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We prove existence and uniqueness of a reaction-diffusion equation whose diffusivity is a non-linear functional of the boundary temperature. We do this by studying systems of one-dimensional reflecting diffusions whose noise is a function…
We study the single-species diffusion-annihilation process with a time-dependent reaction rate, lambda(t)=lambda_0 t^-omega. Scaling arguments show that there is a critical value of the decay exponent omega_c(d) separating a…
The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate…
In this article, for a time-fractional diffusion-wave equation $\pppa u(x,t) = -Au(x,t)$, $0<t<T$ with fractional order $\alpha \in (1,2)$, we consider the backward problem in time: determine $u(\cdot,t)$, $0<t<T$ by $u(\cdot,T)$ and…
We consider a particle living in $\mathbb{R}_+$, whose velocity is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in $(0,\infty)$. When it hits the boundary $x=0$, the particle restarts with…
In this paper, we present an approach to characterising self-similar fast-reaction limits of systems with nonlinear diffusion. For appropriate initial data, in the fast-reaction limit as k tends to infinithy,spatial segregation results in…
The present theory considers the transient evolution of the temperature and fuel mass fraction distributions, and it can determine the critical heating power and minimum ignition energy for successful ignition. The flame initiation process…
A simple relation is established between the zero-$T$ penetration depth $\lambda (0)$ and the slope of $\lambda^{-2}(T)$ near $T_c$, similar to Helfand-Werthamer's relation for $H_{c2}(0)$ and the slope of $H_{c2}(T)$ at $T_c$ for the…
Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of…
We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the…
This paper is focused on the behavior near the extinction time of solutions of systems of ordinary differential equations with a sublinear dissipation term. Suppose the dissipation term is a product of a linear mapping $A$ and a positively…
In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-\Delta)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-\Delta)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the…
We introduce a method of intervals for the analysis of diffusion-limited annihilation, A+A -> 0, on the line. The method leads to manageable diffusion equations whose interpretation is intuitively clear. As an example, we treat the…
We consider positive radial decreasing blow-up solutions of the semilinear heat equation \begin{equation*} u_t-\Delta u=f(u):=e^{u}L(e^{u}),\quad x\in \Omega,\ t>0, \end{equation*} where $\Omega=\mathbb{R}^n$ or $\Omega=B_R$ and $L$ is a…
I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for…
This paper is devoted to the study of some qualitative and quantitative aspects of nonlinear propagation phenomena in diffusive media. More precisely, we consider the case a reaction-diffusion equation in a periodic medium with…
A boundary value problem for a stationary nonlinear dispersive equation of order $2l+1\;\; l\in \mathbb{N}$ with a convective term in the form $u^ku_x\;\; k\in \mathbb{N}$ was considered on an interval $(0,L)$. The existence, uniqueness and…
This paper is devoted to propagation phenomena for a reaction-diffusion-advection equation in a one-dimensional heterogeneous environment, where heterogeneity is reflected by the nonlinearity term -- being KPP type on $(-\infty, -L]$ and…
We hereby develop the theory of Turing instability for reaction-diffusion systems defined on complex networks assuming finite propagation. Extending to networked systems the framework introduced by Cattaneo in the 40's, we remove the…
In this paper, we propose a new mathematical model nonlinear reaction-diffusion PDE's describing the dynamics of propagation of cancer. Here the mixed problem for the proposed PDE's is investigated and by applying obtained results…