Uniqueness and large-time continuous dependence in a doubly degenerate nutrient taxis system
Abstract
ut = ∇ · (uv∇ u) − χ∇· (u2 v∇ v) + ℓ uv, vt = Δv − uv, is considered in a smoothly bounded domain Ω⊂ ℝn and with χ≥ 0 and ℓ≥ 0. Within a suitable concept of solvability, a statement on uniqueness of solutions emanating from initial data (u0,v0) is derived under the mere assumption that u0∈ L∞(Ω) and v0∈ W1,∞(Ω) are such that u0≥ 0 and v0>0 in Ω̅; in particular, no condition on continuity of u0 is imposed. This result is thereafter used to reveal that in physcally meaningful scenarios, for any initial data which are additionally assumed to be continuous and nontrivial, such solutions can always be found, and that these solutions depend continuously on the initial distributions not only in bounded sets of the form Ω× (0,T), but actually throughout Ω× (0,∞), in a suitable sense.
Related
- Spatial inhomogeneity for a three-dimensional doubly degenerate nutrient system with indirect consumption
- Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
- Existence, Uniqueness, and Continuous Dependence on Initial/Final Values for Liouville–Caputo Fractional Difference Equations
- The Frohmanian Symplectic Tether: Canonical Uniqueness on Coadjoint Orbits of SDiff(T3), Non-Degenerate Casimir Coupling, and Conditional Global Regularity via Quartic Young Absorption in Lean 4
- Boundedness and contractive estimates for orthotropic, widely degenerate, doubly nonlinear diffusion equations
BibTeX
@misc{winkler2026uniqueness,
title = {{Uniqueness and large-time continuous dependence in a doubly degenerate nutrient taxis system}},
author = {Michael Winkler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36117},
eprint = {2610.36117},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36117},
}
RIS
TY - UNPB
TI - Uniqueness and large-time continuous dependence in a doubly degenerate nutrient taxis system
AU - Michael Winkler
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36117
UR - https://arxiv.org/abs/2610.36117
AB - u_t = ∇ · (uv∇ u) − χ∇· (u^2 v∇ v) + ℓ uv, v_t = Δv − uv, is considered in a smoothly bounded domain Ω⊂ ℝ^n and with χ≥ 0 and ℓ≥ 0. Within a suitable concept of solvability, a statement on uniqueness of solutions emanating from initial data (u_0,v_0) is derived under the mere assumption that u_0∈ L^∞(Ω) and v_0∈ W^1,∞(Ω) are such that u_0≥ 0 and v_0>0 in Ω̅; in particular, no condition on continuity of u_0 is imposed. This result is thereafter used to reveal that in physcally meaningful scenarios, for any initial data which are additionally assumed to be continuous and nontrivial, such solutions can always be found, and that these solutions depend continuously on the initial distributions not only in bounded sets of the form Ω× (0,T), but actually throughout Ω× (0,∞), in a suitable sense.
ER -