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

Duong, Giao Ky

doi:10.48550/arxiv.1712.07183

Cited by 1 publication

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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.…”

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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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