Asymptotically self‐similar blow‐up of semilinear heat equations

Giga, Yoshikazu; Kohn, Robert V.

doi:10.1002/cpa.3160380304

Cited by 501 publications

(446 citation statements)

References 9 publications

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“…As in [7], the uniqueness of the limit (d 1 , ξ 1∞ , ξ 2∞ ) implies that u(x, t), ξ i (t) (i = 1, 2) all converge uniformly as t → ∞.…”

Section: Asymptotic Limit For the Area-preserving Casementioning

confidence: 89%

“…By following the standard argument as in [7], we find that the subsequential limit function must be a solution to the steady state equation [a(u x )] x = 0 with the boundary conditions (1.2), ( With the D(t) defined in (1.6), it is clear that D (t) = 0 in the area-preserving case. In particular, D(t) ≡ D(0) for all t > 0.…”

Section: Asymptotic Limit For the Area-preserving Casementioning

confidence: 99%

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On a two-point free boundary problem

Guo

2006

Quart. Appl. Math.

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Abstract. We study a two-point free boundary problem for a quasilinear parabolic equation. This problem arises in the model of flame propagation in combustion theory. It also arises in the study of the motion of interface moving with curvature in which the studied problem is confined in the conical region bounded by two straight lines and the interface has prescribed touching angles with these two straight lines. Depending on these two touching angles, there are three different cases, namely, area-expanding, area-preserving, and area-shrinking cases. We first give a proof of the global existence in the expanding and preserving cases. Then the convergence to a line in the preserving case is derived. Finally, in the shrinking case, we show the finite-time vanishing and the convergence of the solution to a self-similar solution.

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“…As in [7], the uniqueness of the limit (d 1 , ξ 1∞ , ξ 2∞ ) implies that u(x, t), ξ i (t) (i = 1, 2) all converge uniformly as t → ∞.…”

Section: Asymptotic Limit For the Area-preserving Casementioning

confidence: 89%

Section: Asymptotic Limit For the Area-preserving Casementioning

confidence: 99%

On a two-point free boundary problem

Guo

2006

Quart. Appl. Math.

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“…(1.1) with initial data u(r, 0) = u * T 0 (r, 0) + h(r ). By time translation invariance we may assume T 0 = 1 and we introduce similarity coordinates s = − log(T − t) + log T , y = r √ T −t which go back to [24]. Here, T > 0 is a free parameter which will be adjusted later.…”

Section: Outline Of the Proofmentioning

confidence: 99%

Stable self-similar blowup in the supercritical heat flow of harmonic maps

2017

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We consider the heat flow of corotational harmonic maps from R 3 to the threesphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear stability analysis of self-similar shrinkers to the spectral analysis of the associated self-adjoint linearized operators. Mathematics Subject Classification

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“…[12][13][14][15][16][17]) and more recently for their higherorder generalizations (cf. [18][19][20]).…”

Section: Similaritymentioning

confidence: 99%

“…[27,28]) and their analysis requires a synthesis of similarity, dynamical systems, perturbation, and numerical methods; -rigorous analysis of the above issues (cf. [12][13][14][15][16][17]29]). In many cases such analysis builds directly on the formal results;…”

Section: Similaritymentioning

confidence: 99%