Related papers: Boundedness and asymptotics of a reaction-diffusio…
In this paper we study the following one-dimensional reaction-diffusion problem $$ u_t+(-\Delta)^s u=f(x-c t, u) \;\:\textrm{ in } \mathbb{R}\times (0,+\infty), $$ where $s>\frac{1}{2}$, $c \in \mathbb{R}$ is a prescribed velocity, and $f$…
This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $\Omega$; with $u$ and $v$ being both positive…
Particle number fluctuations are considered within the van der Waals (VDW) equation, which contains both attractive (mean-field) and repulsive (eigenvolume) interactions. The VDW equation is used to calculate the scaled variance of particle…
We deal with heteroclinic planar fronts for parameter-dependent reaction-diffusion equations with bistable reaction and saturating diffusive term like $$ u_t=\epsilon \, \textrm{div}\, \left(\frac{\nabla u}{\sqrt{1+\vert \nabla u…
We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=\Delta u-\nabla \cdot(u\nabla v),& x\in\Omega, t>0,\\ v_t=\Delta v+wz,& x\in\Omega, t>0,\\ w_t=-wz,& x\in\Omega, t>0,\\ z_t=\Delta z-z+u,…
We examine the two elliptic systems given by [(G)_{\lambda,\gamma} \quad -\Delta u = \lambda f'(u) g(v), \quad -\Delta v = \gamma f(u) g'(v) \quad in $ \Omega$,] and [(H)_{\lambda,\gamma} \quad -\Delta u = \lambda f(u) g'(v), \quad -\Delta…
In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity…
In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ \hbox{ in }\Omega,…
Let $\Omega\subset\mathbb R^2$ be a bounded domain of class $C^{2+\alpha}$, $0<\alpha<1$. We show that if $u$ is the solution of $\Delta u = 4\exp(2u)$ which tends to $+\infty$ as $(x,y)\to\partial\Omega$, then the hyperbolic radius…
In this work, we study the initial boundary value problem for a non-strictly hyperbolic $2\times2$ system of equations in the quarter plane $x>0,t>0$ which is derived from Eulerian droplet model for air particle flow for velocity and volume…
This paper concerns the nonautonomous reaction-diffusion equation \[ u_t=u_{xx}+ug(t,x-ct,u), \quad t>0,x\in\mathbb{R}, \] where $c\in\mathbb{R}$ is the shifting speed, and the time periodic nonlinearity $ug(t,\xi,u)$ is asymptotically of…
We study a reaction-diffusion process that involves two species of atoms, immobile and diffusing. We assume that initially only immobile atoms, uniformly distributed throughout the entire space, are present. Diffusing atoms are injected at…
The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula…
We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is…
We analyze a reaction coefficient identification problem for the spectral fractional powers of a symmetric, coercive, linear, elliptic, second-order operator in a bounded domain $\Omega$. We realize fractional diffusion as the…
We consider the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 3$: \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot…
This paper deals with collisionless transport equations in bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonally invariant velocity measure $\bm{m}(\d v)$ with support…
We consider a class of stochastic reaction-diffusion equations also having a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction, it is possible to replace the SPDE by a…
In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& \Delta u - u +…
We consider a chemotaxis-fluid system involving nonlinear cell diffusion of porous medium type, signal consumption by cells, and rather general, possibly matrix-valued, chemotactic sensitivities. It is shown that if the corresponding…