Global existence and stability of three species predator-prey system with prey-taxis

In this paper, we study the following initial-boundary value problem of a three species predator-prey system with prey-taxis which describes the indirect prey interactions through a shared predator, i.e., \begin{align*} \begin{cases} u_t = d\Delta u+u(1-u)- \frac{a_1uw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ v_t = \eta d\Delta v+rv(1-v)- \frac{a_4vw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ w_t = \nabla\cdot(\nabla w-\chi_1 w\nabla u-\chi_2 w\nabla v) -\mu w+ \frac{a_5uw}{1+a_2u+a_3v}+\frac{a_6vw}{1+a_2u+a_3v}, & \mbox{in}\ \ \Omega, t>0, \ \ \label{II} \end{cases} \end{align*} under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^n (n \geqslant 1)$ with smooth boundary, where the parameters d, \eta, r, \mu, \chi_1, \chi_2, a_i > 0, i = 1, \ldots, 6. We first establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain under certain conditions. Moreover, we prove the global stability of the prey-only state and coexistence steady state by using Lyapunov functionals and LaSalle's invariance principle

Optimisation de forme pour quelques problemes de mecanique des fluides

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Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative

Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

summary:In this work, we study the existence and uniqueness of weak solutions of fourth-order degenerate parabolic equation with variable exponent using the difference and variation methods

Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

In this work, we study the existence and uniqueness of weak solu- tions of fourth-order degenerate parabolic equation with variable exponent using the di erence and variation methods

Optimal shape design of an electromagnet

Optimal shape design of an electromagnet is considered. The aim is to find the shape of the poles in such way that the magnetic field is constant and equal to a given value in the interpolar region. A sensitivity analysis is given. Two numerical examples illustrate the results.peerReviewe