In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence...In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence,asymptotic behavior and blow-up of solutions with initial energy J(u_(0))≤d.Moreover,we estimate the upper bound of the blow-up time for J(u_(0))≤0.展开更多
The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to ...The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle,Strichartz estimates,Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.展开更多
We investigate the alpha helical protein structure characterized by fourth-order interspine coupling,focusing on a three-coupled fourth-order nonlinear Schr??dinger system.We introduce a generalized Darboux transforma...We investigate the alpha helical protein structure characterized by fourth-order interspine coupling,focusing on a three-coupled fourth-order nonlinear Schr??dinger system.We introduce a generalized Darboux transformation,departing from the classical Darboux transformation.Based on this,we construct the two-and three-degenerate soliton solutions and four-degenerate asymptotic soliton solutions.Based on the asymptotic analysis,we find that the amplitudes of interacting solitons are retained upon the interactions.Elastic interactions between two degenerate solitons exhibiting four curve-type asymptotic solitons are depicted.When the lattice parameterβchanges,the velocities of the two degenerate solitons also change.Elastic interaction among three degenerate solitons comprising four curve-type asymptotic solitons and two line-type solitons is presented.Interaction among one soliton and two degenerate solitons with different velocities is shown.Elastic interaction among four degenerate solitons comprising eight curve-type asymptotic solitons is also presented.Interaction among two two-degenerate solitons with two spectral parameters is shown.The relative distance between two asymptotic solitons exhibits logarithmic growth with|t|,where t represents the retarded time.Acceleration of soliton separation decays exponentially with relative distance,and eventually approaches zero.Phase shifts depend on t.展开更多
In this work, an efficient spectral method is proposed to solve the fourth-order eigenvalue problem in cylinder domain. Firstly, the key point of this method is to decompose the original model into a kind of decoupled...In this work, an efficient spectral method is proposed to solve the fourth-order eigenvalue problem in cylinder domain. Firstly, the key point of this method is to decompose the original model into a kind of decoupled two-dimensional eigenvalue problem by cylindrical coordinate transformation and Fourier series expansion, and deduce the crucial essential pole conditions. Secondly, we define a kind of weighted Sobolev spaces, and establish a suitable variational formula and its discrete form for each two-dimensional eigenvalue problem. Furthermore, we derive the equivalent operator formulas and obtain some prior error estimates of spectral theory of compact operators. More importantly, we further obtained error estimates for approximating eigenvalues and eigenfunctions by using two newly constructed projection operators. Finally,some numerical experiments are performed to validate our theoretical results and algorithm.展开更多
We present a Dancer‐type unilateral global bifurcation result for a class of fourth‐order two‐point boundary value problem x′″+kx″+lx=λh(t)x+g(t,x,λ),0〈t〈1,x(0)=x(1)=x′(0)=x′(1)=0.Under some natural hypoth...We present a Dancer‐type unilateral global bifurcation result for a class of fourth‐order two‐point boundary value problem x′″+kx″+lx=λh(t)x+g(t,x,λ),0〈t〈1,x(0)=x(1)=x′(0)=x′(1)=0.Under some natural hypotheses on the perturbation function g:(0,1)×R^(2)→R,we show that(λk,0)is a bifurcation point of the above problem.And there are two distinct unbounded continuas,C+k and Ck-,consisting of the bifurcation branch Ck from(λk,0),whereλk is the k‐th eigenvalue of the linear problem corresponding to the above problems.As an applica‐tion of the above result,the global behavior of the components of nodal solutions of the following problem x′″+kx″+lx=rh(t)f(x),0〈t〈1,x(0)=x(1)=x′(0)=x′(1)=0 is studied.We obtain the existence of multiple nodal solutions for the problem if f_(0)=∞,f_(∞)∈(0,∞),f_(0)=lim|s|→0 f(s)/s,f_(∞)=|lims|→∞f(s)/s.展开更多
This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some...This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some previous result is extended.展开更多
This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classificati...This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.展开更多
In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractio...In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.展开更多
We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evoluti...We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.展开更多
In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated ...In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.展开更多
In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal...In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.展开更多
In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian（Φp（u000））0（t）＋a（t）f（t, u（t？τ）, u0（t））=0, 0〈t〈1;u000 （0）=u00 （0）=0,...In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian（Φp（u000））0（t）＋a（t）f（t, u（t？τ）, u0（t））=0, 0〈t〈1;u000 （0）=u00 （0）=0, u0 （1）=αu0 （η）;u（t）=0, ？τ ≤t≤0. By using Schauder fixed-point theorem, some su？cient conditions are obtained which guar-antee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.展开更多
This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x...This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.展开更多
This paper is a sequel to a previous paper （Yang, Y. and Zhang, J. H. Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69（2）, 1364-1375 （2008）） in which the n...This paper is a sequel to a previous paper （Yang, Y. and Zhang, J. H. Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69（2）, 1364-1375 （2008）） in which the nontrivial solutions to the fourthorder boundary value problems were studied. In the current work with the same conditions near infinity but different near zero, the positive, negative, and sign-changing solutions are obtained by the critical point theory, retracting property, and invariant sets.展开更多
In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem：{u（4）（t） = f（t, u（t））-b, 0 〈 t 〈 1,u（0） = u′（0） = u′（1） = ...In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem：{u（4）（t） = f（t, u（t））-b, 0 〈 t 〈 1,u（0） = u′（0） = u′（1） = u（3）（1） = 0,where f ∈ C（[0,1] ×（-∞,＋∞）,（-∞, ＋∞））,b ≥ 0 is a constant. The novelty of this paper is that the boundary value problem is a new type and the method is a new fixed point theorem ofφ-（h,e）-concave operators.展开更多
This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p...This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.展开更多
This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation.A bi-Hamiltonian formulation is ...This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation.A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out,which exhibits the Liouville integrability of each model in the resulting hierarchy.Two specific examples,consisting of novel generalized combined nonlinear Schrodinger equations and modified Korteweg-de Vries equations,are given.展开更多
By mixed monotone method, we establish the existence and uniqueness of positive solutions for fourth-order nonlinear singular Sturm-Liouville problems. The theorems obtained are very general and complement previously ...By mixed monotone method, we establish the existence and uniqueness of positive solutions for fourth-order nonlinear singular Sturm-Liouville problems. The theorems obtained are very general and complement previously known results.展开更多
In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
In this paper we construct the finite-difference scheme for the singularly perturbed boundary value problem for the fourth-order elliptic differential equation on the basis of paper [1], and prove the uniform converge...In this paper we construct the finite-difference scheme for the singularly perturbed boundary value problem for the fourth-order elliptic differential equation on the basis of paper [1], and prove the uniform convergence of this scheme with respect to the small parameter Ε in the discrete energy norm. Finally, we give a numerical example.展开更多
基金Supported by NSFC(No.12101482)the Natural Science Foundation of Shaanxi Province,China(No.2018JQ1052)。
文摘In this paper,we consider the fourth-order parabolic equation with p(x)Laplacian and variable exponent source ut+∆^(2)u−div(|■u|^(p(x)−2■u))=|u|^(q(x))−1u.By applying potential well method,we obtain global existence,asymptotic behavior and blow-up of solutions with initial energy J(u_(0))≤d.Moreover,we estimate the upper bound of the blow-up time for J(u_(0))≤0.
基金Supported by National Natural Science Foundation of China(Grant No.11601122).
文摘The paper considers the initial value problem of inhomogeneous fourth-order Schr¨odinger equation with potential in energy space H^(2)(R^(d)).The global well-posedness is obtained in dimensions d≥5 resorting to contractive mapping principle,Strichartz estimates,Caffarelli-Kohn-Nirenberg-type inequality and the continuity method.
基金supported by the Natural Science Foundation of Shandong Province(Grant No.ZR2025QC30)。
文摘We investigate the alpha helical protein structure characterized by fourth-order interspine coupling,focusing on a three-coupled fourth-order nonlinear Schr??dinger system.We introduce a generalized Darboux transformation,departing from the classical Darboux transformation.Based on this,we construct the two-and three-degenerate soliton solutions and four-degenerate asymptotic soliton solutions.Based on the asymptotic analysis,we find that the amplitudes of interacting solitons are retained upon the interactions.Elastic interactions between two degenerate solitons exhibiting four curve-type asymptotic solitons are depicted.When the lattice parameterβchanges,the velocities of the two degenerate solitons also change.Elastic interaction among three degenerate solitons comprising four curve-type asymptotic solitons and two line-type solitons is presented.Interaction among one soliton and two degenerate solitons with different velocities is shown.Elastic interaction among four degenerate solitons comprising eight curve-type asymptotic solitons is also presented.Interaction among two two-degenerate solitons with two spectral parameters is shown.The relative distance between two asymptotic solitons exhibits logarithmic growth with|t|,where t represents the retarded time.Acceleration of soliton separation decays exponentially with relative distance,and eventually approaches zero.Phase shifts depend on t.
基金Supported by the National Natural Science Foundation of China(Grant No.12261017)the Scientific Research Foundation of Guizhou University of Finance and Economics(Grant No.2022ZCZX077)。
文摘In this work, an efficient spectral method is proposed to solve the fourth-order eigenvalue problem in cylinder domain. Firstly, the key point of this method is to decompose the original model into a kind of decoupled two-dimensional eigenvalue problem by cylindrical coordinate transformation and Fourier series expansion, and deduce the crucial essential pole conditions. Secondly, we define a kind of weighted Sobolev spaces, and establish a suitable variational formula and its discrete form for each two-dimensional eigenvalue problem. Furthermore, we derive the equivalent operator formulas and obtain some prior error estimates of spectral theory of compact operators. More importantly, we further obtained error estimates for approximating eigenvalues and eigenfunctions by using two newly constructed projection operators. Finally,some numerical experiments are performed to validate our theoretical results and algorithm.
基金Supported by the National Natural Science Foundation of China(11561038)the Gansu Provincial Natural Science Foundation(145RJZA087)
文摘We present a Dancer‐type unilateral global bifurcation result for a class of fourth‐order two‐point boundary value problem x′″+kx″+lx=λh(t)x+g(t,x,λ),0〈t〈1,x(0)=x(1)=x′(0)=x′(1)=0.Under some natural hypotheses on the perturbation function g:(0,1)×R^(2)→R,we show that(λk,0)is a bifurcation point of the above problem.And there are two distinct unbounded continuas,C+k and Ck-,consisting of the bifurcation branch Ck from(λk,0),whereλk is the k‐th eigenvalue of the linear problem corresponding to the above problems.As an applica‐tion of the above result,the global behavior of the components of nodal solutions of the following problem x′″+kx″+lx=rh(t)f(x),0〈t〈1,x(0)=x(1)=x′(0)=x′(1)=0 is studied.We obtain the existence of multiple nodal solutions for the problem if f_(0)=∞,f_(∞)∈(0,∞),f_(0)=lim|s|→0 f(s)/s,f_(∞)=|lims|→∞f(s)/s.
基金The 985 Program of Jilin Universitythe Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University
文摘This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some previous result is extended.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.SJ08A05
文摘This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries （GCSs）. Complete group classification results are presented, and some examples are given to show the main reduction procedure.
基金supported by the National Natural Science Foundation of China (No. 10671182)
文摘In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations.
文摘In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.
文摘In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.
基金Foundation item： Supported by the National Natural Science Foundation of China（10801001） Supported by the Natural Science Foundation of Anhui Province（1208085MA13, KJ2009A005Z）
文摘In this work, we investigate the following fourth-order delay differential equation of boundary value problem with p-Laplacian（Φp（u000））0（t）＋a（t）f（t, u（t？τ）, u0（t））=0, 0〈t〈1;u000 （0）=u00 （0）=0, u0 （1）=αu0 （η）;u（t）=0, ？τ ≤t≤0. By using Schauder fixed-point theorem, some su？cient conditions are obtained which guar-antee the fourth-order delay differential equation of boundary value problem with p-Laplacian has at least one positive solution. Some corresponding examples are presented to illustrate the application of our main results.
基金supported by the NSF of China(10571079,10671085)and the program of NCET
文摘This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.
基金Project supported by the National Natural Science Foundation of China (No. 10871096)the Foun-dation of Major Project of Science and Technology of Chinese Education Ministry (No. 205056)+2 种基金the Project of Graduate Education Innovation of Jiangsu Province (No. CX09B_284Z)the Foundation for Outstanding Doctoral Dissertation of Nanjing Normal Universitythe Foundation for Young Teachers of Jiangnan University (No. 2008LQN008)
文摘This paper is a sequel to a previous paper （Yang, Y. and Zhang, J. H. Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69（2）, 1364-1375 （2008）） in which the nontrivial solutions to the fourthorder boundary value problems were studied. In the current work with the same conditions near infinity but different near zero, the positive, negative, and sign-changing solutions are obtained by the critical point theory, retracting property, and invariant sets.
基金Supported by the Shanxi Scholarship Council of China(Grant No.2016-009)the Natural Science Foundation of Shanxi Province(Grant No.201701D121004)
文摘In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem：{u（4）（t） = f（t, u（t））-b, 0 〈 t 〈 1,u（0） = u′（0） = u′（1） = u（3）（1） = 0,where f ∈ C（[0,1] ×（-∞,＋∞）,（-∞, ＋∞））,b ≥ 0 is a constant. The novelty of this paper is that the boundary value problem is a new type and the method is a new fixed point theorem ofφ-（h,e）-concave operators.
文摘This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.
基金supported in part by NSFC under Grants 12271488, 11975145 and 11972291the Ministry of Science and Technology of China (G2021016032L and G2023016011L)the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17 KJB 110020)
文摘This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation.A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out,which exhibits the Liouville integrability of each model in the resulting hierarchy.Two specific examples,consisting of novel generalized combined nonlinear Schrodinger equations and modified Korteweg-de Vries equations,are given.
文摘By mixed monotone method, we establish the existence and uniqueness of positive solutions for fourth-order nonlinear singular Sturm-Liouville problems. The theorems obtained are very general and complement previously known results.
文摘In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.
文摘In this paper we construct the finite-difference scheme for the singularly perturbed boundary value problem for the fourth-order elliptic differential equation on the basis of paper [1], and prove the uniform convergence of this scheme with respect to the small parameter Ε in the discrete energy norm. Finally, we give a numerical example.
