. 12, pp. 7523-7579. https://doi.org/10.1016/j.jde.2018.02.022

}

TY - JOUR

T1 - Blowup solutions for a reaction–diffusion system with exponential nonlinearities

AU - Ghoul, Tej-eddine

AU - Nguyen, Van Tien

AU - Zaag, Hatem

PY - 2018/6/15

Y1 - 2018/6/15

N2 - We consider the following parabolic system whose nonlinearity has no gradient structure: {∂tu=Δu+epv,∂tv=μΔv+equ,u(⋅,0)=u0,v(⋅,0)=v0,p,q,μ>0, in the whole space RN. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

AB - We consider the following parabolic system whose nonlinearity has no gradient structure: {∂tu=Δu+epv,∂tv=μΔv+equ,u(⋅,0)=u0,v(⋅,0)=v0,p,q,μ>0, in the whole space RN. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

KW - Blowup profile

KW - Blowup solution

KW - Semilinear parabolic system

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85042421665&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042421665&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2018.02.022

DO - 10.1016/j.jde.2018.02.022

M3 - Article

VL - 264

SP - 7523

EP - 7579

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 12

ER -