### Abstract

We consider the following exponential reaction–diffusion equation involving a nonlinear gradient term: ∂_{t}U=ΔU+α|∇U|^{2}+e^{U},(x,t)∈R^{N}×[0,T),α>−1. We construct for this equation a solution which blows up in finite time T>0 and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time T simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some sense, resulting in the change of the final blowup profile in comparison with the case α=0. The proof of the construction is inspired by the method of Merle and Zaag in 1997. It relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the blowup region, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations in the initial data.

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

Pages (from-to) | 4517-4564 |

Number of pages | 48 |

Journal | Journal of Differential Equations |

Volume | 263 |

Issue number | 8 |

DOIs | |

State | Published - Oct 15 2017 |

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### Keywords

- Blowup profile
- Finite-time blowup
- Semilinear heat equations
- Stability

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

*263*(8), 4517-4564. https://doi.org/10.1016/j.jde.2017.05.023

**Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term.** / Ghoul, Tej-eddine; Nguyen, Van Tien; Zaag, Hatem.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 263, no. 8, pp. 4517-4564. https://doi.org/10.1016/j.jde.2017.05.023

}

TY - JOUR

T1 - Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

AU - Ghoul, Tej-eddine

AU - Nguyen, Van Tien

AU - Zaag, Hatem

PY - 2017/10/15

Y1 - 2017/10/15

N2 - We consider the following exponential reaction–diffusion equation involving a nonlinear gradient term: ∂tU=ΔU+α|∇U|2+eU,(x,t)∈RN×[0,T),α>−1. We construct for this equation a solution which blows up in finite time T>0 and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time T simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some sense, resulting in the change of the final blowup profile in comparison with the case α=0. The proof of the construction is inspired by the method of Merle and Zaag in 1997. It relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the blowup region, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations in the initial data.

AB - We consider the following exponential reaction–diffusion equation involving a nonlinear gradient term: ∂tU=ΔU+α|∇U|2+eU,(x,t)∈RN×[0,T),α>−1. We construct for this equation a solution which blows up in finite time T>0 and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time T simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some sense, resulting in the change of the final blowup profile in comparison with the case α=0. The proof of the construction is inspired by the method of Merle and Zaag in 1997. It relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the blowup region, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations in the initial data.

KW - Blowup profile

KW - Finite-time blowup

KW - Semilinear heat equations

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85020215701&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020215701&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2017.05.023

DO - 10.1016/j.jde.2017.05.023

M3 - Article

AN - SCOPUS:85020215701

VL - 263

SP - 4517

EP - 4564

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 8

ER -