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 potentia...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.展开更多
In this paper,we consider a class of quasilinear equations involving a nonlinearity term having critical exponential growth.By using Mountain Pass Theorem,Ekeland's variational principle and inequalities of the ty...In this paper,we consider a class of quasilinear equations involving a nonlinearity term having critical exponential growth.By using Mountain Pass Theorem,Ekeland's variational principle and inequalities of the type Trudinger-Moser,we obtain the existence of at least two positive weak solutions.展开更多
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 p...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.展开更多
In this paper,(ⅰ)we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation:-Δ_(N)^(u)+V(x)|u|^(N-2)u=f(u)in R~N,N≥2 in...In this paper,(ⅰ)we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation:-Δ_(N)^(u)+V(x)|u|^(N-2)u=f(u)in R~N,N≥2 in three different cases allowing the potential V∈C(R^(N),R)to be periodic,radially symmetric,or asymptotically constant,whereΔ_(Nu):=div(|?u|~(N-2)?u)and f has critical exponential growth;(ⅱ)two new compactness lemmas in W^(1,N)(R^(N))for general nonlinear functionals are established,which generalize the ones obtained in the radially symmetric space W_(rad)^(1,N)(R^(N));(ⅲ)based on some key observations,we construct a special path allowing us to control the mountain-pass minimax level by a fine threshold under which the compactness can be restored for the critical case.In particular,some delicate analyses are developed to overcome non-standard difficulties due to both the quasilinear characteristic of the equation and the lack of compactness aroused by the critical exponential growth of f.Our results extend and improve the ones of Alves et al.(2012),Ibrahim et al.(2015)(N=2),and Masmoudi and Sani(2015)(N≥3)for the constant potential case,Alves and Figueiredo(2009)for the periodic potential case,Lam and Lu(2012)and Yang(2012)for the coercive potential case,and Chen et al.(Sci China Math,2021)for the degenerate potential case,which are totally new even for the simpler semilinear case of N=2.We believe that our approaches and strategies may be adapted and modified to attack more variational problems with critical exponential growth.展开更多
By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε...By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.展开更多
In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 rep...In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.展开更多
In this study,we investigate the well-posedness of exponential growth backward stochastic differcntial cquations(BSDEs)drivcn by a markcd point process(MPP)under unbounded terminal conditions.Our analysis utilizes a f...In this study,we investigate the well-posedness of exponential growth backward stochastic differcntial cquations(BSDEs)drivcn by a markcd point process(MPP)under unbounded terminal conditions.Our analysis utilizes a fixed-point argument,the O-method,and an approximation procedurc.Additionally,wc cstablish the solvability of mean-reflected exponential growth BSDEs driven by the MPP using the-method.展开更多
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 pos...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.展开更多
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 cu...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.展开更多
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 e...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.展开更多
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 op...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.展开更多
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 radi...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.展开更多
This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of norm...This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of normalized positive solutions for this equation via the Trudinger-Moser inequality and variational methods.Moreover,these solutions are also ground state solutions.Additionally,the article also characterized the asymptotic behavior of solutions.The results of this article expand the research of relevant literature.展开更多
This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem{-d i v(ω(x)|∇u(x)|^(N-2)∇u(x))=f(x,u)+εh(x),in B;u=0,on ∂B,where B is the unit ball i...This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem{-d i v(ω(x)|∇u(x)|^(N-2)∇u(x))=f(x,u)+εh(x),in B;u=0,on ∂B,where B is the unit ball in R^(N),the radial positive weight ω(x)is of logarithmic type function,the functional f(x,u)is continuous in B×R and has double exponential critical growth,which behaves like exp{e^(α|u|^(N/N-1))}as|u|→∞ for some α>0.Moreover,ϵ>0,and the radial function h belongs to the dual space of W_(0,rad)^(1,N)(B)h≠0.展开更多
We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions,dispersion,source,and variable-exponents.We discovered that the solution of the system is global and con...We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions,dispersion,source,and variable-exponents.We discovered that the solution of the system is global and constrained under the right assumptions about the relaxation functions and initial conditions.After that,it is demonstrated that the blow-up has negative initial energy.Subsequently,the growth of solutions is demonstrated with positive initial energy,and the general decay result in the absence of the source term is achieved by using an integral inequality due to Komornik.展开更多
In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> and f has critical exponential growth. By using a version of Trudinger...In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> 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.展开更多
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 ...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展开更多
We study the nonlinear process of second harmonic generation in photonic time-crystals,materials with refractive index that varies abruptly and periodically in time,and obtain the phase matching condition for this pro...We study the nonlinear process of second harmonic generation in photonic time-crystals,materials with refractive index that varies abruptly and periodically in time,and obtain the phase matching condition for this process.We find conditions for which the second harmonic generation is highly enhanced even in the absence of phase matching,governed by the exponential growth of the modes residing in the momentum gap of the photonic time crystal.Additionally,under these conditions,a cascade of higher-order harmonics is generated at growing exponential rates.The process is robust,with no requirement for phase-matching,the presence of a resonance or a threshold,drawing energy from the modulation.展开更多
With the rapid development of quantum devices across various platforms[1–4],reconstructing quantum many-body states from experimentally measured data posts a crucial challenge.Straightforward quantum state tomography...With the rapid development of quantum devices across various platforms[1–4],reconstructing quantum many-body states from experimentally measured data posts a crucial challenge.Straightforward quantum state tomography(QST)is only applicable for small systems[5],since the required classical computing resources,such as the number of measurements and the memory size,grow exponentially as the system size increases.展开更多
文摘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.
基金Supported by the Natural Science Foundation of Anhui Province(Grant No.1808085QA15)the Scientific Research Project of Anhui University of Finance and Economics(Grant No.ACKYC19050)the Natural Science Foundation of China(Grant No.11571093)。
文摘In this paper,we consider a class of quasilinear equations involving a nonlinearity term having critical exponential growth.By using Mountain Pass Theorem,Ekeland's variational principle and inequalities of the type Trudinger-Moser,we obtain the existence of at least two positive weak solutions.
基金Supported by the National Natural Science Foundation of China(Grant No.11801145)。
文摘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.
基金supported by National Natural Science Foundation of China(Grant Nos.11971485,12001542,12171486,and 12371181)the Project for Young Backbone Teachers of Hunan Province(Grant No.10900-150220002)+1 种基金Natural Science Foundation for Excellent Young Scholars of Hunan Province(Grant No.2023JJ20057)supported by the Grant“Nonlinear Differential Systems in Applied Sciences”of the Romanian Ministry of Research,Innovation,and Digitization(Grant No.PNRR-Ⅲ-C9-2022-I8/22)。
文摘In this paper,(ⅰ)we present unified approaches to studying the existence of ground state solutions and mountain-pass type solutions for the following quasilinear equation:-Δ_(N)^(u)+V(x)|u|^(N-2)u=f(u)in R~N,N≥2 in three different cases allowing the potential V∈C(R^(N),R)to be periodic,radially symmetric,or asymptotically constant,whereΔ_(Nu):=div(|?u|~(N-2)?u)and f has critical exponential growth;(ⅱ)two new compactness lemmas in W^(1,N)(R^(N))for general nonlinear functionals are established,which generalize the ones obtained in the radially symmetric space W_(rad)^(1,N)(R^(N));(ⅲ)based on some key observations,we construct a special path allowing us to control the mountain-pass minimax level by a fine threshold under which the compactness can be restored for the critical case.In particular,some delicate analyses are developed to overcome non-standard difficulties due to both the quasilinear characteristic of the equation and the lack of compactness aroused by the critical exponential growth of f.Our results extend and improve the ones of Alves et al.(2012),Ibrahim et al.(2015)(N=2),and Masmoudi and Sani(2015)(N≥3)for the constant potential case,Alves and Figueiredo(2009)for the periodic potential case,Lam and Lu(2012)and Yang(2012)for the coercive potential case,and Chen et al.(Sci China Math,2021)for the degenerate potential case,which are totally new even for the simpler semilinear case of N=2.We believe that our approaches and strategies may be adapted and modified to attack more variational problems with critical exponential growth.
基金supported by National Natural Science Foundation of China(No.12171152)。
文摘By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671403 and 11671236)Henan Provincial General Natural Science Foundation Project(Grant No.232300420113)。
文摘In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.
基金supported by NSFC(Grant No.12371473)by the Tianyuan Fund for Mlathematics of NSFC(Grant No.12326603)。
文摘In this study,we investigate the well-posedness of exponential growth backward stochastic differcntial cquations(BSDEs)drivcn by a markcd point process(MPP)under unbounded terminal conditions.Our analysis utilizes a fixed-point argument,the O-method,and an approximation procedurc.Additionally,wc cstablish the solvability of mean-reflected exponential growth BSDEs driven by the MPP using the-method.
文摘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.
基金This research is partially supported by a Natural Sciences and Engineering Research Council Canada discovery grant,and National Natural Science Foundation of China(No.11771075).
文摘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.
基金supported by National Natural Science Foundation of China(Grant Nos.11301305 and 11571207)supported by the State Scholarship Fund from China Scholarship Council(CSC)+2 种基金supported by National Natural Science Foundation of China(Grant No.11701559)the Fundamental Research Funds for the Central Universities(Grant No.2018QC054)supported by National Natural Science Foundation of China(Grant No.11571387)。
文摘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.
文摘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.
基金Supported by National Natural Science Foundation of China(Grant Nos.11971485 and 12001542)。
文摘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.
基金Supported by National Natural Science Foundation of China(11671403,11671236)Henan Provincial General Natural Science Foundation Project(232300420113)。
文摘This article studies a class of nonlinear Kirchhoff equations with exponential critical growth,trapping potential,and perturbation.Under appropriate assumptions about f and h,the article obtained the existence of normalized positive solutions for this equation via the Trudinger-Moser inequality and variational methods.Moreover,these solutions are also ground state solutions.Additionally,the article also characterized the asymptotic behavior of solutions.The results of this article expand the research of relevant literature.
基金supported by Natural Science Foundation of Chongqing,China(Grant Nos.CSTB2024N and SCQ-LZX0038).
文摘This paper is devoted to studying the existence and multiplicity of nontrivial solutions for the following boundary value problem{-d i v(ω(x)|∇u(x)|^(N-2)∇u(x))=f(x,u)+εh(x),in B;u=0,on ∂B,where B is the unit ball in R^(N),the radial positive weight ω(x)is of logarithmic type function,the functional f(x,u)is continuous in B×R and has double exponential critical growth,which behaves like exp{e^(α|u|^(N/N-1))}as|u|→∞ for some α>0.Moreover,ϵ>0,and the radial function h belongs to the dual space of W_(0,rad)^(1,N)(B)h≠0.
文摘We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions,dispersion,source,and variable-exponents.We discovered that the solution of the system is global and constrained under the right assumptions about the relaxation functions and initial conditions.After that,it is demonstrated that the blow-up has negative initial energy.Subsequently,the growth of solutions is demonstrated with positive initial energy,and the general decay result in the absence of the source term is achieved by using an integral inequality due to Komornik.
文摘In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R<sup>2</sup> 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.
基金partially supported by PROCAD/UFG/Un B and FAPDF(Grant No.PRONEX 193.000.580/2009)partially supported by NSFC(Grant Nos.11571317,11101374,11271331)ZJNSF(Grant No.Y15A010026)
文摘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
基金supported by the Israel Science Foundation through the MAPATS programby the US Air Force Office for Scientific Research,AFOSR.
文摘We study the nonlinear process of second harmonic generation in photonic time-crystals,materials with refractive index that varies abruptly and periodically in time,and obtain the phase matching condition for this process.We find conditions for which the second harmonic generation is highly enhanced even in the absence of phase matching,governed by the exponential growth of the modes residing in the momentum gap of the photonic time crystal.Additionally,under these conditions,a cascade of higher-order harmonics is generated at growing exponential rates.The process is robust,with no requirement for phase-matching,the presence of a resonance or a threshold,drawing energy from the modulation.
基金supported by the National Natural Science Foundation of China(11925404,92165209,92265210,92365301,T2225008,12075128,and 62173201)the Innovation Program for Quantum Science and Technology(2021ZD0300203,2021ZD0302203,and 2021ZD0300201)+1 种基金the National Key Research and Development Program of China(2017YFA0304303)the Tsinghua University Dushi Program,and the Shanghai Qi Zhi Institute Innovation Program(SQZ202318)。
文摘With the rapid development of quantum devices across various platforms[1–4],reconstructing quantum many-body states from experimentally measured data posts a crucial challenge.Straightforward quantum state tomography(QST)is only applicable for small systems[5],since the required classical computing resources,such as the number of measurements and the memory size,grow exponentially as the system size increases.
