Existence of Multidimensional Travelling Waves and Systems of Waves

Volpert, A. I.; Volpert, Vitaly

doi:10.1081/pde-100002239

Cited by 20 publications

(16 citation statements)

References 19 publications

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“…Next we prove the Fredholm property and the existence of the topological degree for the integro-differential operator A associated to (7) with α(y) ≡ α > 0 constant. We make the following additional assumption on the function κ…”

Section: Abridged English Versionmentioning

confidence: 99%

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Systèmes de réaction–diffusion sans propriété de Fredholm

Ducrot

Marion

Volpert

2005

Comptes Rendus. Mathématique

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Section: Abridged English Versionmentioning

confidence: 99%

“…For Λ = 1 this system can be reduced to a single equation, which has been extensively studied [1][2][3][4][5]7]. In the case Λ = 1 the operator associated to these equations does not satisfy the Fredholm property.…”

Section: Abridged English Versionmentioning

confidence: 99%

Systèmes de réaction–diffusion sans propriété de Fredholm

Ducrot

Marion

Volpert

2005

Comptes Rendus. Mathématique

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“…A numerous number of results have also been obtained in higher dimensions. We refer for instance to Berestycki and Nirenberg [7] for travelling wave on cylinders, Hamel and Omrani [19] and Volpert and Volpert [27] for multistable nonlinearities on cylinders.…”

Section: Remark 14mentioning

confidence: 99%

A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

Ducrot¹

2015

Math. Ann.

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In this work we study the existence of wave solutions for a scalar reactiondiffusion equation of bistable type posed in a multi-dimensional periodic medium. Roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term. Here the term wave is used to describe either pulsating travelling wave or standing transition solution. As a special case we study a two-dimensional heterogeneous Allen-Cahn equation in both cases of slowly varying medium and rapidly oscillating medium. We prove that bistability occurs in these two situations and we conclude to the existence of waves connecting u = 0 and u = 1. Moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction.

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“…In fact, his method of [12] is rather different from ours. See [10] on traveling wave solutions in bistable maps, [2] time-periodic nonlocal bistable equations, [1] time-periodic bistable reaction-diffusion equations, e.g., [3,4,7,9,15] discrete bistable equations, [8] nonlocal Burgers equations and [13,14,16] multistable reaction-diffusion equations.…”

Section: ∂U ∂T (T X) = U(t X − 1) − U(t X) − λU(t X)(u(t X) − α)mentioning

confidence: 99%

Existence of Traveling Wave Solutions for a Nonlocal Bistable Equation: An Abstract Approach

Yagisita¹

2010

Publ. Res. Inst. Math. Sci.

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We consider traveling fronts to the nonlocal bistable equationwhere μ is a Borel-measure on R with μ(R) = 1 and f satisfies f (0) = f (1) = 0, f < 0 in (0, α) and f > 0 in (α, 1) for some constant α ∈ (0, 1). We do not assume that μ is absolutely continuous with respect to the Lebesgue measure. We show that there are a constant c and a monotone function φ with φ(−∞) = 0 and φ(+∞) = 1 such that u(t, x) := φ(x+ct) is a solution to the equation, provided f (α) > 0. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.

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Existence of Multidimensional Travelling Waves and Systems of Waves

Cited by 20 publications

References 19 publications

Systèmes de réaction–diffusion sans propriété de Fredholm

Systèmes de réaction–diffusion sans propriété de Fredholm

A multi-dimensional bistable nonlinear diffusion equation in a periodic medium

Existence of Traveling Wave Solutions for a Nonlocal Bistable Equation: An Abstract Approach

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