Domains Of Arbitrary Dimension Research Articles

A degenerate migration-consumption model in domains of arbitrary dimension

Abstract 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].

Convergence rates of solutions in apredator-preysystem withindirect pursuit-evasion interaction in domains of arbitrary dimension

<p style='text-indent:20px;'>In this paper, we deal with the following indirect pursuit-evasion model</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+ u(\lambda-u+av), \quad x\in \Omega, t>0, \\ v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+ v(\mu-v-bu), \quad x\in \Omega, t>0, \\ { }{0 = \Delta w- w+v}, \quad x\in \Omega, t>0, \\ { }{0 = \Delta z- z+ u}, \quad x\in \Omega, t>0, \\ \end{array}\right. \ \ \ \ \ \ \ \ \ \ (\star) \end{align} $ \end{document} </tex-math></disp-formula></p> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \xi, \lambda, \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M4">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ b $\end{document}</tex-math></inline-formula> are positive parameters. This system is used to achieve some insight into possible dynamical properties of pursuit-evasion processes, in which the respective tactic movements are oriented along gradients of some indirectly produced stimuli, rather than following individuals directly. One main purpose of the present paper is to remove the restriction of <inline-formula><tex-math id="M6">\begin{document}$ N\leq3 $\end{document}</tex-math></inline-formula>. Indeed, by using a iteration argument combined with suitable a priori estimates, we conclude that for any <inline-formula><tex-math id="M7">\begin{document}$ N\geq1 $\end{document}</tex-math></inline-formula>, an associated initial-boundary value problem <inline-formula><tex-math id="M8">\begin{document}$ (\star) $\end{document}</tex-math></inline-formula> admits a unique global bounded classical solution. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \chi<\left\{\begin{array}{ll} 4\sqrt{\frac{a(1+ab)}{b(\lambda+a\mu)}}, \quad\; \; \mbox{if}\; \; \lambda>b\mu, \\ 4\sqrt{\frac{a}{b\lambda}}, \quad\; \; \mbox{if}\; \; \lambda\leq b\mu\ \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and <inline-formula><tex-math id="M9">\begin{document}$ \xi<4\sqrt{\frac{b(1+ab)}{a(\mu-b\lambda)_+}} $\end{document}</tex-math></inline-formula>, the corresponding solution <inline-formula><tex-math id="M10">\begin{document}$ (u, v, w, z) $\end{document}</tex-math></inline-formula> of the system decays to <inline-formula><tex-math id="M11">\begin{document}$ (u_*, v_*, v_*, u_*) $\end{document}</tex-math></inline-formula> exponentially (or algebraically), where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ u_* = \left\{\begin{array}{ll} \frac{\lambda+a\mu}{1+ab}, \quad\; \; \mbox{if}\; \; \lambda>b\mu, \\ \lambda, \quad\; \; \mbox{if}\; \; \lambda\leq b\mu\ \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and <inline-formula><tex-math id="M12">\begin{document}$ v_* = \frac{(\lambda-b\mu)_+}{1+ab} $\end{document}</tex-math></inline-formula>. To the best of our knowledge, there is the first result on convergence rates of solutions of the system.</p>

Global solutions to a Keller-Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension

This manuscript considers the no-flux initial-boundary problem for the migration-consumption system{ut=Δ(uϕ(v)),vt=Δv−uv,(⋆) in a smoothly bounded domain Ω⊂Rn, n≥1, where ϕ suitably generalizes the singular prototype given byϕ(ξ)=ξ−α,ξ>0, with α>0.It is firstly shown that for such diffusion singularities of arbitrary strength, and for any given initial data u0 and v0 from W1,∞(Ω) with u0≥0 and v0>0 in Ω‾, a so-called very weak-strong solution (u,v) with (u,v)|t=0=(u0,v0) can be constructed. Under the additional restrictions that 2≤n≤5 and α>n−26−n, and under an additional assumption on ϕ′, it is furthermore asserted that the flux ∇(uϕ(v)) belongs to some reflexive Lebesgue space, and that (⋆) is satisfied in a standard weak sense.Finally, in the one-dimensional case a statement on global classical solvability is derived under the mere condition that ϕ∈C3((0,∞)) be positive.

Global smooth solvability of a parabolic–elliptic nutrient taxis system in domains of arbitrary dimension

This paper deals with the nutrient taxis system{ut=Δu−∇⋅(u∇v),0=Δv−uv−μv+r(x,t), in a bounded domain Ω⊂Rn, n≥1, with smooth boundary, where μ≥0 is a parameter and r∈C1(Ω‾×[0,∞)) is a given nonnegative function.It is shown that for any prescribed initial data u0∈W1,∞(Ω) with u0>0 in Ω‾, the corresponding Neumann initial–boundary problem admits a global classical solution. With regard to qualitative aspects, it is moreover, inter alia, seen that if r additionally satisfies∫tt+1∫Ω|∇r|2→0as t→∞, then in the large time limit the solution component u stabilizes toward the constant 1|Ω|∫Ωu0 with respect to the norm in L1(Ω), and that if furthermoresupt>0⁡‖r(⋅,t)‖Lq(Ω)<∞ for some q≥1 fulfilling q>n2, then u is uniformly bounded.

Consistency on Domain Boundaries for Linear PDAE Systems

This paper considers the determination of consistent variations of the dependent variables over the boundary of the domain of definition of a linear system of partial differential and algebraic equations (PDAEs). Particular emphasis is placed on the specifications (“boundary conditions”) imposed on different parts of the boundary and their consistency with the underlying PDAE system and with each other. Specifications imposed on overlapping parts of the boundary (e.g., faces with common edges, or edges with common vertices) often lead to inconsistencies (corner singularities) that are not trivial to detect, especially in PDAEs involving three or more dimensions. A symbolic/numerical algorithm is proposed for the analysis of PDAE systems defined over finite hyperrectangular domains of arbitrary dimensions.

Global Attractors for Cross Diffusion Systems on Domains of Arbitrary Dimension

Global Attractors for Cross Diffusion Systems on Domains of Arbitrary Dimension

Global existence for a class of triangular parabolic systems on domains of arbitrary dimension

A class of triangular parabolic systems given on bounded domains of R n \mathbb {R}^n with arbitrary n n is investigated. Sufficient conditions on the structure of the systems are found to assure that weak solutions exist globally.

Cross Diffusion Systems on 2 Spatial Dimensional Domains

A triangular cross diffusion system given on a domain of arbitrary dimension n is considered. It will be shown that (L∞, L n ) boundedness implies uniformly boundedness. For n = 2, the general result is then applied to a system with Lotka-Volterra reaction and the existence of the global attractor is proven.