We study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and reveal some fundamental differences from its one dimensional version considered in cite{cdjfa} recently. Technically, this high dimensiona
l problem is much more difficult to treat since it involves two kernel functions which arise from the original kernel function $J(|x|)$ in rather implicit ways. By introducing new techniques, we are able to determine the long-time dynamics of the model, including firstly finding the threshold condition on the kernel function that governs the onset of accelerated spreading, and the determination of the spreading speed when it is finite. Moreover, for two important classes of kernel functions, sharp estimates of the spreading profile are obtained. More precisely, for kernel functions with compact support, we show that logarithmic shifting occurs from the finite wave speed propagation, which is strikingly different from the one dimension case; for kernel functions $J(|x|)$ behaving like $|x|^{-beta}$ for $xinR^N$ near infinity, we obtain the rate of accelerated spreading when $betain (N, N+1]$, which is the exact range of $beta$ where accelerated spreading is possible. These sharp estimates are obtained by constructing subtle upper and lower solutions, based on careful analysis of the involved kernel functions.
The extent and the morphology of ice forming in a differentially heated cavity filled with water is studied by means of experiments and numerical simulations. We show that the main mechanism responsible for the ice shaping is the existence of a cold
upward convective current in the system. Such a current is ascribed to the peculiar equation of state of water, i.e., the non-monotonous dependence of density with temperature. The precise form of the ice front depends on several factors, first the temperature difference across the cell which drives the convection, second the wall inclination with respect to the vertical, both of which are here explored. We propose a boundary-layer model and a buoyancy-intensity model which account for the main features of the ice morphology.
We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in
some of the equations in the system. The problem is monostable in nature, resembling the well known Fisher-KPP equation. Such a system covers various models arising from mathematical biology, with the Fisher-KPP equation as the simplest special case, where a spreading-vanishing dichotomy is known to govern the long time dynamical behaviour. The question of spreading speed is widely open for such systems except for the scalar case. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and traveling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave, and obtain sharp estimates of the semi-wave profile and the spreading speed. For kernel functions that behave like $|x|^{-gamma}$ near infinity, we are able to obtain better estimates of the spreading speed for both the finite speed case, and the infinite speed case, which appear to be the first for this kind of free boundary problems, even for the special Fisher-KPP equation.
We study a family of reaction-diffusion equations that present a doubly nonlinear character given by a combination of the $p$-Laplacian and the porous medium operators. We consider the so-called slow diffusion regime, corresponding to a degenerate be
haviour at the level 0, ormalcolor in which nonnegative solutions with compactly supported initial data have a compact support for any later time. For some results we will also require $pge2$ to avoid the possibility of a singular behaviour away from 0. Problems in this family have a unique (up to translations) travelling wave with a finite front. When the initial datum is bounded, radially symmetric and compactly supported, we will prove that solutions converging to 1 (which exist, as we show, for all the reaction terms under consideration for wide classes of initial data) do so by approaching a translation of this unique traveling wave in the radial direction, but with a logarithmic correction in the position of the front when the dimension is bigger than one. As a corollary we obtain the asymptotic location of the free boundary and level sets in the non-radial case up to an error term of size $O(1)$. In dimension one we extend our results to cover the case of non-symmetric initial data, as well as the case of bounded initial data with supporting sets unbounded in one direction of the real line. A main technical tool of independent interest is an estimate for the flux. Most of our results are new even for the special cases of the porous medium equation and the $p$-Laplacian evolution equation.
We consider the nonlinear Stefan problem $$ left { begin{array} {ll} -d Delta u=a u-b u^2 ;; & mbox{for } x in Omega (t), ; t>0, u=0 mbox{ and } u_t=mu| abla_x u |^2 ;;&mbox{for } x in partialOmega (t), ; t>0, u(0,x)=u_0 (x) ;; & mbox{for } x
in Omega_0, end{array}right. $$ where $Omega(0)=Omega_0$ is an unbounded smooth domain in $mathbb R^N$, $u_0>0$ in $Omega_0$ and $u_0$ vanishes on $partialOmega_0$. When $Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood by cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $Omega_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $Omega_0$.
We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the non
local diffusion problem with free boundaries considered in [4,8]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form $J_epsilon(x)=frac 1epsilon J(frac xepsilon)$ for small $epsilon>0$, where $J(x)$ has compact support. We also give an estimate of the error term of the approximation by some positive power of $epsilon$.
In Cao, Du, Li and Li [8], a nonlocal diffusion model with free boundaries extending the local diffusion model of Du and Lin [12] was introduced and studied. For Fisher-KPP type nonlinearities, its long-time dynamical behaviour is shown to follow a s
preading-vanishing dichotomy. However, when spreading happens, the question of spreading speed was left open in [8]. In this paper we obtain a rather complete answer to this question. We find a condition on the kernel function such that spreading grows linearly in time exactly when this condition holds, which is achieved by completely solving the associated semi-wave problem that determines this linear speed; when the kernel function violates this condition, we show that accelerating spreading happens.
We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but
here the population dispersal is described by nonlocal diffusion instead of local diffusion. We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in cite{CDLL}, and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several extra difficulties, which are overcome by the use of some new techniques.
In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth func
tion in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.
In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The k
ey observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.
