Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal

The chemotaxis-Stokes system [Formula: see text] is considered subject to the boundary condition [Formula: see text] with [Formula: see text] and a given nonnegative function [Formula: see text]. In contrast to the well-studied case when the second requirement herein is replaced by a homogeneous Neumann boundary condition for [Formula: see text], the Dirichlet condition imposed here seems to destroy a natural energy-like property that has formed a core ingredient in the literature by providing comprehensive regularity features of the latter problem. This paper attempts to suitably cope with accordingly poor regularity information in order to nevertheless derive a statement on global existence within a generalized framework of solvability which involves appropriately mild requirements on regularity, but which maintains mass conservation in the first component as a key solution property.

Decay properties for the incompressible Navier-Stokes flows in a half space

In this article, we give a comprehensive characterization of $L^1$ -summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that $L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$ ; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$ -estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition.

A delayed reaction–diffusion viral infection model with nonlinear incidences and cell-to-cell transmission

In this paper, we propose a reaction–diffusion viral infection model with nonlinear incidences, cell-to-cell transmission, and a time delay. We impose the homogeneous Neumann boundary condition. For the case where the domain is bounded, we first study the well-posedness. Then we analyze the local stability of homogeneous steady states. We establish a threshold dynamics which is completely characterized by the basic reproduction number. For the case where the domain is the whole Euclidean space, we consider the existence of traveling wave solutions by using the cross-iteration method and Schauder’s fixed point theorem. Finally, we study how the speed of spread in space affects the spread of cells and viruses. We obtain the existence of the wave speed, which is dependent on the diffusion coefficient.

Dynamics of Spatial Epidemic Model with Infected Population Repulsion

In this paper, we are concerned with the spatial epidemic model with infected-taxis in which the susceptible individuals could avoid the infected ones. The spatial pattern for the resulted model is investigated under homogeneous Neumann boundary condition. We gain the condition for spatial pattern induced by diffusion term and infected-taxis term. Moreover, we obtain the condition for the occurrence of pattern formations induced by infected-taxis, in which the diffusion-driven Turing instability case is excluded. We give numerical examples to support the theoretical scheme.

Turing-Hopf Bifurcation in the Predator-prey Model with Cross-diffusion Considering Two Different Prey Behaviours Transition

In this paper, we study the Turing-Hopf bifurcation in the predator-prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analyzing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we investigate the dynamical classification near the Turing-Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

Spatiotemporal Dynamics and Pattern Formations of an Activator-Substrate Model with Double Saturation Terms

By using center manifold theory, Poincaré–Bendixson theorem, spatiotemporal spectrum and dispersion relation of linear operators, the spatiotemporal dynamics of an activator-substrate model with double saturation terms under the homogeneous Neumann boundary condition are considered in the present paper. It is surprising to find that the system can induce new dynamics, such as subcritical Hopf bifurcation and the coexistence of two limit cycles. Moreover, Turing instability in equilibrium mainly generates stripe patterns, while homogeneous periodic solutions mainly generate spot patterns or spot-stripe patterns, where the pattern formations are enormously consistent with the theoretical results. Interestingly, Turing instability can create equilibrium and periodic solution simultaneously in the subcritical Hopf bifurcation, which is the new finding of the diffusion-driven instability. In fact, those theoretical methods are also valid for finding the patterns of other models in one-dimensional space.

On a Steklov eigenvalue problem associated with the (p,q)-Laplacian

"Consider in a bounded domain \Omega \subset \mathbb{R}^N, N\ge 2, with smooth boundary \partial \Omega, the following eigenvalue problem (1) \begin{eqnarray*} &~&\mathcal{A} u:=-\Delta_p u-\Delta_q u=\lambda a(x) \mid u\mid ^{r-2}u\ \ \mbox{ in} ~ \Omega, \nonumber \\ &~&\big(\mid \nabla u\mid ^{p-2}+\mid \nabla u\mid ^{q-2}\big)\frac{\partial u}{\partial\nu}=\lambda b(x) \mid u\mid ^ {r-2}u ~ \mbox{ on} ~ \partial \Omega, \nonumber \end{eqnarray*} where 1<r<q<p<\infty or 1<q<p<r<\infty; r\in \Big(1, \frac{p(N-1)}{N-p}\Big) if p<N and r\in (1, \infty) if p\ge N; a\in L^{\infty}(\Omega),~ b\in L^{\infty}(\partial\Omega) are given nonnegative functions satisfying \[ \int_\Omega a~dx+\int_{\partial\Omega} b~d\sigma >0. \] Under these assumptions we prove that the set of all eigenvalues of the above problem is the interval [0, \infty). Our result complements those previously obtained by Abreu, J. and Madeira, G., [Generalized eigenvalues of the (p, 2)-Laplacian under a parametric boundary condition, Proc. Edinburgh Math. Soc., 63 (2020), No. 1, 287–303], Barbu, L. and Moroşanu, G., [Full description of the eigenvalue set of the (p,q)-Laplacian with a Steklov-like boundary condition, J. Differential Equations, in press], Barbu, L. and Moroşanu, G., [Eigenvalues of the negative (p,q)– Laplacian under a Steklov-like boundary condition, Complex Var. Elliptic Equations, 64 (2019), No. 4, 685–700], Fărcăşeanu, M., Mihăilescu, M. and Stancu-Dumitru, D., [On the set of eigen-values of some PDEs with homogeneous Neumann boundary condition, Nonlinear Anal. Theory Methods Appl., 116 (2015), 19–25], Mihăilescu, M., [An eigenvalue problem possesing a continuous family of eigenvalues plus an isolated eigenvale, Commun. Pure Appl. Anal., 10 (2011), 701–708], Mihăilescu, M. and Moroşanu, G., [Eigenvalues of -\triangle_p-\triangle_q under Neumann boundary condition, Canadian Math. Bull., 59 (2016), No. 3, 606–616]."

Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establish a priori estimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge (proportion of both the species) and harvesting of prey species in this contribution. Criteria for asymptotic stability (local and global) and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point. For partial differential equation (PDE), the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated. The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species. A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space, that is, spots, stripe-spot mixtures, labyrinthine, holes, stripe-hole mixtures and stripes replication. Finally, we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.