Profile for the imaginary part of a blowup solution for a complex-valued semilinear heat equation

Duong, Giao Ky

doi:10.1016/j.jfa.2019.05.009

Cited by 17 publications

(35 citation statements)

References 28 publications

(61 reference statements)

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“…Although, in [5], we believe we made an important achievement, we acknowledge that we left unanswered the case where p > 1 and p / ∈ N. From the limitation of the above works, it motivates us to study model (1.1) in general even for irrational p. The following theorem is considered as a generalization of [5] for all p > 1.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

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A blowup solution of a complex semi-linear heat equation with an irrational power

Duong

2019

Journal of Differential Equations

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In this paper, we consider the following semi-linear complex heat equation ∂tu = ∆u + u p , u ∈ C in R n , with an arbitrary power p, p > 1. In particular, p can be non integer and even irrational, unlike our previous work [5], dedicated to the integer case. We construct for this equation a complex solution u = u 1 + iu 2 , which blows up in finite time T and only at one blowup point a. Moreover, we also describe the asymptotics of the solution by the following final profiles:2) This model is connected to the viscous Constantin-Lax-Majda equation with a viscosity term, which is a one dimensional model for the vorticity equation in fluids. For more details, the readers are addressed to the following works: Constantin, Lax, Majda [2], Guo, Ninomiya and Yanagida in [8], Okamoto, Sakajo and Wunsch [25], Sakajo in [26] and [27], Schochet [28]. In [5], we treated the case p ∈ N. Indeed, handling the 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

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Section: Statement Of the Resultsmentioning

confidence: 99%

“…In particular, in that works, we are able to derive a sharp blowup profile for the imaginary part of the solution. In 2018, in [5], we handled equation (1.1) when p is an integer.…”

Section: Ealier Workmentioning

confidence: 99%

A blowup solution of a complex semi-linear heat equation with an irrational power

Duong

2019

Journal of Differential Equations

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“…As a matter of fact, this term plays the most important role in our analysis. Therefore, we show here some main properties on the linear operator L and the potential V (see more details in [1], [2]) -Operator L: This operator is self-adjoint in D(L) ⊂ L 2 ρ (R n ), where L 2 ρ is defined as follows…”

Section: )mentioning

confidence: 99%

“…There exists K 2 > 0 such that for all K 0 ≥ K 2 and δ 2 > 0, there exist α 2 (K 0 , δ 2 ) > 0 and C 2 (K 0 ) > 0 such that for every α 0 ∈ (0, α 2 ], there exists ǫ 2 (K 0 , δ 2 , α 0 ) > 0 such that for every ǫ 0 ∈ (0, ǫ 2 ] and A ≥ 1, there exists T 2 (K 0 , δ 2 , ǫ 0 , A, C 2 ) > 0 such that for all T ≤ T 2 and s 0 = − ln T . The following holds: 2] n such that if we define the following mapping…”

Section: Initial Datamentioning

confidence: 99%

Profile of a touch-down solution to a nonlocal MEMS model

Duong

Zaag

2019

Math. Models Methods Appl. Sci.

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In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on (0, T ) × Ω :where Ω is a bounded domain in R n and λ, γ > 0. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point a ∈ Ω. In particular, we give a description of the quenching behavior according to the following final profile1 3 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

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“…• Solving the finite dimensional problem thanks to a classical topological argument based on the index theory. This general two-step procedure has been extended to various situations such as the case of the complex Ginzgburg-Landau equation by Masmoudi-Zaag [21], Nouaili-Zaag [27] (see also Zaag [31] for an earlier work); the complex semilinear heat equation with no variational structure by Duong [7], Nouaili-Zaag [26]; non-scaling invariant semilinear heat equations by Ebde-Zaag [9], Nguyen-Zaag [25], Duong-Nguyen-Zaag [8]. We also mention the work of Tayachi-Zaag [29,30] and Ghoul-Nguyen-Zaag [16] dealing with a nonlinear heat equation with a double source depending on the solution and its gradient in some critical setting.…”

Section: Introductionmentioning

confidence: 99%

Construction of type I blowup solutions for a higher order semilinear parabolic equation

Ghoul

Nguyen

Zaag

2019

Advances in Nonlinear Analysis

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We consider the higher-order semilinear parabolic equationin the whole space R N , where p > 1 and m ≥ 1 is an odd integer. We exhibit type I non selfsimilar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m = 1, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [15].where u(t) : R N → R with N ≥ 1, ∆ stands for the standard Laplace operator in R N , and the exponents p and m are fixed, p > 1 and m ∈ N, m ≥ 1 odd.The higher-order semilinear parabolic equation (1.1) is a natural generation of the classical semilinear heat equation (m = 1). It arises in many physical applications such as theory of thin film, lubrication, convection-explosion, phase translation, or applications to structural mechanics (see the Petetier-Troy book [28] and references therein).By standard results the local Cauchy problem for equation (1.1) can be solved in L 1 ∩ L ∞ thanks to the integral representationt 0 K m (t − s) * u(s)|u(s)| p−1 ds, (1.2) 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

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Profile for the imaginary part of a blowup solution for a complex-valued semilinear heat equation

Cited by 17 publications

References 28 publications

A blowup solution of a complex semi-linear heat equation with an irrational power

A blowup solution of a complex semi-linear heat equation with an irrational power

Profile of a touch-down solution to a nonlocal MEMS model

Construction of type I blowup solutions for a higher order semilinear parabolic equation

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