Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918354182209536 |
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| author | Herrero-Hervás, Federico |
| author_facet | Herrero-Hervás, Federico |
| contents | This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases}
u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex]
\hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $Ω\subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0) = u_0(x) \geq 0$, where $f(x,t) \geq 0$ is known external supply of the nutrient.
It is shown that for any nonnegative $u_0 \in W^{1,\infty}(Ω)$ and $f \in C^1\big(\barΩ \times [0,\infty) \big)$, $f \not \equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach.
The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzelà-Ascoli theorem and the Aubin-Lions lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21164 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system Herrero-Hervás, Federico Analysis of PDEs 35K65, 35K59, 35K57, 35Q92 This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $Ω\subset \mathbb{R}$, under no-flux boundary conditions and nonnegative initial value $u(x,0) = u_0(x) \geq 0$, where $f(x,t) \geq 0$ is known external supply of the nutrient. It is shown that for any nonnegative $u_0 \in W^{1,\infty}(Ω)$ and $f \in C^1\big(\barΩ \times [0,\infty) \big)$, $f \not \equiv 0$, a global weak solution of the problem can be constructed by means of a regularization approach. The core of the analysis lies on a Harnack-type inequality for the second that allows us to overcome the lack of uniform coercivity. Together with time regularity properties, we obtain relative compactness through a combination of the Arzelà-Ascoli theorem and the Aubin-Lions lemma. |
| title | Weak global solvability of a doubly degenerate parabolic-elliptic nutrient taxis system |
| topic | Analysis of PDEs 35K65, 35K59, 35K57, 35Q92 |
| url | https://arxiv.org/abs/2602.21164 |