Global existence of weak solutions in a Keller-Segel system with local anisotropy and nonlinear diffusion
Abstract
We show global existence of weak solutions to the following system of partial differential equations with nonlinear diffusion in both equations and anisotropic signal production ut=∇·((u+1)m1−1∇ u)−∇·(u∇ v), vt=∇·((v+1)m2−1∇ v)+∇·(u∇ v)−v+u, in a smooth bounded domain Ω⊆ℝn, n∈ℕ with nonnegative initial data and homogenuous Neumann boundary conditions. Moreover, we show boundedness of u in L∞((0,∞); Lm1(Ω)) and v in L∞(Ω×(0,∞)). The strategy will be to show global existence and boundedness of classical solutions (uε,vε) to the approximating system uε t=∇·((uε+1)m1−1∇ uε) −∇·((uε∇ vε)/(1+ε|∇ vε|)), vε t=∇·((vε+1)m2−1∇ vε)+∇·(uε∇ vε)−vε+uε. Passing to the limit εsearrow 0, we obtain convergence of weak solutions of the second to weak solutions of the first system. One of the main challenges lie in the proof of weak convergence of (∇ vε)/(1+ε|∇ vε|) to ∇ v in L2(Ω×(0,T)) for T∈(0,∞).
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BibTeX
@misc{hegenbart2026global,
title = {{Global existence of weak solutions in a Keller-Segel system with local anisotropy and nonlinear diffusion}},
author = {Max Hegenbart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11462},
eprint = {2610.11462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11462},
}
RIS
TY - UNPB
TI - Global existence of weak solutions in a Keller-Segel system with local anisotropy and nonlinear diffusion
AU - Max Hegenbart
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11462
UR - https://arxiv.org/abs/2610.11462
AB - We show global existence of weak solutions to the following system of partial differential equations with nonlinear diffusion in both equations and anisotropic signal production u_t=∇·((u+1)^m_1−1∇ u)−∇·(u∇ v), v_t=∇·((v+1)^m_2−1∇ v)+∇·(u∇ v)−v+u, in a smooth bounded domain Ω⊆ℝ^n, n∈ℕ with nonnegative initial data and homogenuous Neumann boundary conditions. Moreover, we show boundedness of u in L^∞((0,∞); L^m_1(Ω)) and v in L^∞(Ω×(0,∞)). The strategy will be to show global existence and boundedness of classical solutions (u_ε,v_ε) to the approximating system u_ε t=∇·((u_ε+1)^m_1−1∇ u_ε) −∇·((u_ε∇ v_ε)/(1+ε|∇ v_ε|)), v_ε t=∇·((v_ε+1)^m_2−1∇ v_ε)+∇·(u_ε∇ v_ε)−v_ε+u_ε. Passing to the limit εsearrow 0, we obtain convergence of weak solutions of the second to weak solutions of the first system. One of the main challenges lie in the proof of weak convergence of (∇ v_ε)/(1+ε|∇ v_ε|) to ∇ v in L^2(Ω×(0,T)) for T∈(0,∞).
ER -