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Register a new userDecay rate and radial symmetry of the exponential elliptic equation
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Let $ngeq 3$, $alpha$, $betainmathbb{R}$, and let $v$ be a solution $Delta v+alpha e^v+beta xcdot abla e^v=0$ in $mathbb{R}^n$, which satisfies the conditions $lim_{Rtoinfty}frac{1}{log R}int_{1}^{R}rho^{1-n} (int_{B_{rho}}e^v,dx)drhoin (0,infty)$ and $|x|^2e^{v(x)}le A_1$ in $R^n$. We prove that $frac{v(x)}{log |x|}to -2$ as $|x|toinfty$ and $alpha>2beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.
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It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which applies to infinite cylindrical (channel) domains, and also generalizes previous work on radial domains as we impose no symmetry assumptions on the equation or its solutions.
In this paper we consider the initial value {problem $partial_{t} u- Delta u=f(u),$ $u(0)=u_0in exp,L^p(mathbb{R}^N),$} where $p>1$ and $f : mathbb{R}tomathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|sim mbox{e}^{|u|^q}$ as $|u|to infty$,} $|f(u)|sim |u|^{m}$ as $uto 0,$ $0<qleq pleq,m,;{N(m-1)over 2}geq p>1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.
We are interested in the Moore-Gibson-Thompson(MGT) equation with memory begin{equation} onumber tau u_{ttt}+ alpha u_{tt}+c^2A u+bA u_t -int_0^tg(t-s)A w(s)ds=0. end{equation} We first classify the memory into three types. Then we study how a memory term creates damping mechanism and how the memory causes energy decay.
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 dimensional 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.
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