A degenerate migration-consumption model in domains of arbitrary dimension
In a smoothly bounded convex domain with ≥ 1, a no-flux initial-boundary value problem for is considered under the assumption that near the origin, the function suitably generalizes the prototype given by By means of separate approaches, it is shown that in both cases ∈ (0, 1) and ∈ [1, 2] some glob...
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| Published in | Advanced nonlinear studies Vol. 24; no. 3; pp. 592 - 615 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
De Gruyter
01.07.2024
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| Subjects | |
| Online Access | Get full text |
| ISSN | 2169-0375 2169-0375 |
| DOI | 10.1515/ans-2023-0131 |
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| Abstract | In a smoothly bounded convex domain
with
≥ 1, a no-flux initial-boundary value problem for
is considered under the assumption that near the origin, the function
suitably generalizes the prototype given by
By means of separate approaches, it is shown that in both cases
∈ (0, 1) and
∈ [1, 2] some global weak solutions exist which, inter alia, satisfy
with sup
) < ∞ if
∈ [1, 2]. |
|---|---|
| AbstractList | In a smoothly bounded convex domain
with
≥ 1, a no-flux initial-boundary value problem for
is considered under the assumption that near the origin, the function
suitably generalizes the prototype given by
By means of separate approaches, it is shown that in both cases
∈ (0, 1) and
∈ [1, 2] some global weak solutions exist which, inter alia, satisfy
with sup
) < ∞ if
∈ [1, 2]. In a smoothly bounded convex domain Ω⊂Rn ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem forut=Δuϕ(v),vt=Δv−uv, $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given byϕ(ξ)=ξα,ξ∈[0,ξ0]. $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfyC(T)≔ess supt∈(0,T)∫Ωu(⋅,t)lnu(⋅,t)<∞for all T>0, $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){< }\infty \qquad \text{for\,all\,}T{ >}0,$$ with supT>0 C(T) < ∞ if α ∈ [1, 2]. In a smoothly bounded convex domain Ω ⊂ R n ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ ( v ) , v t = Δ v − u v , $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype given by ϕ ( ξ ) = ξ α , ξ ∈ [ 0 , ξ 0 ] . $$\phi \left(\xi \right)={\xi }^{\alpha },\qquad \xi \in \left[0,{\xi }_{0}\right].$$ By means of separate approaches, it is shown that in both cases α ∈ (0, 1) and α ∈ [1, 2] some global weak solutions exist which, inter alia, satisfy C ( T ) ≔ ess sup t ∈ ( 0 , T ) ∫ Ω u ( ⋅ , t ) ln u ( ⋅ , t ) < ∞ for all T > 0 , $$C\left(T\right){:=}\underset{t\in \left(0,T\right)}{\text{ess\,sup}}{\int }_{{\Omega}}u\left(\cdot ,t\right)\mathrm{ln}u\left(\cdot ,t\right){< }\infty \qquad \text{for\,all\,}T{ >}0,$$ with sup T >0 C ( T ) < ∞ if α ∈ [1, 2]. |
| Author | Winkler, Michael |
| Author_xml | – sequence: 1 givenname: Michael surname: Winkler fullname: Winkler, Michael email: michael.winkler@math.uni-paderborn.de organization: Universität Paderborn, Institut für Mathematik, 33098 Paderborn, Germany |
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| Snippet | In a smoothly bounded convex domain
with
≥ 1, a no-flux initial-boundary value problem for
is considered under the assumption that near the origin, the... In a smoothly bounded convex domain Ω ⊂ R n ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ ( v ) , v t... In a smoothly bounded convex domain Ω⊂Rn ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem forut=Δuϕ(v),vt=Δv−uv,... |
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| SubjectTerms | 35K51 35K57 (secondary) 35K65 (primary) 35Q92 92C17 a priori estimate chemotaxis degenerate diffusion |
| Title | A degenerate migration-consumption model in domains of arbitrary dimension |
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