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dc.contributor.authorTrofimchuk, Elena
dc.contributor.authorTkachenko, Victor
dc.contributor.authorTrofimchuk, Sergei I.
dc.date.accessioned2007-03-20T12:00:08Z
dc.date.accessioned2016-10-05T14:14:22Z
dc.date.available2007-03-20T12:00:08Z
dc.date.available2016-10-05T14:14:22Z
dc.date.issued2007-03-28
dc.identifier.urihttp://publications.mfo.de/handle/mfo/1207
dc.descriptionResearch in Pairs 2007en_US
dc.description.abstractWe study positive bounded wave solutions $u(t, x) = \phi(\nu \cdot x+ct)$, $\phi(-\infty)=0$, of equation $u_t(t, x)=\delta u(t,x)−u(t,x) +g(u(t−h,x))$, $x \in \mathbb{R}^m$(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: $u_1 \equiv 0$ and $u_2 \equiv \kappa>9$. The birth function $g \in C(\mathbb{R}_+, \mathbb{R}_+)$ satisfies a few mild conditions: it is unimodal and differentiable at $0,\kappa$. Some results also require the positive feedback of $g:[g(\text{max} g),\text{max} g] \to \mathbb{R}_+$ with respect to $\kappa$. If additionally $\phi(+\infty) =\kappa$, the above wave solution $u(t,x)$ is called a travelling front. We prove that every wave $\phi(\nu \cdot x+ct)$ is eventually monotone or slowly oscillating about $\kappa$. Furthermore, we indicate $c^∗ \in \mathbb{R}_+ \cup {+\infty}$ such that (∗) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity $c > c^∗$. Our results are based on a detailed geometric description of the wave profile $\phi$. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass $\mathcal{G}$ of ’asymmetric’ tent maps such that given $g \in \mathcal{G}$, there exists exactly one travelling front for each fixed admissible speed.en_US
dc.language.isoenen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2007,12
dc.titleSlowly oscillating wave solutions of a single species reaction-diffusion equation with delayen_US
dc.typePreprinten_US
dc.identifier.doi10.14760/OWP-2007-12
local.scientificprogramResearch in Pairs 2007
local.series.idOWP-2007-12
local.subject.msc34
local.subject.msc35
local.subject.msc92


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