Some remarks on segregation of $k$ species in strongly competing systems

Lanzara, Flavia; Montefusco, Eugenio

doi:10.4171/ifb/458

Cited by 3 publications

(5 citation statements)

References 17 publications

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“…To solve (22), we first transform it into a periodic problem, and then use separation of variables to write the solution in Fourier series. To this aim, we notice that v solves (22) if and only if…”

Section: Solutions In the Half-planementioning

confidence: 99%

“…We state the main result of this section. Its assumptions should be compared to those of Proposition 3.8, in particular we point out that they imply the existence of a unique solution v of (22). We recall that for Φ ∈ Lip([0, 2π]) we denote the Fourier coefficients of e −αx Φ(x) as…”

Section: Nodal Sets In the Half-planementioning

confidence: 99%

“…Proof. We recall that v satisfies (22), v is analytic in R × R + and continuous up to the boundary {(x, y) : y = 0} (see Proposition 3.8). By well known results of Hartman and Wintner [16],…”

Section: Lemma 42 Under the Above Notationmentioning

confidence: 99%

“…The homogeneous Dirichlet problem. We now turn our attention to the homogeneous version of (22), that is we look for conditions under which there exists a non zero solution v of…”

Section: 1mentioning

confidence: 99%

“…Hence, in comparison with the literature, our work tackles the segregation problem from a new perspective, that is the existence of limit segregated profiles satisfying additional qualitative properties or shadowing some given shapes. On the other hand, the literature on other aspects of segregation triggered by strong competition, starting from pioneering works by Dancer and Du [10,11], is now very vast and it is impossible to give a complete account of it here; besides the papers quoted above, we quote a few more recent ones such as [30,2,4,5,21,22] and we refer the interested reader to the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Rotating Spirals in segregated reaction-diffusion systems

Salort¹,

Terracini²,

Verzini³

et al. 2022

Preprint

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We give a complete characterization of the boundary traces ϕ i (i = 1, . . . , K) supporting spiraling waves, rotating with a given angular speed ω, which appear as singular limits of competition-diffusion systems of the typeHere Ω is a rotationally invariant planar set and a ij > 0 for every i and j. We tackle also the homogeneous Dirichlet and Neumann boundary conditions, as well as entire solutions in the plane. As a byproduct of our analysis we detect explicit families of eternal, entire solutions of the pure heat equation, parameterized by ω ∈ R, which reduce to homogeneous harmonic polynomials for ω = 0.

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Section: Solutions In the Half-planementioning

confidence: 99%

Section: Nodal Sets In the Half-planementioning

confidence: 99%

Section: Lemma 42 Under the Above Notationmentioning

confidence: 99%

“…The homogeneous Dirichlet problem. We now turn our attention to the homogeneous version of (22), that is we look for conditions under which there exists a non zero solution v of…”

Section: 1mentioning

confidence: 99%