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PreprintCC BY 4.0arXiv

Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent qr=(p−1)(s+1)

  • Xianfa Song
  • Zeen Song
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Abstract

In this paper, we study the following Gierer--Meinhardt system ut=d1Δu−a1u+up/vq+δ1(x), x∈ Ω, t>0, vt=d2Δv−a2v+ur/vs+δ2(x), x∈ Ω, t>0, ∂νu=∂νv=0, x∈ ∂ Ω, t>0, u(x,0)=u0(x), v(x,0)=v0(x), x∈ Ω, where Ω⊂ ℝN is a bounded domain, ν is the outer norm vector with respect to the smooth boundary ∂ Ω, d1,d2, p,q,r>0, s>−1, p−1<r, qr=(p−1)(s+1), δ1(x), δ2(x), u0(x) and v0(x) are continuous functions. For every pair d1,d2>0, we respectively establish the global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent qr=(p−1)(s+1) under certain conditions.

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Cite

BibTeXRIS
BibTeX
@misc{song2026global,
  title = {{Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent $qr=(p-1)(s+1)$}},
  author = {Xianfa Song and Zeen Song},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01968},
  eprint = {2610.01968},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01968},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent qr=(p−1)(s+1)
AU  - Xianfa Song
AU  - Zeen Song
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01968
UR  - https://arxiv.org/abs/2610.01968
AB  - In this paper, we study the following Gierer--Meinhardt system u_t=d_1Δu−a_1u+u^p/v^q+δ_1(x), x∈ Ω, t>0, v_t=d_2Δv−a_2v+u^r/v^s+δ_2(x), x∈ Ω, t>0, ∂_νu=∂_νv=0, x∈ ∂ Ω, t>0, u(x,0)=u_0(x), v(x,0)=v_0(x), x∈ Ω, where Ω⊂ ℝ^N is a bounded domain, ν is the outer norm vector with respect to the smooth boundary ∂ Ω, d_1,d_2, p,q,r>0, s>−1, p−1<r, qr=(p−1)(s+1), δ_1(x), δ_2(x), u_0(x) and v_0(x) are continuous functions. For every pair d_1,d_2>0, we respectively establish the global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent qr=(p−1)(s+1) under certain conditions.
ER  -