This paper deals with a quasilinear parabolic-parabolic-elliptic attraction-repulsion chemotaxis system. Boundedness, stabilization and blow-up in this system of the fully parabolic parabolic-elliptic-elliptic versions have already been proved. The purpose is to derive boundedness version.

We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal MongeKantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [6] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the...

where u and φ1, φ2 are, respectively, the complexand the real-valued functions of spatial coordinates x, y and the time t. The parameters σ, κ, γ,m1,m2, λ, n are real constants and σ is normalized as | σ |= 1. The GDS system has been derived to model 2 + 1 dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses [1]. The parametric relation (λ − 1)(m2 −...

This paper deals with the existence and multiplicity of solutions for a class Kirchhoff type elliptic systems involving nonlinearities Trudinger-Moser exponential growth. We first study following system: $$\begin{aligned} \left\{ \begin{array}{ll} -\big (a_1+b_1\Vert u\Vert ^{2(\theta _1-1)}\big )\Delta u= \lambda H_u(x,u,v) &{} \quad \text{ in }\ \ \Omega ,\\ (a_2+b_2\Vert v\Vert _2-1)}\big v=...

The paper describes the automatic motor vehicle video stream based classification system. The system determines vehicle type at payment collection plazas on toll roads. Classification is performed in accordance with a preconfigured set of rules which determine type by number of wheel axles, vehicle length, height over the first axle and full height. These characteristics are calculated using va...

In this paper, the authors study an optimal control problem for quasilinear elliptic PDEs with pointwise state constraints. Weak and strong optimality conditions of Pontryagin maximum principle type are derived. In proving these results, we penalized the state constraints and respectively use the Ekeland variational principle and an exact penalization method. In this paper, our aim is to prove ...

We study a system of elliptic equations from the Abelian BornInfeld system coupled with the Einstein equations under the boundary condition of the symmetric vacuum(nontopological type). When the total string number satisfies 1 ≤ N < 1 4πG , where G is the gravitational constant, we construct a family of solutions to the system. The qualitative properties of the solutions are quite different fro...

We describe a new algebraically completely integrable system, whose integral manifolds are co-elliptic subvarieties of Jacobian varieties. This is a multi-periodic extension of the Krichever-Treibich-Verdier system, which consists of elliptic solitons. The goal of this work is to generalize the theory of elliptic solitons, which was developed by A. Treibich and J.-L. Verdier based on earlier wo...