Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

Meyer, John C.; Needham, D. J.

doi:10.1098/rspa.2014.0632

Cited by 3 publications

(4 citation statements)

References 27 publications

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“…We remark that the approach adopted here in the proof of theorem 4.3 was primarily motivated by the specific problems in [1,21,23]. However, the methodology is remarkably similar to that developed by Carathéodory in the context of ordinary differential equations in [4].…”

Section: Proofmentioning

confidence: 99%

“…It is the aim of this paper to consider the corresponding questions for (BRDC) in detail for two broader classes of nonlinear term f : R → R. In particular we will consider the situation when f : R → R is Hölder continuous and when f : R → R is upper Lipschitz continuous, after which these cases may be combined to consider the situation when f : R → R is both Hölder and upper Lipschitz continuous. The relevance and motivation for considering nonlinear terms f : R → R in these classes is detailed in the specific studies of Aguirre and Escobedo [1], Grundy and Peletier [9], Herrero and Velázquez [11], Meyer and Needham [21], Needham [23], McCabe et al . [13][14][15][16] and discussed in detail in Meyer [19, ch.…”

Section: Introductionmentioning

confidence: 99%

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Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

Meyer¹,

Needham²

2016

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

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Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

Meyer¹,

Needham²

2016

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Qualitative properties of non-negative classical bounded solutions to boundary value problems for (1.1), have been considered in [1], [13], [17], [19], [20], [22] and [27] with 0 < p < 1 and non-negative initial data, and until [CP] in [22] with n = 1, two-signed solutions were not considered. We highlight here that the spatially inhomogeneous solutions constructed in this paper are two signed onD T because any non-negative classical bounded solution to [CP] must be spatially homogeneous [1,Corollary 2.6].…”

Section: Introductionmentioning

confidence: 99%

On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity

Clark

Meyer

2020

Journal of Differential Equations

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In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish wellposedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as η → ∞.2 Self-similar solution structure to [CP] In this section, we establish a priori bounds for solutions to [CP]. Consequently, we consider a radial self-similar solution structure of solutions to [CP] which yields (P). To begin, we have, Lemma 2.1. Let u :D T → R be a solution to [CP]. Then,Proof. Since [CP] has spatially homogeneous initial data, the maximal solution u + :D T → R and minimal solution u − :D T → R to [CP] are spatially homogeneous for t ∈ [0, T ] (see, for example, [21, Proposition 8.31] for n = 1). We note that u ± :D T → R given byare the maximal and minimal solutions to [CP], and hence any solution u :D T → R to [CP] satisfies u − ≤ u ≤ u + onD T , as required.

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“…One of the main applications of the sublinear KdV equation ( 4) with α ∈ (0, 1) is in the theory of granular chains near the harmonic limit [28,29], for which the solitary waves has double-exponential decay to zero at infinity. Other applications of the sublinear nonlinearity is in modeling of the chemical reactions (autocatalysis), for which the diffusion equation with the sublinear nonlinear terms is a common model [30,31].…”

Section: Introductionmentioning

confidence: 99%

Stability and interaction of compactons in the sublinear KdV equation

Pelinovsky,

Slunyaev,

Kokorina

et al. 2020

Preprint

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Compactons are studied in the framework of the Korteweg-de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to compact perturbations with the same support is proven analytically. Dynamics of compactons is studied numerically, including evolution of pulse-like disturbances and interactions of compactons of the same or opposite polarities. Compactons interact inelastically, though almost restore their shapes after collisions. Compactons play a two-fold role of the long-living solitonlike structures and of the small-scale waves which spread the wave energy.

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Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis

Cited by 3 publications

References 27 publications

Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity

Stability and interaction of compactons in the sublinear KdV equation

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