Related papers: Scaling Variables and Stability of Hyperbolic Fron…
For convex co-compact hyperbolic quotients $X=\Gamma\backslash\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the…
We consider a kinetic model for a system of two species of particles interacting through a longrange repulsive potential and a reservoir at given temperature. The model is described by a set of two coupled Vlasov-Fokker-Plank equations. The…
We consider the one-dimensional Fisher-KPP equation with step-like initial data. Nolen, Roquejoffre, and Ryzhik showed that the solution $u$ converges at long time to a traveling wave $\phi$ at a position $\tilde \sigma(t) = 2t - (3/2)\log…
This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical…
Consider the elliptic operator given by $$ \mathscr{L}_{\epsilon}f= {b} \cdot \nabla f + \epsilon \Delta f $$ for some smooth vector field $ b\colon \mathbb R^d \to\mathbb R^d$ and a small parameter $\epsilon>0$. Consider the initial-valued…
The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N \tau v_t=\Delta…
The nonlinear Schroedinger equation possesses three distinct six-parameter families of complex-valued quasi-periodic travelling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their…
This paper deals with the stability of semi-wavefronts to the following delay non-local monostable equation: $\dot{v}(t,x) = \Delta v(t,x) - v(t,x) + \int_{\R^d}K(y)g(v(t-h,x-y))dy, x \in \R^d,\ t >0;$ where $h>0$ and $d\in\Z_+$. We give…
In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction…
In this paper we concern ourselves with an incompressible, viscous, isotropic, and periodic micropolar fluid. We find that in the absence of forcing and microtorquing there exists an infinite family of well-behaved solutions, which we call…
We prove that traveling waves in viscous compressible liquids are a generic phenomenon. The setting for our result is a horizontally infinite, finite depth layer of compressible, barotropic, viscous fluid, modeled by the free boundary…
The free boundary problem\[ \begin{cases} \partial_tu=\frac{1}{2}\Delta u+u,\quad &t>0, \, x>L_t,\\ u(t,x)=0,\quad &t>0,\, x\le L_t,\\ \int_{L_t}^{\infty}u(t,y)dy=1,\quad &t> 0,\\ u(t,x)dx \to u_0(dx)&\text{weakly as }t\to 0, \end{cases}\]…
We analyze ``pulled'' or ``linearly marginally stable'' fronts propagating into unstable states. While ``pushed'' fronts into meta- and unstable states relax exponentially, pulled fronts relax algebraically, and simultaneously the standard…
We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction…
We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp…
Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Yi et. al. (2021) has demonstrated the existence of such waves for two classes of regularisations, including…
We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically smal dissipation and convex potential. We prove that if the damping term behaves like K/t^\alpha for t large enough, k>0…
A modification of the parabolic Allen-Cahn equation, determined by the substitution of Fick's diffusion law with a relaxation relation of Cattaneo-Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two…
In this article, we study the 1 + 3 dimensional relativistic Euler equations on a pre-specified conformally flat expanding spacetime background with spatial slices that are diffeomorphic to $\mathbb{R}^3.$ We assume that the fluid verifies…
We study multiplicity of the supercritical traveling front solutions for scalar reaction-diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions…