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Sergei Trofimchuk

Publications and source records attributed to Sergei Trofimchuk.

At least 37 records · Page 2Linked to original sources
arXiv2015

Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions

We consider the nonlocal KPP-Fisher equation $u_t(t,x) = u_{xx}(t,x) + u(t,x)(1-(K *u)(t,x))$ which describes the evolution of population density $u(t,x)$ with respect to time $t$ and location $x$. The non-locality is expressed in terms of the convolution of $u(t, \cdot)$ with kernel $K(\cdot) \geq 0,$ $\int_{\mathbb{R}} K(s)ds =1$. The restrictions $K(s), s \geq 0,$ and $K(s), s \leq 0,$ are responsible for interactions of an individual with his left and right neighbors, respectively. We show that these two parts of $K$ play quite different roles as for the existence and uniqueness of traveling fronts to the KPP-Fisher equation. In particular, if the left interaction is dominant, the uniqueness of fronts can be proved, while the dominance of the right interaction can induce the co-existence of monotone and oscillating fronts. We also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on $K$.

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arXiv2014

Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations

We study the asymptotic stability of traveling fronts and front's velocity selection problem for the time-delayed monostable equation $(*)$ $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),\ x \in \mathbb{R},\ t >0$, considered with Lipschitz continuous reaction term $g: \mathbb{R}_+ \to \mathbb{R}_+$. We are also assuming that $g$ is $C^{1,α}$-smooth in some neighbourhood of the equilibria $0$ and $κ>0$ to $(*)$. In difference with the previous works, we do not impose any convexity or subtangency condition on the graph of $g$ so that equation $(*)$ can possess pushed traveling fronts. Our first main result says that the non-critical wavefronts of $(*)$ with monotone $g$ are globally nonlinearly stable. In the special and easier case when the Lipschitz constant for $g$ coincides with $g'(0)$, we present a series of results concerning the exponential [asymptotic] stability of non-critical [respectively, critical] fronts for the monostable model $(*)$. As an application, we present a criterion of the absolute global stability of non-critical wavefronts to the diffusive Nicholson's blowflies equation.

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arXiv2014

Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction

We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this equation possesses pushed traveling fronts. First we prove that the pushed wavefronts are nonlinearly stable with asymptotic phase. Moreover, combinations of these waves attract, uniformly on $R$, every solution of equation $(*)$ with the initial datum sufficiently rapidly decaying at one (or at the both) infinities of the real line. These results provide a sharp form of the theory of spreading speeds for equation $(*)$.

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arXiv2014

Construction of minimal non-invertible skew-product maps on 2-manifolds

Applying the Hric-Jäger blow up technique, we give an affirmative answer to the question about the existence of non-invertible minimal circle-fibered self-maps of the Klein bottle. In addition, we present a simpler construction of a non-invertible minimal self-map of two dimensional torus.

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arXiv2013

A note on the existence of non-monotone non-oscillating wavefronts

In this note, we present a monostable delayed reaction-diffusion equation with the unimodal birth function which admits only non-monotone wavefronts. Moreover, these fronts are either eventually monotone (in particular, such is the minimal wave) or slowly oscillating. Hence, for the Mackey-Glass type diffusive equations, we answer affirmatively the question about the existence of non-monotone non-oscillating wavefronts. As it was recently established by Hasik {\it et al.} and Ducrot {\it et al.}, the same question has a negative answer for the KPP-Fisher equation with a single delay.

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arXiv2013

Minimal sets of fibre-preserving maps in graph bundles

Topological structure of minimal sets is studied for a dynamical system $(E,F)$ given by a fibre-preserving, in general non-invertible, continuous selfmap $F$ of a graph bundle $E$. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set of $F$ with full projection onto the base space $B$ of the bundle. We show that $M$ is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere dense, we prove that either a typical fibre of $M$ is a Cantor set, or there is a positive integer $N$ such that a typical fibre of $M$ has cardinality $N$. If $M$ has nonempty interior we prove that there is a positive integer $m$ such that a typical fibre of $M$, in fact even each fibre of $M$ over a \emph{dense open} set $\mathcal O \subseteq B$, is a disjoint union of $m$ circles. Moreover, we show that each of the fibres of $M$ over $B\setminus \mathcal O$ is a union of circles properly containing a disjoint union of $m$ circles. Surprisingly, some of the circles in such "non-typical" fibres of $M$ may intersect. We also give sufficient conditions for $M$ to be a sub-bundle of $E$.

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arXiv2013

Traveling waves for a model of the Belousov-Zhabotinsky reaction

Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov-Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay $h\geq 0$. As we show, the BZ system has a dual character: it is monostable when its key parameter $r \in (0,1]$ and it is bistable when $r >1$. For $h=0, r\not=1$, and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case $r=1$ which to some extent resembles to the Zeldovich equation.

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arXiv2013

An extension of the Wright's 3/2-theorem for the KPP-Fisher delayed equation

We present a short proof of the following natural extension of the famous Wright's 3/2-stability theorem: the conditions $τ\leq 3/2, \ c \geq 2$ imply the presence of the positive traveling fronts (not necessarily monotone) $u = ϕ(x\cdot ν+ct), \ |ν| =1,$ in the delayed KPP-Fisher equation $u_t(t,x) = Δu(t,x) + u(t,x)(1-u(t-τ,x))$, $u\geq 0,$ $x \in \R^m.$

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arXiv2012

Global continuation of monotone wavefronts

In this paper, we answer the question about the criteria of existence of monotone travelling fronts $u = ϕ(ν\cdot x+ct), ϕ(-\infty) =0, ϕ(+\infty) = κ,$ for the monostable (and, in general, non-quasi-monotone) delayed reaction-diffusion equations $u_t(t,x) - Δu(t,x) = f(u(t,x), u(t-h,x)).$ $C^{1,γ}$-smooth $f$ is supposed to satisfy $f(0,0) = f(κ,κ) =0$ together with other monostability restrictions. Our theory covers the two most important cases: Mackey-Glass type diffusive equations and KPP-Fisher type equations. The proofs are based on a variant of Hale-Lin functional-analytic approach to the heteroclinic solutions where Lyapunov-Schmidt reduction is realized in a `mobile' weighted space of $C^2$-smooth functions. This method requires a detailed analysis of a family of associated linear differential Fredholm operators: at this stage, the discrete Lyapunov functionals by Mallet-Paret and Sell are used in an essential way.

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arXiv2012

Slowly oscillating wavefronts of the KPP-Fisher delayed equation

This paper concerns the semi-wavefronts (i.e. bounded solutions $u=ϕ(x ν+ct) >0,$ $ |ν|=1, $ satisfying $ϕ(-\infty)=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = Δu(t,x) + u(t,x)(1-u(t-τ,x)), \ u \geq 0,\ x \in \R^m. \eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $ϕ(+\infty)=1$) if $c \geq 2$ and $τ\leq 1$; our proof uses dynamical properties of some auxiliary one-dimensional map with the negative Schwarzian. The analysis of the fronts' asymptotic expansions at infinity is another key ingredient of our approach. It allows to indicate the maximal domain ${\mathcal D}_n$ of $(τ,c)$ where the existence of non-monotone wavefronts can be expected. Here we show that the problem of wavefront's existence is closely related to the Wright's global stability conjecture.

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arXiv2012

Separation dichotomy and wavefronts for a nonlinear convolution equation

This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation should either be asymptotically separated from zero or it should converge (exponentially) to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our abstract results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess at the same time the stationary, the expansion and the extinction waves.

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arXiv2011

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = Δu(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.

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arXiv2010

On uniqueness of semi-wavefronts (Diekmann-Kaper theory of a nonlinear convolution equation re-visited)

Motivated by the uniqueness problem for monostable semi-wavefronts, we propose a revised version of the Diekmann and Kaper theory of a nonlinear convolution equation. Our version of the Diekmann-Kaper theory allows 1) to consider new types of models which include nonlocal KPP type equations (with either symmetric or anisotropic dispersal), non-local lattice equations and delayed reaction-diffusion equations; 2) to incorporate the critical case (which corresponds to the slowest wavefronts) into the consideration; 3) to weaken or to remove various restrictions on kernels and nonlinearities. The results are compared with those of Schumacher (J. Reine Angew. Math. 316: 54-70, 1980), Carr and Chmaj (Proc. Amer. Math. Soc. 132: 2433-2439, 2004), and other more recent studies.

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arXiv2010

Positive travelling fronts for reaction-diffusion systems with distributed delay

We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction-diffusion systems with distributed delay. To prove the existence of travelling waves, we give an abstract formulation of the equation for the wave profiles in some suitable Banach spaces, and apply known results about the index of some associated Fredholm operators. After a Liapunov-Schmidt reduction, these waves are obtained via the Banach contraction principle, as perturbations of a positive heteroclinic solution for the associated system without diffusion, whose existence is proven under some requirements. By a careful analysis of the exponential decay of the travelling wave profiles at $-\infty$, their positiveness is deduced. The existence of positive travelling waves is important in terms of applications to biological models. Our method applies to systems of delayed reaction-diffusion equations whose nonlinearities are not required to satisfy a quasi-monotonicity condition. Applications are given, and include the delayed Fisher-KPP equation.

math.CA↗
arXiv2010

Monotone traveling wavefronts of the KPP-Fisher delayed equation

In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov-Petrovskii-Piskunov-Fisher equation. Since then, this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction-diffusion equations. In this paper, we give a complete solution to the problem of existence and uniqueness of monotone waves in the KPP-Fisher equation. We show that each monotone traveling wave can be found via an iteration procedure. The proposed approach is based on the use of special monotone integral operators (which are different from the usual Wu-Zou operator) and appropriate upper and lower solutions associated to them. The analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach.

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arXiv2008

Uniqueness of fast travelling fronts in reaction-diffusion equations with delay

We consider positive travelling fronts of the time-delayed reaction-diffusion equation with the monostable birth function. Our main result says that for every fixed and sufficiently large velocity c, the positive travelling front is unique (modulo translations). To prove the uniqueness, we introduce a small parameter 1/c and realize the Lyapunov-Schmidt reduction in a scale of Banach spaces.

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arXiv2006

Admissible wavefront speeds for a single species reaction-diffusion equation with delay

We consider equation $u_t(t,x) = Δu(t,x)- u(t,x) + g(u(t-h,x)) (*) $, when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=κ>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(κ)) = [c_*,c^*]$ such that $(*)$ has at least one (possibly, nonmonotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here $c_*>0$ is finite and $c^* \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound $c_*$ is sharp so that there are not wavefronts moving with speed $c < c_*$. In contrast to reported results, the interval $\mathcal{C}$ can be compact, and we conjecture that some of equations $(*)$ can indeed have an upper bound for propagation speeds of travelling fronts. As particular cases, Eq. $(*)$ includes the diffusive Nicholson's blowflies equation and the Mackey-Glass equation with nonmonotone nonlinearity.

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