Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients

Remy, Pascal

doi:10.14232/actasm-020-571-9

Cited by 6 publications

(10 citation statements)

References 50 publications

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“…We must prove that the coefficients u j,˚p xq P OpD ρ1 q of the formal solution r upt, xq satisfy similar inequalities. The approach we present below is analoguous to the ones already developed in [3,[20][21][22] in the framework of linear partial and integro-differential equations, and in [17,18,23,24] in the case of certain nonlinear equations. It is based on the Nagumo norms [5,13,28] and on a technique of majorant series.…”

Section: Proposition 22 ([2 20])mentioning

confidence: 98%

“…We shall now bound the Nagumo norms }u j,˚} ps`1qj,ρ for any j ě 0. To do that, we shall proceed similarly as in [3,17,18,[20][21][22][23][24] by using a technique of majorant series.…”

Section: 22mentioning

confidence: 99%

“…In this article, we propose to prove that these two results remain true in the case of the generalized Boussinesq equation (1.2). To do this, we shall use an approach similar to those already developed by the author in [17][18][19][23][24][25][26] for some nonlinear partial differential equations (see also [3,[20][21][22] for an approach in the linear case). Let us point out here that, as we shall see below, the terms u m pB x uq 2 of Eq.…”

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confidence: 99%

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Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

Remy

2022

Asymptotic Analysis

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In this article, we investigate Gevrey and summability properties of the formal power series solutions of the inhomogeneous generalized Boussinesq equations. Even if the case that really matters physically is an analytic inhomogeneity, we systematically examine here the cases where the inhomogeneity is s-Gevrey for any s ⩾ 0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy: for any s ⩾ 1, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < 1, the formal solutions are generically 1-Gevrey. In the latter case, we give in particular an explicit example in which the formal solution is s ′ -Gevrey for no s ′ < 1, that is exactly 1-Gevrey. Then, we give a necessary and sufficient condition under which the formal solutions are 1-summable in a given direction arg ( t ) = θ. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proofs of our various results.

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Section: Proposition 22 ([2 20])mentioning

confidence: 98%

“…We shall now bound the Nagumo norms }u j,˚} ps`1qj,ρ for any j ě 0. To do that, we shall proceed similarly as in [3,17,18,[20][21][22][23][24] by using a technique of majorant series.…”

Section: 22mentioning

confidence: 99%

mentioning

confidence: 99%

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Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

Remy

2022

Asymptotic Analysis

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“…Note that, for general initial data, A 0 (x) is Gevrey regular of order k = 1 2 on an appropriate domain implies that the homogeneous solution A h (x, µ) is analytic, by standard theory for homogeneous heat equations. See for example [54], and recall that a function f (z) is Gevrey regular of order k on a set |z| < r, if there exist positive constants…”

Section: The Homogeneous and Particular Solutionsmentioning

confidence: 99%

“…This follows from standard theory for the analyticity of solutions. We refer to [6,43,54] for the general theory of analyticity of solutions and Gevrey regularity of order k for homogeneous and inhomogeneous heat equations.…”

Section: The Homogeneous and Particular Solutionsmentioning

confidence: 99%

Delayed Hopf bifurcation and space-time buffer curves in the Complex Ginzburg-Landau equation

Goh,

Kaper,

2020

Preprint

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In this article, the recently-discovered phenomenon of delayed Hopf bifurcations (DHB) in reaction-diffusion PDEs is analyzed in the cubic Complex Ginzburg-Landau equation, as an equation in its own right, with a slowly-varying parameter. We begin by using the classical asymptotic methods of stationary phase and steepest descents to show that solutions which have approached the attracting quasi-steady state (QSS) before the Hopf bifurcation remain near that state for long times after the instantaneous Hopf bifurcation and the QSS has become repelling. In the complex time plane, the phase function of the linear PDE has a saddle point, and the Stokes and anti-Stokes lines are central to the asymptotics. The nonlinear terms are treated by applying an iterative method to the mild form of the PDE given by perturbations about the linear particular solution. This tracks the closeness of solutions near the attracting and repelling QSS. Next, we show that beyond a key Stokes line through the saddle there is a curve in the space-time plane along which the particular solution of the linear PDE ceases to be exponentially small, causing the solution of the nonlinear PDE to diverge from the repelling QSS and exhibit large-amplitude oscillations. This curve is called the space-time buffer curve. The homogeneous solution also stops being exponentially small in a spatially dependent manner, as determined also by the initial time. Hence, a competition arises between these two solutions, as to which one ceases to be exponentially small first, and this competition governs spatial dependence of the DHB. We find four different cases of DHB, depending on the outcomes of the competition, and we quantify to leading order how these depend on the main system parameters, including the Hopf frequency, initial time, initial data, source terms, and diffusivity. Examples are presented for each case, with source terms that are uni-modal, spatially-periodic, smooth step function, and algebraically-growing. Also, rich spatio-temporal dynamics are observed in the post-DHB oscillations. Finally, it is shown that large-amplitude source terms can be designed so that solutions spend substantially longer times near the repelling QSS, and hence region-specific control over the delayed onset of oscillations can be achieved.

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Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity

Remy

2021

RACSAM

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In this article, we are interested in the Gevrey properties of the formal power series solution in time of the partial differential equations with a polynomial semilinearity and with analytic coefficients at the origin of C n`1 . We prove in particular that the inhomogeneity of the equation and the formal solution are together s-Gevrey for any s ě sc, where sc is a nonnegative rational number fully determined by the Newton polygon of the associated linear PDE. In the opposite case s ă sc, we show that the solution is generically sc-Gevrey while the inhomogeneity is s-Gevrey, and we give an explicit example in which the solution is s 1 -Gevrey for no s 1 ă sc.

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Gevrey index theorem for the inhomogeneous n-dimensional heat equation with a power-law nonlinearity and variable coefficients

Cited by 6 publications

References 50 publications

Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

Delayed Hopf bifurcation and space-time buffer curves in the Complex Ginzburg-Landau equation

Gevrey regularity of the solutions of the inhomogeneous partial differential equations with a polynomial semilinearity

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