Global weak solutions to a doubly degenerate nutrient taxis system on the whole real line

2025/08/10 by Herrero-Hervás, Federico
#35K57 #35K65 (Primary) 35K59 #35Q92 #92C17 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.07503

Abstract

This work addresses the one-dimensional Cauchy problem for the doubly degenerate nutrient taxis model \begincases (∂ u)/(∂ t) = (∂)/(∂ x)(u v u_x) - (∂)/(∂ x)(u² v v_x) + u v, amp; x∈ ℝ, ~tgt;0,
(∂ v)/(∂ t) = (∂² v)/(∂ x²) - u v, amp; x∈ ℝ, ~tgt;0,
u(x,0) = u₀(x) ≥ 0, v(x,0) = v₀(x)gt;0, ~ amp; x∈ ℝ, \endcases which models pattern formation in bacterial populations. The global existence of weak solutions is established for initial data satisfying appropriate regularity and integrability conditions. To account for the degeneracy caused by u₀ not being strictly positive and the difficulties arising from the unboundedness of the domain, we consider a family of regularized problems posed on bounded intervals (-(1)/(ε), (1)/(ε)), for ε ∈ (0,1). Through adequate estimates uniform in ε, we construct global solutions to the system by passing to the limit using the Aubin-Lions lemma.

Citations

• Boundedness in a two-dimensional doubly degenerate nutrient taxis system
• Global-in-time Boundedness of solution for Cauchy problem to the Parabolic-Parabolic Keller-Segel system with logistic growth
• Derivation of a bacterial nutrient-taxis system with doubly degenerate\n cross-diffusion as the parabolic limit of a velocity-jump process
• The effects of nutrient chemotaxis on bacterial aggregation patterns\n with non-linear degenerate cross diffusion

Related

• Global weak solutions to a two-dimensional doubly degenerate nutrient taxis system with logistic source
• Global boundedness for a two-dimensional doubly degenerate nutrient taxis system
• Boundedness in a two-dimensional doubly degenerate nutrient taxis system
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• L^∞ bounds and asymptotic behavior in a doubly degenerate chemotaxis system below six dimensions