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This article is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical…
In this paper, we investigate the location of the spreading front and convergence to traveling wave profile of solutions to the Fisher-KPP equation in the following two cases: (i) in unbounded domains with an expanding boundary; (ii) on the…
We present results for a finite variant of the one-dimensional Toom model with closed boundaries. We show that the steady state distribution is not of product form, but is nonetheless simple. In particular, we give explicit formulas for the…
We consider the road-field reaction-diffusion model introduced by Berestycki, Roquejoffre, and Rossi. By performing a "thin-front limit," we are able to deduce a Hamilton-Jacobi equation with a suitable effective Hamiltonian on the road…
The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary is considered. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behavior of the global solution.…
We investigate generalized Aubry-Andr\'{e} models featuring tunable quasidisordered potentials and a mobility edge that separates extended and localized states, with critical states for the mobility edge confirmed through finite-size…
We consider here a logistic equation, modeling processes of nonlocal character both in the diffusion and proliferation terms. More precisely, for populations that propagate according to a L\'evy process and can reach resources in a…
We investigate numerically the dynamics of crack propagation along a weak plane using a model consisting of fibers connecting a soft and a hard clamp. This bottom-up model has previously been shown to contain the competition of two crack…
We describe the accelerated propagation wave arising from a non-local reaction-diffusion equation. This equation originates from an ecological problem, where accelerated biological invasions have been documented. The analysis is based on…
In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a…
Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For active dispersion process in…
We have obtained the universal conductance distribution of two-dimensional disordered systems in the strongly localized limit. This distribution is directly related to the Tracy-Widom distribution, which has recently appeared in many…
We consider a class of Cahn-Hilliard equation with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on…
Experimental data on the propagation of wildfires show that its short-time spread has a double semi-elliptical shape. Our main goal is to show that this shape can be accurately approximated in polar coordinates by choosing suitable…
We are interested in the threshold phenomena for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such…
The analysis of contagion-diffusion processes in metapopulations is a powerful theoretical tool to study how mobility influences the spread of communicable diseases. Nevertheless, many metapopulation approaches use indistinguishable agents…
We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both…
We investigate the asymptotic speed of spread of the solutions of a non-autonomous Fisher-KPP equation with nonlocal diffusion, driven by a thin-tailed kernel. In this paper, we are concerned with both compactly supported and exponentially…
Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…