## Tunisian Journal of Mathematics

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

#### Abstract

We consider the semilinear heat equation

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

in the whole space $ℝn$, where $p>1$ and $α∈ℝ$. 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

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
Revised: 6 September 2017
Accepted: 21 September 2017
First available in Project Euclid: 2 March 2019

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

#### 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

#### References

• A. Bressan, “Stable blow-up patterns”, J. Differential Equations 98:1 (1992), 57–75.
• J. Bricmont and A. Kupiainen, “Universality in blow-up for nonlinear heat equations”, Nonlinearity 7:2 (1994), 539–575.
• M. A. Ebde and H. Zaag, “Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term”, SeMA J. 55:1 (2011), 5–21.
• T.-E. Ghoul, V. T. Nguyen, and H. Zaag, “Construction and stability of blowup solutions for a non-variational semilinear parabolic system”, preprint, 2016.
• T.-E. Ghoul, V. T. Nguyen, and H. Zaag, “Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term”, J. Differential Equations 263:8 (2017), 4517–4564.
• T.-E. Ghoul, V. T. Nguyen, and H. Zaag, “Blowup solutions for a reaction-diffusion system with exponential nonlinearities”, preprint, 2017.
• Y. Giga and R. V. Kohn, “Characterizing blowup using similarity variables”, Indiana Univ. Math. J. 36:1 (1987), 1–40.
• Y. Giga and R. V. Kohn, “Nondegeneracy of blowup for semilinear heat equations”, Comm. Pure Appl. Math. 42:6 (1989), 845–884.
• M. A. Herrero and J. J. L. Velázquez, “Blow-up profiles in one-dimensional, semilinear parabolic problems”, Comm. Partial Differential Equations 17:1-2 (1992), 205–219.
• N. Masmoudi and H. Zaag, “Blow-up profile for the complex Ginzburg–Landau equation”, J. Funct. Anal. 255:7 (2008), 1613–1666.
• F. Merle, “Solution of a nonlinear heat equation with arbitrarily given blow-up points”, Comm. Pure Appl. Math. 45:3 (1992), 263–300.
• F. Merle, “Solution of a nonlinear heat equation with arbitrarily given blow-up points”, Comm. Pure Appl. Math. 45:3 (1992), 263–300.
• F. Merle and H. Zaag, “Stabilité du profil à l'explosion pour les équations du type $u_t=\Delta u+|u|^{p-1}u$”, C. R. Acad. Sci. Paris Sér. I Math. 322:4 (1996), 345–350.
• F. Merle and H. Zaag, “Stability of the blow-up profile for equations of the type $u_t=\Delta u+|u|^{p-1}u$”, Duke Math. J. 86:1 (1997), 143–195.
• V. T. Nguyen and H. Zaag, “Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 16:4 (2016), 1275–1314.
• V. T. Nguyen and H. Zaag, “Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation”, Ann. Sci. Éc. Norm Supér. $(4)$ 50:5 (2017), 1241–1282.
• N. Nouaili and H. Zaag, “Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation”, Comm. Partial Differential Equations 40:7 (2015), 1197–1217.
• S. Tayachi and H. Zaag, “Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term”, preprint, 2015.
• S. Tayachi and H. Zaag, “Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term”, preprint, 2015.
• H. Zaag, “Blow-up results for vector-valued nonlinear heat equations with no gradient structure”, Ann. Inst. H. Poincaré Anal. Non Linéaire 15:5 (1998), 581–622.