اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدGrowth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity
184
0
0.0
(
0
)
تأليف
Kin Ming Hui
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove the growth rate of global solutions of the equation $u_t=Delta u-u^{- u}$ in $R^ntimes (0,infty)$, $u(x,0)=u_0>0$ in $R^n$, where $ u>0$ is a constant. More precisely for any $0<u_0in C(R^n)$ satisfying $A_1(1+|x|^2)^{alpha_1}le u_0le A_2(1+|x|^2)^{alpha_2}$ in $R^n$ for some constants $1/(1+ u)lealpha_1<1$, $alpha_2gealpha_1$ and $A_2ge A_1= (2alpha_1(1-3)(n+2alpha_1-2))^{-1/(1+ u)}$ where $0<3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{alpha_1}le u(x,t)le A_2(1+|x|^2+b_2t)^{alpha_2}$ in $R^ntimes (0,infty)$ where $b_1=2(n+2alpha_1-2)3$ and $b_2=2n$ if $0<alpha_2le 1$ and $b_2=2(n+2alpha_2-2)$ if $alpha_2>1$. We also find various conditions on the initial value for the solution to extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time.
قيم البحث
اقرأ أيضاً
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.
Let $0le u_0(x)in L^1(R^2)cap L^{infty}(R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|ge r_1$ and is monotone decreasing for all $|x|ge r_1$ for some constant $r_1>0$ and ${ess}inf_{2{B}_{r_1}(0)}u_0ge{ess} sup_{R^2setminus B_{r_2}(0)}u_0$ for so
me constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum cite{DP2}, cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=Deltalog u$ in $R^2times (0,T)$, $u(x,0)=u_0(x)$ in $R^2$, near its extinction time $T=int_{R^2}u_0dx/4pi$.
We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-
time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semi-group. We obtain existence and uniqueness of solution for nonnegative intial conditions, results on the invariant measures, and on the reflection measures.
We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual eigenfunction of the growth-fragmentation operator. This space is the la
rgest one in which we can expect convergence to the steady size distribution. Although this convergence is known to occur under fairly general conditions on the coefficients of the equation, we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded. First we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the Dyson-Phillips series. Then we extend it to general kernels by using the notion of quasi-compactness and the fact that it is a topological invariant.
Invasion phenomena for heterogeneous reaction-diffusion equations are contemporary and challenging questions in applied mathematics. In this paper we are interested in the question of spreading for a reaction-diffusion equation when the subdomain whe
re the reaction term is positive is shifting/contracting at a given speed $c$. This problem arises in particular in the modelling of the impact of climate change on population dynamics. By placing ourselves in the appropriate moving frame, this leads us to consider a reaction-diffusion-advection equation with a heterogeneous in space reaction term, in dimension $Ngeq1$. We investigate the behaviour of the solution $u$ depending on the value of the advection constant~$c$, which typically stands for the velocity of climate change. We find that, when the initial datum is compactly supported, there exists precisely three ranges for $c$ leading to drastically different situations. In the lower speed range the solution always spreads, while in the upper range it always vanishes. More surprisingly, we find that that both spreading and vanishing may occur in an intermediate speed range. The threshold between those two outcomes is always sharp, both with respect to $c$ and to the initial condition. We also briefly consider the case of an exponentially decreasing initial condition, where we relate the decreasing rate of the initial condition with the range of values of~$c$ such that spreading occurs.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
