In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The prop...In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.展开更多
On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear ...On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].展开更多
This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal(convolution)dispersals:{u_(t)=k*u-u+u(1-u-av),x∈R,t∈R,vt=d(k*v-v)+rv(1-v-bu),c∈R,t∈...This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal(convolution)dispersals:{u_(t)=k*u-u+u(1-u-av),x∈R,t∈R,vt=d(k*v-v)+rv(1-v-bu),c∈R,t∈R.(0.1)Here a≠1,b≠1,d,and r are positive constants.By studying the eigenvalue problem of(0.1)linearized at(ϕc(ξ),0),we construct a pair of super-and sub-solutions for(0.1),and then establish the existence of entire solutions originating from(ϕc(ξ),0)as t→−∞,whereϕc denotes the traveling wave solution of the nonlocal Fisher-KPP equation ut=k*u−u+u(1−u).Moreover,we give a detailed description on the long-time behavior of such entire solutions as t→∞.Compared to the known works on the Lotka-Volterra competition system with classical diffusions,this paper overcomes many difficulties due to the appearance of nonlocal dispersal operators.展开更多
In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where ...In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.展开更多
In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex d...In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial ...In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial in f of degree d（≤ n-2）, p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.展开更多
In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonline...In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.展开更多
In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general in...In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.展开更多
This paper mainly focuses on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function J may not be symmetric here. The asymmetry o...This paper mainly focuses on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function J may not be symmetric here. The asymmetry of J has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of f′(0) = 0, we can no longer use the common method which mainly depends on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, so we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.展开更多
In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous fun...In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.展开更多
In this paper,we mainly investigate entire solutions of the following two non-linear differential-difference equations f^(n)(z)+ωf^(n-1)(z)f′(z)+f^((k))(z+c)=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥5 and f^(n)(z)+ωf^(n-1)...In this paper,we mainly investigate entire solutions of the following two non-linear differential-difference equations f^(n)(z)+ωf^(n-1)(z)f′(z)+f^((k))(z+c)=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥5 and f^(n)(z)+ωf^(n-1)(z)f′(z)+q(z)f^((k))(z+c)e^(Q(z))=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥4,where k≥0 is an integer,c,ω,p_(1),p_(2),α_(1),α_(2)are non-zero constants,q(z)is a non-vanishing polynomial and Q(z)is a non-constant polynomial.Under some additional hypotheses,we analyze the existence and expressions of transcendental entire solutions of the above equations.展开更多
In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）&#...In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）×[0,∞）is a continuous tunctlon. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of ｜x｜^（p-N）/（p-1） for p〉N or log｜x｜ for N-p as ｜x｜→∞.展开更多
This paper is concerned with the existence of entire solutions of Lotka Volterra competition-diffusion model. Using the comparing argument and sub-super solutions method, we obtain the existence of entire solutions wh...This paper is concerned with the existence of entire solutions of Lotka Volterra competition-diffusion model. Using the comparing argument and sub-super solutions method, we obtain the existence of entire solutions which behave as two wave fronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables.展开更多
By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn...By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.展开更多
This paper is concerned with the existence of entire solutions of some reaction-diffusion systems. We first consider Belousov-Zhabotinskii reaction model. Then we study a general model. Using the comparing argument an...This paper is concerned with the existence of entire solutions of some reaction-diffusion systems. We first consider Belousov-Zhabotinskii reaction model. Then we study a general model. Using the comparing argument and sub-super-solutions method, we obtain the existence of entire solutions which behave as two wavefronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. At last, we give some examples to explain our results for the general models.展开更多
This paper deals with the solutions defined for all time of the degenerate Fisher equation. Some solutions are obtained by considering two traveling fronts with critical speed that come from both sides of the X-axis a...This paper deals with the solutions defined for all time of the degenerate Fisher equation. Some solutions are obtained by considering two traveling fronts with critical speed that come from both sides of the X-axis and mix. Unfortunately, the entire solutions which behave as two opposite wave fronts with non-critical speed approaching each other from both sides of the X-axis can not be obtained, because the essential difficulty originates from the algebraic decay rate of the fronts with non-critical speed.展开更多
In this paper, the existence of positive solutions for a class of quasilinear elliptic differential equation systems are established by Schauder-TychonofF fixed point theorem.
We consider the N-dimensional singular p-Laplace systems and obtain some sufficient conditions which guarantee that the system has infinitely many radial positive entire asymptotic solutions.
基金supported partly by the National Natural Science Foundation of China(11926201,12171050)the National Science Foundation of Guangdong Province(2018A030313508)。
文摘In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.
基金Supported by the National Natural Science Foundation of China(12261023,11861023)the Foundation of Science and Technology project of Guizhou Province of China([2018]5769-05)。
文摘On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].
基金supported by the NSF of China (12271226)the NSF of Gansu Province of China (21JR7RA537)+4 种基金the Fundamental Research Funds for the Central Universities (lzujbky-2022-sp07)supported by the Basic and Applied Basic Research Foundation of Guangdong Province (2023A1515011757)the National Natural Science Foundation of China (12271494)the Fundamental Research Funds for the Central Universities,China University of Geosciences (Wuhan) (G1323523061)supported by the NSF of China (12201434).
文摘This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal(convolution)dispersals:{u_(t)=k*u-u+u(1-u-av),x∈R,t∈R,vt=d(k*v-v)+rv(1-v-bu),c∈R,t∈R.(0.1)Here a≠1,b≠1,d,and r are positive constants.By studying the eigenvalue problem of(0.1)linearized at(ϕc(ξ),0),we construct a pair of super-and sub-solutions for(0.1),and then establish the existence of entire solutions originating from(ϕc(ξ),0)as t→−∞,whereϕc denotes the traveling wave solution of the nonlocal Fisher-KPP equation ut=k*u−u+u(1−u).Moreover,we give a detailed description on the long-time behavior of such entire solutions as t→∞.Compared to the known works on the Lotka-Volterra competition system with classical diffusions,this paper overcomes many difficulties due to the appearance of nonlocal dispersal operators.
文摘In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.
文摘In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金supported by the National Natural Science Foundation of China(11661044)
文摘In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial in f of degree d（≤ n-2）, p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.
基金Supported by the National Natural Science Foundation of China (11171184)the Scientific ResearchFoundation of CAUC,China (2011QD10X)
文摘In this paper, we shall utilize Nevanlinna value distribution theory and normality theory to study the solvability of a certain type of functional-differential equations. We also consider the solutions of some nonlinear differential equations.
文摘In this paper, we are concerned with positive entire solutions to elliptic equations of the form Δu+ f(x,u)= 0 x∈ RN N ≥ 3 where u →f(x,u) is not assumed to be regular near u = 0 and f(x,u) may be more general involving both singular and sublinear terms. Some sufficient conditions are given with the aid of the barrier method and ODE approach, which guarantee the existence of positive entire solutions that tend to any sufficiently large constants arbitrarily prescribed in advance.
基金supported by National Natural Science Foundation of China(Grant Nos.11671180 and 11371179)the Fundamental Research Funds for the Central Universities(Grant No.lzujbky2016-ct12)
文摘This paper mainly focuses on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function J may not be symmetric here. The asymmetry of J has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of f′(0) = 0, we can no longer use the common method which mainly depends on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, so we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.
文摘In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.
基金the National Natural Science Foundation of China(11971344)。
文摘In this paper,we mainly investigate entire solutions of the following two non-linear differential-difference equations f^(n)(z)+ωf^(n-1)(z)f′(z)+f^((k))(z+c)=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥5 and f^(n)(z)+ωf^(n-1)(z)f′(z)+q(z)f^((k))(z+c)e^(Q(z))=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥4,where k≥0 is an integer,c,ω,p_(1),p_(2),α_(1),α_(2)are non-zero constants,q(z)is a non-vanishing polynomial and Q(z)is a non-constant polynomial.Under some additional hypotheses,we analyze the existence and expressions of transcendental entire solutions of the above equations.
基金The work is supported by the National Natural Science Foundation of China (10271056)the Natural Science Foundation of Fujian Province (F00018).
文摘In this paper N-dimensional singular, p-Laplace equations of the following form △pu：=N↑∑↑i=1Di（｜Du｜^p-2Diu）=f（｜x｜,u,｜Du｜u^-β,x∈R^N（N≥3） are considered, where p≥N,β〉0,and f：[0,∞）×[0,∞）×[0,∞）is a continuous tunctlon. Some sufficient conditions are obtained for the existence of infinitely many radially positive entire solutions of the equation which are asymptotic to positive constant multiples of ｜x｜^（p-N）/（p-1） for p〉N or log｜x｜ for N-p as ｜x｜→∞.
文摘This paper is concerned with the existence of entire solutions of Lotka Volterra competition-diffusion model. Using the comparing argument and sub-super solutions method, we obtain the existence of entire solutions which behave as two wave fronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables.
基金supported partly by the National Natural Science Foundation of China(12171050,11871260)National Science Foundation of Guangdong Province(2018A030313508)。
文摘By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.
文摘This paper is concerned with the existence of entire solutions of some reaction-diffusion systems. We first consider Belousov-Zhabotinskii reaction model. Then we study a general model. Using the comparing argument and sub-super-solutions method, we obtain the existence of entire solutions which behave as two wavefronts coming from the both sides of x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. At last, we give some examples to explain our results for the general models.
基金Supported by the National Natural Science Foundation of China(Nos.11501016,11471221)Beijing Municipal Education Commission(KZ201310028030)Research Foundation for Youth Scholars of Beijing Technology and Business University(QNJJ2015-16)
文摘This paper deals with the solutions defined for all time of the degenerate Fisher equation. Some solutions are obtained by considering two traveling fronts with critical speed that come from both sides of the X-axis and mix. Unfortunately, the entire solutions which behave as two opposite wave fronts with non-critical speed approaching each other from both sides of the X-axis can not be obtained, because the essential difficulty originates from the algebraic decay rate of the fronts with non-critical speed.
基金Project Supported by the Foundations of Henan Education Committee and Henan Science Technology Committee (984050400).
文摘In this paper, the existence of positive solutions for a class of quasilinear elliptic differential equation systems are established by Schauder-TychonofF fixed point theorem.
文摘We consider the N-dimensional singular p-Laplace systems and obtain some sufficient conditions which guarantee that the system has infinitely many radial positive entire asymptotic solutions.
