Tunisian Journal of Mathematics

Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation

Giao Ky Duong, Van Tien Nguyen, and Hatem Zaag

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Abstract

We consider the semilinear heat equation

t u = Δ u + | u | p 1 u ln α ( u 2 + 2 )

in the whole space n, where p>1 and α. Unlike the standard case α=0, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time T only at one blowup point a, according to the asymptotic dynamic

u ( x , t ) ψ ( t ) ( 1 + ( p 1 ) | x a | 2 4 p ( T t ) | ln ( T t ) | ) 1 ( p 1 )  as  t T ,

where ψ(t) is the unique positive solution of the ODE

ψ = ψ p ln α ( ψ 2 + 2 ) , lim t T ψ ( t ) = + .

The construction relies on the reduction of the problem to a finite-dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite-dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale-invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.

Article information

Source
Tunisian J. Math., Volume 1, Number 1 (2019), 13-45.

Dates
Received: 25 July 2017
Revised: 6 September 2017
Accepted: 21 September 2017
First available in Project Euclid: 2 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.tunis/1551495679

Digital Object Identifier
doi:10.2140/tunis.2019.1.13

Mathematical Reviews number (MathSciNet)
MR3907732

Zentralblatt MATH identifier
07027515

Subjects
Primary: 35K50 35B40: Asymptotic behavior of solutions
Secondary: 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations

Keywords
blowup solution blowup profile stability semilinear heat equation nonscaling invariant heat equation

Citation

Duong, Giao Ky; Nguyen, Van Tien; Zaag, Hatem. Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation. Tunisian J. Math. 1 (2019), no. 1, 13--45. doi:10.2140/tunis.2019.1.13. https://projecteuclid.org/euclid.tunis/1551495679


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