Related papers: Global convergence towards pushed travelling front…
The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind.…
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…
In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution…
Cannone \cite{Cannone} proved the global well-posedness of the incompressible Navier-Stokes equations for a class of highly oscillating data. In this paper, we prove the global well-posedness for the compressible Navier-Stokes equations in…
We give an integral variational characterization for the speed of fronts of the nonlinear diffusion equation $u_t = u_{xx} + f(u)$ with $f(0)=f(1)=0$, and $f>0$ in $(0,1)$, which permits, in principle, the calculation of the exact speed for…
We investigate the pushed-to-pulled transition for a minimal model for invasive fronts influence by ``aerotaxis,'' that is, when organisms follow oxygen gradients. We consider two singular reaction-advection-diffusion models for this. The…
We establish nonlinear stability of fronts that describe the creation of a periodic pattern through the invasion of an unstable state. Our results concern pushed fronts, that is, fronts whose propagation is driven by a localized mode at the…
The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small…
We consider an initial-boundary value problem for a fully nonlinear coupled parabolic system with nonlinear boundary conditions modelling hygro-thermal behavior of concrete at high temperatures. We prove a global existence of a weak…
We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a compact "obstacle". The model we study is motivated by applications in biology and…
We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations \begin{equation*} \partial_t u + u \cdot \nabla u = \Delta u - \nabla p + \zeta + \xi \;, \quad u (0, \cdot) = u_{0}(\cdot) \;, \quad…
The non-monotonic propagation of fronts is considered. When the speed function $F:\mathbb{R}^{n} \times [0,T]\rightarrow \mathbb{R}$ is prescribed, the non-linear advection equation $\phi_{t}+F|\nabla \phi|=0$ is a Hamilton-Jacobi equation…
A reaction-diffusion model which is called the field-road model was introduced by Berestycki, Roquejoffre and Rossi [9] to describe biological invasion with fast diffusion on a line. In this paper, we investigate this model in a…
We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed $c\in{I\!\!R}$, we seek a control with minimum cost, which achieves a…
We consider a class of parabolic variational inequalities with time dependent obstacle of the form $|{\boldsymbol u}(x,t)| \le p(x,t)$, where ${\boldsymbol u}$ is the velocity field of a fluid governed by the Navier--Stokes variational…
This work studies the following doubly degenerate parabolic-elliptic nutrient taxis system $$ \begin{cases} u_t = (uvu_x)_x -(u^2 vv_x)_x + uv, \\[1.5 ex] \hspace{0.2 cm}0 = v_{xx} - uv + f(x,t), \end{cases} $$ in a bounded interval $\Omega…
n this paper we show the existence of traveling waves $w: [0,+\infty) \times \mathbb{R}^2 \to \mathbb{R}^k$ ($k \geq 2$) for the parabolic Allen-Cahn system \begin{equation} \partial_t w - \Delta w = -\nabla_u V(w) \mbox{ in } [0,+\infty)…
We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant…
We study the existence of traveling wave solutions for a numerical counterpart of the KPP equation. We obtain the existence of monostable fronts for all super-critical speeds in the regime where the spatial step size is small. The key…
This article considers equations of Kolmogorov Petrovskii Piscunov type in one space dimension, with stochastic perturbation: \partial_t u = \left (\frac{\kappa}{2} u_{xx} + u(1-u) \right) dt + \epsilon u \partial_t \zeta where the…