2018
DOI: 10.1016/j.jde.2018.01.033
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Abstract: In this paper, we present and apply a computer-assisted method to study steady states of a triangular cross-diffusion system. Our approach consist in an a posteriori validation procedure, that is based on using a fixed point argument around a numerically computed solution, in the spirit of the Newton-Kantorovich theorem. It allows us to prove the existence of various non homogeneous steady states for different parameter values. In some situations, we get as many as 13 coexisting steady states. We also apply th… Show more

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Cited by 15 publications
(32 citation statements)
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References 50 publications
(91 reference statements)
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“…In the OOPDE setting, we can employ the routine assema to compute those matrices, but this needs c(v) andc(u) on each element center, which is obtained interpolating u and v from the nodes to the element centers, as it can be seen in Listings 1 and 2, which show the main files implementing the triangular cross-diffusion system (1.5) in pde2path. Remark : Note that the parameter set used in [7,9,30,32] and reported in Table 1 gives α > 0. In Table 2 we report the bifurcation values d B (λ k ) corresponding to the eigenvalue λ k of the Laplacian for the 1D domain (0, 1) and obtained by formula (B.3), for different values of n, compared with the numerical values estimated by the software pde2path.…”
Section: Discussionmentioning
confidence: 99%
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“…In the OOPDE setting, we can employ the routine assema to compute those matrices, but this needs c(v) andc(u) on each element center, which is obtained interpolating u and v from the nodes to the element centers, as it can be seen in Listings 1 and 2, which show the main files implementing the triangular cross-diffusion system (1.5) in pde2path. Remark : Note that the parameter set used in [7,9,30,32] and reported in Table 1 gives α > 0. In Table 2 we report the bifurcation values d B (λ k ) corresponding to the eigenvalue λ k of the Laplacian for the 1D domain (0, 1) and obtained by formula (B.3), for different values of n, compared with the numerical values estimated by the software pde2path.…”
Section: Discussionmentioning
confidence: 99%
“…The software setup required for system (1.5) can be found in Appendix A. For the numerical results we use the set of parameter values widely used in literature for the weak competition regime [7,9,30,32], here reported in Table 1. We set d 1 = d 2 =: d and use d as one main bifurcation parameter.…”
Section: Numerical Continuation On a 1d Domainmentioning
confidence: 99%
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“…Such cases have been studied extensively, see e.g. [4,11,10,24,28]. Furthermore, we do not split into real and imaginary parts (cf.…”
Section: Remark 8 the Interpolations Chosen In The Definitions Of G Kmentioning
confidence: 99%