GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0 〈 E（0） 〈 d. At last we show that the energy will grow up as an exponential function as time goes to infinity, provided the initial data is large enough or E（0） 〈 0.

Exponential Growth of Solutions for Nonlinear Coupled Viscoelastic Wave Equations with Degenerate Nonlocal Damping and Source Terms

This paper is concerned with a system of nonlinear viscoelastic wave equations with degenerate nonlocal damping and memory terms.We will prove that the energy associated to the system is unbounded.In fact,it will be proved that the energy will grow up as an exponential function as time goes to infinity,provided that the initial data are positive initial energy.

EXPONENTIAL GROWTH AND OSCILLATION OF NONLINEAR HOMOGENEOUS DIFFERENCEEQUATIONS

For a certain class of nonlinear homogeneous difference equations, it is shown that every nonoscillatory entire solution xn has exponential bounds on Z and that the oscillation is equivalent to the nonexistence of positive real characteristic roots. Explicit conditions for oscillation in terms of coefficients are also obtained.

Estimating epidemic exponential growth rate and basic reproduction number

The initial exponential growth rate of an epidemic is an important measure of the severeness of the epidemic,and is also closely related to the basic reproduction number.Estimating the growth rate from the epidemic curve can be a challenge,because of its decays with time.For fast epidemics,the estimation is subject to over-fitting due to the limited number of data points available,which also limits our choice of models for the epidemic curve.We discuss the estimation of the growth rate using maximum likelihood method and simple models.

The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth

In this article,we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth.We prove that on each energy level E(x,v)=k with k>c(L),where c(L)is Mane’s critical value,the EulerLagrange flow has positive topological entropy.This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.

Exponential Growth Solutions of Elliptic Equations

We show that for a class of second order divergence form elliptic equations on an infinite strip with the Dirichlet boundary condition,the space of fixed order exponential growth solutions is of finite dimension.An optimal estimation of the dimension is given.Examples also show that the finiteness property may not be true if one drops some of the conditions we make in our result.

On the Chern-Simons-Schrodinger Equation with Critical Exponential Growth

This paper is concerned with the following Chern-Simons-Schrodinger equation -Δu+V(|x|)u+(∫_(|x|)^(∞)h(s)/su^(2)(s)ds+h^(2)(|x|)/|x|^(2))u=a(|x|)f(u)in R^(2),where h(s)=∫_(0)^(s)l/2u^(2)(l)dl,V,a:R^(+)→R are radially symmetric potentials and the nonlinearity f:R→R is of subcritical or critical exponential growth in the sense of Trudinger-Moser.We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions.In particular,some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.

Existence of Nontrivial Weak Solutions to Quasi-Linear Elliptic Equations with Exponential Growth

In this paper, we study the existence of nontrivial weak solutions to the following quasi-linear elliptic equations

The Existence of Ground State Solutions for Schrödinger-Kirchhoff Equations Involving the Potential without a Positive Lower Bound

In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R2 and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.

Existence of Semiclassical States for a Quasilinear Schr?dinger Equation on R^N with Exponential Critical Growth

We study a quasilinear Schrodinger equation {-εN△Nu＋V（x）｜u｜N-2= Q（x）f（u） in R^N,0〈u∈W1,N（RN）,u（x）^｜x｜→∞0,where V, Q are two continuous real functions on R^N and c 〉 0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods. Keywords Exponential critical growth, semiclassical solutions, variational methods

The Ground State Solutions for Kirchhoff-Schrodinger Type Equations with Singular Exponential Nonlinearities in R^(N)

In this paper,the authors consider the following singular Kirchhoff-Schrodinger problem M(∫_(R^(N))|∇u|^(N)+V(x)|u|^(N)dx)(−Δ_(N)u+V(x)|u|^(N-2)u)=f(x,u)/|x|^(η)in R^(N),(P_(η))where 0<η<N,M is a Kirchhoff-type function and V(x)is a continuous function with positive lower bound,f(x,t)has a critical exponential growth behavior at infinity.Combining variational techniques with some estimates,they get the existence of ground state solution for(P_(η)).Moreover,they also get the same result without the A-R condition.

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.

Deviation Property of Periodic Measures in C^1 Non-uniformly Hyperbolic Systems with Limit Domination

In a C1 non-uniformly hyperbolic systems with limit domination, we consider the periodic measures that supported on the Pesin set and keep a distance at least 6 to a hyperbolic ergodic measure μ given before. And then, we bound from top the exponential growth rate of such periodic measures by the supremum of measure theoretic entropy on a closed set.

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in R^n

We establish sufficient conditions under which the quasilinear equation -div（｜△↓u｜^n-2△↓u）＋V（x）｜u｜^n-2u=f（x,u）/｜x｜^β＋εh（x） in R^n,has at least two nontrivial weak solutions in W^1,n（R^n） when ε 〉 0 is small enough, 0 〈β 〈 n, V is a continuous potential, f（x,u） behaves like exp{γ｜u｜^n/（n-1） } as ｜u｜→∞ for some γ 〉 0 and h 0 belongs to the dual space of W^1,n （Rn）.

A Multiplicity Result for Some Integro-Differential Biharmonic Equation in R^4

In this paper, we prove the existence of at least two nontrivial solutions for some biharmonic elliptic equation involving an integral term. The nonlinear term exhibits an exponential growth at infinity. Our method consists of a combination between variational tools and iterative technique.

The impact of policy timing on the spread of COVID-19

We model COVID-19 data for 89 nations and US states with a recently developed formalism that describes mathematically any pattern of growth with the minimum number of parameters.The results show that the disease has a typical duration of 18 days,with a significant increase in fatality when it lasts longer than about 4 months.Searching for correlations between“flattening of the curve”and preventive public policies,we find strong statistical evidence for the impact of the first implemented policy on decreasing the pandemic growth rate;a delay of one week in implementation nearly triples the size of the infected population,on average.Without any government action,the initial outburst still slows down after 36 days,possibly thanks to changes in public behavior in response to the pandemic toll.Stay-at-home(lockdown)was not the first policy of any sample member,and we could not find statistically meaningful evidence for its added impact,similar to a recent study that employed an entirely different approach.However,lockdown was mostly imposed only shortly before the exponential rise was arrested by other measures,too late for a meaningful impact.A third of the sample members that did implement lockdown imposed it only after the outburst had already started to slow down.The possibility remains that lockdown might have significantly shortened the initial exponential rise had it been employed as first resort rather than last.

COVID-19 prevalence estimation:Four most affected African countries

The world at large has been confronted with several disease outbreak which has posed and still posing a serious menace to public health globally.Recently,COVID-19 a new kind of coronavirus emerge from Wuhan city in China and was declared a pandemic by the World Health Organization.There has been a reported case of about 8622985 with global death of 457,355 as of 15.05 GMT,June 19,2020.South-Africa,Egypt,Nigeria and Ghana are the most affected African countries with this outbreak.Thus,there is a need to monitor and predict COVID-19 prevalence in this region for effective control and management.Different statistical tools and time series model such as the linear regression model and autoregressive integrated moving average(ARIMA)models have been applied for disease prevalence/incidence prediction in different diseases outbreak.However,in this study,we adopted the ARIMA model to forecast the trend of COVID-19 prevalence in the aforementioned African countries.The datasets examined in this analysis spanned from February 21,2020,to June 16,2020,and was extracted from theWorld Health Organization website.ARIMA models with minimum Akaike information criterion correction(AICc)and statistically significant parameters were selected as the best models.Accordingly,the ARIMA(0,2,3),ARIMA(0,1,1),ARIMA(3,1,0)and ARIMA(0,1,2)models were chosen as the best models for SA,Nigeria,and Ghana and Egypt,respectively.Forecasting was made based on the best models.It is noteworthy to claim that the ARIMA models are appropriate for predicting the prevalence of COVID-19.We noticed a form of exponential growth in the trend of this virus in Africa in the days to come.Thus,the government and health authorities should pay attention to the pattern of COVID-19 in Africa.Necessary plans and precautions should be put in place to curb this pandemic in Africa.

Estimating the quarantine failure rate for COVID-19

Quarantine is a crucial control measure in reducing imported COVID-19 cases and community transmissions.However,some quarantined COVID-19 patients may show symptoms after finishing quarantine due to a long median incubation period,potentially causing community transmissions.To assess the recommended 14-day quarantine policy,we develop a formula to estimate the quarantine failure rate from the incubation period distribution and the epidemic curve.We found that the quarantine failure rate increases with the exponential growth rate of the epidemic curve.We apply our formula to United States,Canada,and Hubei Province,China.Before the lockdown of Wuhan City,the quarantine failure rate in Hubei Province is about 4.1%.If the epidemic curve flattens or slowly decreases,the failure rate is less than 2.8%.The failure rate in US may be as high as 8.3%-11.5%due to a shorter 10-day quarantine period,while the failure rate in Canada may be between 2.5%and 3.9%.A 21-day quarantine period may reduce the failure rate to 0.3%-0.5%.

Existence of Solutions to Nonlinear Schr?dinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity

We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential critical growth.The approaches used here are based on a version of the Trudinger–Moser inequality and a minimax theorem.