A Geometric Approach to Singularly Perturbed Nonlocal Reaction-Diffusion Equations

Bose, Amitabha

doi:10.1137/s0036141098342556

Cited by 14 publications

(31 citation statements)

References 24 publications

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“…This constant can be expressed in terms of an integral over V by solving the U equation. Hence, one obtains a scalar equation for V with a nonlocal term as leading order approximation, and this equation is known as the shadow system (see also [2] and [3] for more examples of reaction-diffusion equations with nonlocal terms). The shadow system can then be studied by variational techniques, and the results can be extended to the full system.…”

Section: Discussionmentioning

confidence: 99%

“…Several aspects of the contents of this paper are related to existing literature, such as: the existence and stability of singular 'localized' patterns in the Gray-Scott model [9], [6], [7], [41]; the shadow system approach [20], [28], [29], [30], [2]; the stability of multi-pulse solutions [35], [36], [2], [3]; and the SLEP method [32], [33], [31], [19]. A section in which these relations are discussed concludes the paper.…”

Section: (λ ε)mentioning

confidence: 99%

“…Such a reduction is known in the literature on (linear) Schrödinger problems in the case that the potential, in our case V red h (ξ), is explicitly known (see for instance [26]). This is only the case in (1.2) or (1.8), when β 2 = 2 or β 2 = 3 (see [6], which corresponds to β 2 = 2; here V red h (ξ) = c 1 /(cosh c 2 ξ) 2 for certain c 1 , c 2 ). In this paper, we present a general transformation, that does not depend on an explicit expression for V red h (ξ), by which the inhomogeneous problem can be written as an inhomogeneous hypergeometric differential equation.…”

Section: Introductionmentioning

confidence: 99%

“…This equation can be solved by the classical Green's function method (see Appendix B on page 502). This solution is substituted into the expression for t 2 …”

Section: Introductionmentioning

confidence: 99%

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Large stable pulse solutions in reaction-diffusion equations

Doelman¹,

Gardner²,

Kaper³

2001

Indiana Univ. Math. J.

166

361

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ABSTRACT. In this paper we study the existence and stability of asymptotically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diffusion equations. This family includes the generalized Gierer-Meinhardt equation. The existence of N-pulse homoclinic orbits (N ≥ 1) is established by the methods of geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability of these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear eigenvalue problem associated to the stability of the pattern can be decomposed into the product of a slow and a fast transmission function. The NLEP approach determines explicit leading order approximations of these transmission functions. It is shown that the zero/pole cancellation in the decomposition of the Evans function, called the NLEP paradox, is a phenomenon that occurs naturally in singularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability problem, can be studied by the slow transmission function. The key ingredient of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric differential equation. By this transformation it is possible to determine both the number and the position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hopf bifurcation at a certain value µ Hopf of the main parameter µ. The NLEP approach not only yields a leading order approximation of µ Hopf , but it also shows that there is another bifurcation value, µ edge , at which a new (stable) eigenvalue bifurcates from the edge of the essential spectrum. Finally, it is shown that the N-pulse patterns are always unstable when N ≥ 2. 443

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Section: Discussionmentioning

confidence: 99%

Section: (λ ε)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…This equation can be solved by the classical Green's function method (see Appendix B on page 502). This solution is substituted into the expression for t 2 …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Large stable pulse solutions in reaction-diffusion equations

Doelman¹,

Gardner²,

Kaper³

2001

Indiana Univ. Math. J.

166

361

View full text Add to dashboard Cite

show abstract

“…The spectral properties of this non-local operator have been extensively studied in certain cases: see [4,5,14,20] for the case n = 1; [23] for n ≥ 3, and; [1,15,16,24] for general n. It is shown in these papers that the eigenvalues, corresponding eigenfunctions and other spectral properties of the operator L(ǫ) are precisely those of its closed extensionL(ǫ) and hence in the following, we simply refer to the operator L(ǫ) as defined in (5).…”

mentioning

confidence: 99%

Existence of Positive Solutions due to Non-local Interactions in a Class of Nonlinear Boundary Value Problems

Davidson¹,

Dodds²

2007

Methods and Applications of Analysis

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Abstract. We consider a class of non-local boundary value problems of the type used to model a variety of physical and biological processes, from Ohmic heating to population dynamics. Of particular relevance therefore is the existence of positive solutions. We are interested in the existence of such solutions that arise as a direct consequence of the non-local interactions in the problem. Conditions are therefore imposed that preclude the existence of a positive solution for the related local problem. Under these conditions, we prove that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise.

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Spatial Dynamics

Kuehn¹

2014

Applied Mathematical Sciences

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A Geometric Approach to Singularly Perturbed Nonlocal Reaction-Diffusion Equations

Cited by 14 publications

References 24 publications

Large stable pulse solutions in reaction-diffusion equations

Large stable pulse solutions in reaction-diffusion equations

Existence of Positive Solutions due to Non-local Interactions in a Class of Nonlinear Boundary Value Problems

Spatial Dynamics

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