On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

Ghoul, Tej Eddine; Ibrahim, Slim; Nguyen, Van Tien

doi:10.2140/apde.2019.12.113

Cited by 11 publications

(34 citation statements)

References 72 publications

(135 reference statements)

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“…By induction and part (iii) of Lemma 2.9, the conclusion simply follows. For item (ii), we refer to Lemma 2.9 in [27] for an analogous proof.…”

Section: Admissible Functionsmentioning

confidence: 99%

“…2r 2 sin(2u). (1.11) In [27], we construct for equation (1.11) a family of C ∞ solutions which blow up in finite time via concentration of the profile…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, by considering the case when d ≥ 7, we ask whether we can carry out the analysis in [27] to construct solutions for equation (1.3) which blow up in finite time via concentration of the profile Q. The following theorem is our main result.…”

Section: Introductionmentioning

confidence: 99%

“…-Strategy of the proof. We now summary the main ideas of the proof of Theorem 1.1, which follows the road map in [27] and [49].…”

Section: Introductionmentioning

confidence: 99%

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Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Ghoul

Ibrahim

Nguyen

2018

Journal of Differential Equations

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We consider the energy supercritical wave maps from R d into the d-sphere S d with d ≥ 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation1 1-COROTATIONAL ENERGY SUPERCRITICAL WAVE MAPS 5 Remark 1.3. The proof of Theorem 1.1 involves a detailed description of the the set of initial data leading to the type II blowup with the quantization of the blowup rate (1.13). In particular, given ℓ ∈ N * , L ≫ 1 and s ∼ L, our initial data is of the formwhere Q b is a deformation of the ground state Q = (Q, 0), and b = (b 1 , · · · , b L ) correspond to possible unstable directions of the flow in theḢ s ×Ḣ s−1 topology in a suitable neighborhood of Q.We show that for all q 0 ∈ O ⊂ Ḣ σ ∩Ḣ s × Ḣ σ−1 ∩Ḣ s−1 , where the set O is built on the linearized operator (see Definition 3.1 for its precise description of O) and for all b 1 (0), b ℓ+1 (0), · · · , b L (0) small enough, there exists a choice of unstable directions b 2 (0), · · · , b ℓ (0) such that the solution of (1.3) with initial data (1.15) satisfies the conclusion of Theorem 1.1. The control of (ℓ − 1) unstable modes is done through a topological argument based on Brouwer's fixed point theorem. In some sense, the set of blowup solutions we construct lies on a (ℓ − 1) codimension manifold in the radial class whose proof would require some Lipschitz regularity of the set of initial data we consider and it would be addressed separately in detail.Remark 1.4. It is worth mentioning that our analysis relies only on the study of supercritical Sobolev norms built on the linearized operator, thus, the finiteness of the H 1 norm of the initial data is not requested. Roughly speaking, the initial data (u 0 , u 1 ) can be taken smooth and compactly supported, namely that if u = Q+ε, we take ε(r) ∼ −Q(r) for r ≫ 1. Since the energy is conserved, our constructed solution can be taken to be of finite energy or even compactly supported. As a matter of fact, the finite energy together with the constructed manifold mentioned in the previous remark ensures that the original solution Φ to the wave map equation (1.2) has the same the regularity as for the 1-corotational symmetric solution u described in Theorem 1.1.Remark 1.5. We note from (1.12) thatThis implies that our constructed solution is of Type II blowup in the sense of (1.5).Remark 1.6. Following the work by Côte et al. [18,19] where the question of the classification of the flow near the special class of stationary solution Q are considered in the energy critical setting, i.e. d = 2, we would address the same question for the energy supercritical case d ≥ 7. In Theorem 1.1, the constructed blowup solutions exhibit the decomposition of the form (1.12). Here we ask for a converse problem, namely that if blowup does occur for a solution u, in which energy regime and in what sense does such the decomposition (1.12) always hold? Remark 1.7. It is worth mentioning the work of Krieger-Schlag-Tataru [36], where the authors constructed for equation (1.3) in the...

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“…By induction and part (iii) of Lemma 2.9, the conclusion simply follows. For item (ii), we refer to Lemma 2.9 in [27] for an analogous proof.…”

Section: Admissible Functionsmentioning

confidence: 99%

“…2r 2 sin(2u). (1.11) In [27], we construct for equation (1.11) a family of C ∞ solutions which blow up in finite time via concentration of the profile…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…-Strategy of the proof. We now summary the main ideas of the proof of Theorem 1.1, which follows the road map in [27] and [49].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Ghoul

Ibrahim

Nguyen

2018

Journal of Differential Equations

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“…It was the case of the semilinear heat equation treated in [4], [32], [36] (see also [35], [9] for the case of logarithmic perturbations, [2], [3] and [16] for the exponential source, [37] for the complex-valued case), the Ginzburg-Landau equation in [27], [38] (see also [48] for an earlier work). It was also the nonlinear Schrödinger equation both in the mass critical [28,29,30,31] and mass supercritical [34] cases; the energy critical [10], [22] and supercritical [6] wave equation; the mass critical gKdV equation [24,25,26]; the two dimensional Keller-Segel model [42]; the energy critical and supercritical geometric equations: the wave maps [39] and [18], the Schrödinger maps [33] and the harmonic heat flow [40,41] and [17]; the semilinear heat equation in the energy critical [43] and supercritical [5] cases.…”

Section: Introductionmentioning

confidence: 99%

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

Ghoul

Nguyen

Zaag

2019

Advances in Pure and Applied Mathematics

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AbstractIn this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

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Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

Glogić,

Kistner,

Schörkhuber

2024

Calc. Var.

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We study singularity formation for the heat flow of harmonic maps from $$\mathbb {R}^d$$ R d . For each $$d \ge 4$$ d ≥ 4 , we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.

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On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

Cited by 11 publications

References 72 publications

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

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