We study the nonlinear elliptic system: \[\begin{cases} u \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla u|^{p-2}\nabla u) + u = -\operatorname{div} (a(x) u|\nabla \psi|^{p-2}\nabla \psi) + f(x), \\ \psi \in W^{1,p}_0(\Omega): -\operatorname{div} (a(x)|\nabla \psi |^{p-2}\nabla \psi) = u^{\theta} \end{cases}\] in a bounded, open subset of \(\mathbb{R}^N\) for \(N \gt 2\) and \(2 \lt p \lt N\), where \(f\) satisfies: \[0 \leq f, \quad f\in L^{(p^*){'}}(\Omega), \quad p^*= \frac{Np}{N-p},\] \(a \in L^{\infty}(\Omega) \) is a given function such that there exist \(\alpha, \beta \in \mathbb{R}\) satisfying \[0\lt\alpha \leq a(x) \leq \beta, \quad x\in \Omega.\] We prove the existence of weak solutions in \([W^{1,p}_0(\Omega)]^2\) under the assumption \[0\lt \theta\lt 1- \frac{2}{p^*}.\]

Many population models formulated in terms of quasi-linear parabolic systems incorporate taxis, understood as directed movement of animals in response to some stimulus. Taxis is called direct if the animals follow the density gradient of another population, or indirect if they follow the density of a chemical secreted by individuals from another population. We prove that in the so-called fast reaction limit, i.e. when the ratio epsilon of the chemical signal diffusion to the signal production/degradation vanishes asymptotically, the solution of the model with indirect taxis converges to the solution of the model with direct taxis. It is justified then to replace the gradient of the chemical by the gradient in population density, reducing by one the number of equations in the indirect taxis model. The kinetic part of the considered models describes competition or predation, and the taxis term refers to the avoidance strategy assigned to one of the competitors or to a prey-taxis. Under certain compatibility condition satisfied by the initial data for the space dimension N <= 2, convergence occurs in a strong topology of function spaces, and its rate can be controlled by epsilon. Otherwise, the compactness argument only leads to convergence for a subsequence in the appropriate weak sense.

We review a chemotaxis system with flux limitation and present known results for both: the elliptic and the parabolic-elliptic case. We denote by u and v the density of living organisms and the concentration of a chemical substance, respectively, {[ - div(A(x) ∇ u) + u = -div(u D(x) |∇ v|^p-1∇ v ) + f(x), in x∈Ω ,; - div(D(x) ∇ v) +v= u^θ, in x∈Ω ,; u(x)=v(x)=0, in x∈∂Ω , ]. for given A, D and f, under some restrictions on p ∈ (1,2) and θ∈ (0,1) ; {[ u_t-Δ u= - div (χ u|∇ v|^p-2∇ v), x∈Ω , t>0,; -Δ v = u-1/|Ω |∫ _Ω u_0 dx, x∈Ω , t>0,; ∂ u/∂ n = ∂ u/∂ n=0, x∈∂Ω , t>0,; u(0,x)= u_0(x), x∈Ω , ]. where p ∈ (1,2) . For p

In this article, a parabolic-elliptic system of partial differential equations arising in chemotaxis with non-constant monotone chemotactic sensitivity is analyzed. Let $$\Omega $$ Ω be a bounded and regular domain, u the density of a biological species and v the concentration of a chemical satisfying the parabolic-elliptic system $$\begin{aligned} \left\{ \begin{array}{l} \displaystyle u_{t} - \Delta u = - div (u\chi (v) \nabla v) + \mu u (1- u), \; \; t>0, \; x\in \Omega , \\ \displaystyle -\Delta v+ v = u, \; \; t>0, \; x\in \Omega \end{array}\right. \end{aligned}$$ u t - Δ u = - d i v ( u χ ( v ) ∇ v ) + μ u ( 1 - u ) , t > 0 , x ∈ Ω , - Δ v + v = u , t > 0 , x ∈ Ω under Neumann boundary conditions, and bounded and positive initial data. We study the asymptotic behaviour of solutions under suitable assumptions in $$\chi \chi >0; \; \; \chi ^{\prime } \le 0; \; \; \; \chi ^{\prime \prime } \ge 0$$ χ χ > 0 ; χ ′ ≤ 0 ; χ ″ ≥ 0 when $$\mu $$ μ is sufficiently large for a given initial data $$u_0$$ u 0 . The result is obtained by using the system of ordinary differential equations: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{d \overline{u}}{dt} = \chi (\underline{u}) (\overline{u} -\underline{u})\overline{u}- \chi ^{\prime } (\underline{u}) c^2_{\Omega } ( 2+ \max \{1, \Vert u_0\Vert _{L^{\infty }(\Omega )} \}) ( \overline{u} - \underline{u} ) \overline{u} + \mu \overline{u}(1-\overline{u}), & t>0, \\ \displaystyle \frac{d\underline{u}}{dt} = \chi (\underline{u}) (\underline{u} -\overline{u}) \underline{u} + \mu \underline{u}(1-\underline{u}), & t>0; \end{array} \right. \end{aligned}$$ d u ¯ dt = χ ( u ̲ ) ( u ¯ - u ̲ ) u ¯ - χ ′ ( u ̲ ) c Ω 2 ( 2 + max { 1 , ‖ u 0 ‖ L ∞ ( Ω ) } ) ( u ¯ - u ̲ ) u ¯ + μ u ¯ ( 1 - u ¯ ) , t > 0 , d u ̲ dt = χ ( u ̲ ) ( u ̲ - u ¯ ) u ̲ + μ u ̲ ( 1 - u ̲ ) , t > 0 ; and a comparison method to obtain $$\begin{aligned} \underline{u}(t)< u(t,x)<\overline{u}(t); \; \; \underline{u}(t)< v(t,x) < \overline{u}(t), \text{ a.e. } x\in \Omega , \; t>0. \end{aligned}$$ u ̲ ( t ) < u ( t , x ) < u ¯ ( t ) ; u ̲ ( t ) < v ( t , x ) < u ¯ ( t ) , a.e. x ∈ Ω , t > 0 . The asymptotic behaviour of the system is also analyzed to obtain $$\begin{aligned} \lim _{t \rightarrow +\infty } \Vert u-1\Vert _{L^{\infty }(\Omega )} + \Vert v-1\Vert _{L^{\infty }(\Omega )}=0. \end{aligned}$$ lim t → + ∞ ‖ u - 1 ‖ L ∞ ( Ω ) + ‖ v - 1 ‖ L ∞ ( Ω ) = 0 .

Many ecological population models consider taxis as the directed movement of animals in response to a stimulus. The taxis is named direct if the animals are guided by the density gradient of some other population or indirect if they are guided by the density of a chemical secreted by individuals of the other population. Let u and v denote the densities of two populations and w the density of the chemical secreted by individuals in the v population. We consider a bounded, open set Ω⊂ℝ^N with regular boundary and prove that for the space dimension N≤ 2 the solution to the Lotka-Volterra competition model with repulsive indirect taxis and homogeneous Neumann boundary conditions u_t - d_uΔ u = χ∇· u ∇ w +μ_1u(1-u-a_1v) , v_t - d_vΔ v = μ_2v(1-v-a_2u) , ε ( w_t - d_wΔ w )= v- w , converges to the solution of repulsive direct-taxis model: u_t - d_uΔ u = χ∇· u ∇ v +μ_1u(1-u-a_1v) , v_t - d_vΔ v = μ_2v(1-v-a_2u) when ε⟶ 0. For space dimension N≥ 3 we use the compactness argument to show that the result holds in some weak sense. A similar result is also proved for a typical prey-predator model with prey taxis and logistic growth of predators.

In this article, we study the existence of solutions of a parabolic‐elliptic system of partial differential equations describing the behaviour of a biological species “ ” and a chemical stimulus “ ” in a bounded and regular domain of . The equation for is a parabolic equation with a nonlinear second order term of chemotaxis type with flux limitation as for . The chemical substance distribution satisfies the elliptic equation The evolution of is also determined by a logistic type growth term . The system is studied under homogeneous Neumann boundary conditions. The main result of the article is the existence of uniformly bounded solutions for and any .

In this article, we study a nonlinear system of elliptic partial differential equations describing the interaction of two species, “u” and “ ψ ” in a bounded domain Ω of ℝ^N for N ≥ 3 . The equation for “ ψ ” presents a production term defined by a bounded function B(u) and a drift term, which depends on a known function E. The system is presented in the following way model problem{[ u∈ W_0^1, N/N-1(Ω ): -div(A(x)∇ u) + u = -div(u ∇ψ ) + f,; ψ∈ W_0^1,2(Ω ): -div(A(x)∇ψ ) + ψ = B(u) + E ∇ψ . ]. where A:Ω→ℝ^N^2 is a symmetric matrix with bounded coefficients a_ij for i,j =1 … N , and E: Ω→ℝ^N belongs to (L^N(Ω ))^N , f is assumed to be a non-negative function of L^1(Ω ) satisfying ∫ _Ω f log (1+f) <∞ and B is a continuous and bounded function. We obtain the existence of solutions of the model problem, moreover, for N=3 , if f ∈ L^∞(Ω ) and E∈[ L^∞(Ω ) ] ^3 we have that u ∈ W_0^1, 2(Ω ) ∩ L^∞(Ω ).

In this article we study the existence of solutions of a system of partial differential equations of elliptic type, describing the distribution of a biological species "u" and the density of a chemical stimulus "." in a bounded domain O of RN. The equation for u includes a chemotaxis term with nonlinear flux limitation which depends on the exponent p > 1. The equation for u is given by -div(M(x)del u) + u = -chi div(u vertical bar del psi vertical bar(p-2)del psi) + f(x), where psi presents a subcritical production term u. and satisfies the equation The matrix of coefficients, M, is a known, symmetric and positive defined with coefficients m(ij) is an element of C-1((Omega) over bar), chi is a given real constant, f is a non-negative function belonging to L-m(Omega), m > max{1, N/2}. The production term exponent, theta, is assumed to be positive and fulfills one of the following constrains 1 < p < N theta/N theta - 1, 1 < N theta or max{N, p} 0. The problem is completed with Dirichlet boundary conditions for u and psi. The main result of the article includes the existence of positive solutions in H-0(1) (O) n L-infinity(Omega). (C) 2022 Elsevier Ltd. All rights reserved.

We consider a parabolic-elliptic system of partial differential equations with a chemotactic term in a \begin{document}$ N $\end{document}-dimensional unit ball "\begin{document}$ B $\end{document}" describing the behavior of a biological species "\begin{document}$ u $\end{document}" and a chemical stimuli "\begin{document}$ v $\end{document}". The system presents a sub-linear dependence of "\begin{document}$ \nabla v $\end{document}" in the chemotactic coefficient and a nonlinear diffusive term. The evolution of \begin{document}$ u $\end{document} is described by the equation \begin{document}$ u_t - \Delta u^m = - div (\chi u |\nabla v|^{p-2} \nabla v), \quad \mbox{ for } \ m >2, \quad p \in ( 1,2), \quad N \geq 1 $\end{document} for a positive constant \begin{document}$ \chi $\end{document}. The concentration of the chemical substance \begin{document}$ v $\end{document} satisfies the linear elliptic equation \begin{document}$ - \Delta v = u - \frac{1}{|B|} \int_{B} u_0dx. $\end{document} We consider the radially symmetric case and we prove the local existence of weak solutions for the mass accumulation function under assumption \begin{document}$ - \frac{1}{m}+ \frac{1}{N} + 1-\frac{pm}{4(m-1)} \geq 0, $\end{document} for radial and regular initial data. Additionally, if the constrain \begin{document}$ \frac{m }{m- 2} \left[ \frac{pm}{2(m-1)}-1\right] \leq 1 $\end{document} is satisfied, the solution globally exists in time.

We consider a parabolic–elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$\begin{aligned} \left\{ \begin{array}{l} u_t -\Delta (u \gamma (v))= \mu u(1-u), \\ - \Delta v +v=u, \end{array} \right. \end{aligned}$$ under Neumann boundary conditions and appropriate initial data in a bounded and regular domain $$\Omega $$ of $${{\mathbb {R}}}^N$$ (for $$N \ge 1)$$ , where $$\gamma \in C^3([0, \infty ))$$ and satisfies $$\gamma (s) > 0$$ , $$\gamma ^{\prime }(s) \le 0$$ , $$\gamma ^{\prime \prime } (s) \ge 0$$ , $$\gamma ^{\prime \prime \prime }(s) \le 0$$ for any $$s \ge 0$$ $$\begin{aligned}&-2 \gamma ^{\prime }(s) + \gamma ^{\prime \prime }(s)s \le \mu _0< \mu \\&\frac{[\gamma ^{\prime }(s)]^2}{\gamma (s)} \le c, \quad \text{ for } \text{ any } s \in [0, \infty ). \end{aligned}$$ We obtain the global existence and uniqueness of bounded in time solutions and the following asymptotic behavior $$\begin{aligned} \Vert u- 1\Vert _{L^{\infty }(\Omega )} +\Vert v- 1\Vert _{L^{\infty }(\Omega )} \rightarrow 0, \quad \text{ when } t \rightarrow +\infty . \end{aligned}$$

We consider a Parabolic-Elliptic system of PDE's with a chemotactic term in a N-dimensional unit ball describing the behavior of the density of a biological species "u" and a chemical stimulus "v." The system includes a nonlinear chemotactic coefficient depending of "del v," i.e. the chemotactic term is given in the form -div(chi u vertical bar del v|(p-2) del v), for p is an element of(N/N-1,2), N > 2 for a positive constant chi when v satisfies the poisson equation -Delta v = u - 1/vertical bar Omega vertical bar integral(Omega)u(0)dx. We study the radially symmetric solutions under the assumption in the initial mass 1/vertical bar Omega vertical bar integral(Omega)u(0)dx > 6. For chi large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.

This article deals with a fully parabolic chemotaxis system describing the behavior of a biological species with density “u” which follows a chemical gradient with density “v”. The problem presents a nonlocal growth term defined by f(u)=ua0−a1uα+a2∫Ωuαdxand the system is given by the following two second order coupled parabolic equations ut−Δu=−div(χum∇v)+f(u),vt−Δv+v=uγ,in a bounded domain Ω with homogeneous Neumann boundary conditions and appropriate initial data.The parameters α, m, ai (i=1,2) and γ satisfy α≥1,m>1,γ≥1,α+1>m+γ,a1>0,a1−a2|Ω|>0.Under suitable assumptions on the initial data and the coefficients of the system, the global-in-time existence of classical solutions and the convergence to the steady state u∗=a01α(a1−a2|Ω|)1α,v∗=(u∗)γ,when a0>0, are proved in any space dimension.

We study some qualitative properties of a misaligned journal bearing. The device consists of two cylinders closely spaced: an inner rotating cylinder (the shaft) whose symmetry axis is not parallel to the one of the outer cylinder (the bearing). We consider the load capacity of the system, defined as the force exerted by the pressure. It is given by the integral of the pressure times the normal vector to the bearing surface. We obtain finite load capacity, even in the limit case when a point contact occurs. It was also verified by numerical simulations. We used an adapted Preconditioned Conjugate Gradient Method for solving the direct problem, preserving the A-orthogonality property of the search directions, even after a restarting process. The solution of the related inverse problem is based on an interior, trust-region algorithm. To validate the numerical proposal, the predicted pressure values at the bearing mid-plane, are compared to published experimental data.

In this article we study a lubricated system consisting on a slider moving over a smooth surface and a known external force (the load) applied upon the slider. The slider moves at constant velocity and close proximity to the surface and the gap is filled by an incompressible fluid (the lubricant). At the equilibrium, the position of the slider presents one degree of freedom to be determined by the balance of forces acting on the system: the load and the total force exerted by the pressure of the lubricant. The pressure distribution is described by a variational inequality of elliptic type known as Swift–Stieber model and based on Reynolds equation. The distance h between the surfaces in a two dimensional domain Ω is given by h η ( x 1 , x 2 , y ) = h 0 ( x 1 , x 2 ) + h 1 ( y ) + η , ( x 1 , x 2 ) ∈ Ω , y ∈ [ 0 , 1 ] where h 0 ( x 1 , x 2 ) ∼ | x 1 | α for α > 0 and h 1 ( y ) ∼ | y − y 0 | β for y being the homogenization variable. The main result of the article quantify the influence of the roughness in the load capacity of the mechanism in the following way: If α < 3 γ for 0 < γ ⩽ 2 α < min { 1 γ − 2 , 3 γ } for γ > 2 then, the mechanism presents finite load capacity, i.e. lim η → 0 ∫ Ω p η < ∞. Infinite load capacity is obtained for γ > 1 and α > 2 / ( γ − 1 ). A one dimensional particular case is given for γ > 3 / 2 with infinite load capacity.

We consider an initial boundary value problem of the complex Ginzburg-Landau equation with some delayed feedback terms proposed for the control of chemical turbulence in reaction diffusion systems. We consider the equation in a bounded domain \(\Omega\subset\mathbb{R}^{N}\) (\(N\leq3\)), $$ \frac{\partial u}{\partial t}-(1+i\epsilon)\Delta u +(1+i\beta) | u| ^2u-(1-i\omega) u=F(u(x,t-\tau)) $$ for t>0, with $$ F(u(x,t-\tau)) =e^{i\chi_0}\big\{ \frac{\mu}{| \Omega| }\int_{\Omega}u(x,t-\tau) dx+\nu u(x,t-\tau) \big\} , $$ where \(\mu\), \(\nu\geq0\), \(\tau>0\) but the rest of real parameters \(\epsilon\), \(\beta\), \(\omega\) and \(\chi_0\) do not have a prescribed sign. We prove the existence and uniqueness of weak solutions of problem for a range of initial data and parameters. When \(\nu=0\) and \(\mu>0\) we prove that only the initial history of the integral on \(\Omega\) of the unknown on \((-\tau,0)\) and a standard initial condition at t=0 are required to determine univocally the existence of a solution. We prove several qualitative properties of solutions, such as the finite extinction time (or the zero exact controllability) and the finite speed of propagation, when the term \(|u| ^2u\) is replaced by \(|u| ^{m-1}u\), for some \(m\in(0,1)\). We extend to the delayed case some previous results in the literature of complex equations without any delay. For more information see https://ejde.math.txstate.edu/Volumes/2020/40/abstr.html

We study a parabolic–parabolic chemotactic PDE’s system which describes the evolution of a biological population “u” and a chemical substance “v” in a two-dimensional bounded domain with regular boundary. We consider a growth term of logistic type in the equation of “u” in the form $$u (1-u+f(x,t))$$, for a given bounded function “f” which tends to a periodic in time function independent of x when t goes to infinity. We study the global existence of solutions and its asymptotic behavior for a range of parameters and initial data.

The aim of this article is to fill part of the existing gap between the mathematical modeling of a green roof and its computational treatment, focusing on the mathematical analysis. We first introduce a two-dimensional mathematical model of the thermal behavior of an extensive green roof based on previous models and secondly we analyze such a system of partial differential equations. The model is based on an energy balance for buildings with vegetation cover and it is presented for general shapes of roofs. The model considers a vegetable layer and the substratum and the energy exchange between them. The unknowns of the problem are the temperature of each layer described by a coupled system of two partial differential equations of parabolic type. The equation modeling the evolution of the temperature of the substratum also considers the change of phase of water described by a maximal monotone graph. The main result of the article is the proof of the existence of solutions of the system which is given in detail by using a regularization of the maximal monotone graph. Appropriate estimates are obtained to pass to the limit in a weak formulation of the problem. The result goes one step further from modeling to validate future numerical results.

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain . We consider a growth term of logistic type in the equation of “u” in the form μu(1 − u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense urn:x-wiley:mma:media:mma5423:mma5423-math-0002 where f ∗ is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ∗, if the constant chemotactic sensitivity χ satisfies urn:x-wiley:mma:media:mma5423:mma5423-math-0003 we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.

Existence of global classical solutions with a certain periodic asymptotic behavior of a class of reaction diffusion systems with chemotactic terms is demonstrated. This class contains a system consisting of a parabolic equation with a logistic-like term describing the behavior of a biological species and an ordinary differential equation modeling the concentration of a chemical substance throughout a regular function h. The logistic-like term limits the growth of the biological species and it contains a carrying capacity with a time-periodic asymptotic behavior.