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We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the global attractor, and also discuss the convergence of any bounded solutions to steady states. Finally, the existence of an exponential attractor is also demonstrated for sufficiently smooth kernels in the case of non-degenerate diffusion.
We analyze a phase-field system where the energy balance equation is linearly coupled with a nonlinear and nonlocal ODE for the order parameter $chi$. The latter equation is characterized by a space convolution term which models particle interaction
This paper is concerned with the globally exponential stability of traveling wave fronts for a class of population dynamics model with quiescent stage and delay. First, we establish the comparison principle of solutions for the population dynamics mo
In this paper we address the analytical investigation of a model for adhesive contact, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the un
The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation begin{equation}label{eq:discount 1}tag{HJ$_lambda$} lambda v(x)+H( x, Dv(x) )=0 , quad xin mathbb{R}^n.
The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex bal