We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third ord...We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.展开更多
In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillat...In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.展开更多
Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.
By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigen...By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.展开更多
In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A suffic...In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.展开更多
Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically inter...Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physi- cally interesting cases are found by using the partial Lagrangian approach.展开更多
We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corr...We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.展开更多
Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale hig...Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem.展开更多
We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens...We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t 〈 T＊.Then, when T＊ ≤ t 〈 T＇, we get a new shock wave between two rarefactions, and then, when t 〉 T＇, another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t → ∞. The technique does not involve dimensional reduction or coordinate transformation.展开更多
In this paper, we prove the existence of the global smooth solution to the Cauchy problems for a class of diagonalizable high dimensional quasilinear hyperbolic systems consisted of n-equations.
The class of three-dimensional quasilinear hyperbolic systems is studied. The initial boundary value problem for this class of quasilinear hyperbolic systems is given. By constructing the energy’s integral, a priori ...The class of three-dimensional quasilinear hyperbolic systems is studied. The initial boundary value problem for this class of quasilinear hyperbolic systems is given. By constructing the energy’s integral, a priori estimate for the solution of the initial boundary value problem is obtained. Difference scheme is constructed and an a priori estimate for its solution is obtained. Numerical example exhibits the efficiency and accuracy of the method.展开更多
In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotationa...In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.展开更多
In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms:ur+σ(v)x+2α(t)u=0,vr-ux=0 Under the certain of hypothesis,we prove the globally existence theorems of the smooth...In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms:ur+σ(v)x+2α(t)u=0,vr-ux=0 Under the certain of hypothesis,we prove the globally existence theorems of the smooth solutions for is cauchy problem.展开更多
In this article, the author considers the Cauchy problem for quasilinear non-strict ly hyperbolic systems and obtain a blow-up result for the C1 solution to the Cauchy problem with weaker decaying initial data.
For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.
In this paper the authors prove the existence and uniqueness of global classical solutions to some kinds of typical boundary-value problems and typical free-boundary problems for quasilinear hyperbolic systems.
For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an appl...For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.展开更多
基金partially supported by the National Nature Science Foundation of China(12271114)the Guangxi Natural Science Foundation(2023JJD110009,2019JJG110003,2019AC20214)+2 种基金the Innovation Project of Guangxi Graduate Education(JGY2023061)the Key Laboratory of Mathematical Model and Application(Guangxi Normal University)the Education Department of Guangxi Zhuang Autonomous Region。
文摘We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.
基金Supported by Hunan Provincial NSF(05jj400008)of China under Grant.
文摘In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.
基金This work is supported in part by NNSF of China(10571126)and in part by Program for New Century Excellent Talents in University.
文摘Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.
文摘By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.
文摘In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.
文摘Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physi- cally interesting cases are found by using the partial Lagrangian approach.
文摘We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.
文摘Deterministic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by the sigma-convergence method that the sequence of solutions to a class of multi-scale highly oscillatory hyperbolic problems converges to the solution to a homogenized quasilinear hyperbolic problem.
基金supported by the NationalNatural Science Foundation of China(11371042,1471028,11601021)the Beijing Natural Science Foundation(1142001)
文摘We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t 〈 T＊.Then, when T＊ ≤ t 〈 T＇, we get a new shock wave between two rarefactions, and then, when t 〉 T＇, another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t → ∞. The technique does not involve dimensional reduction or coordinate transformation.
文摘In this paper, we prove the existence of the global smooth solution to the Cauchy problems for a class of diagonalizable high dimensional quasilinear hyperbolic systems consisted of n-equations.
文摘The class of three-dimensional quasilinear hyperbolic systems is studied. The initial boundary value problem for this class of quasilinear hyperbolic systems is given. By constructing the energy’s integral, a priori estimate for the solution of the initial boundary value problem is obtained. Difference scheme is constructed and an a priori estimate for its solution is obtained. Numerical example exhibits the efficiency and accuracy of the method.
文摘In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.
文摘In this paper,we discuss a class of the quasillinear hyperbolic equations with the inhomogeneous terms:ur+σ(v)x+2α(t)u=0,vr-ux=0 Under the certain of hypothesis,we prove the globally existence theorems of the smooth solutions for is cauchy problem.
文摘In this article, the author considers the Cauchy problem for quasilinear non-strict ly hyperbolic systems and obtain a blow-up result for the C1 solution to the Cauchy problem with weaker decaying initial data.
基金supported by the National Natural Science Foundation of China(Nos.12122104,11831011)Shanghai Science and Technology Programs(Nos.21ZR1406000,21JC1400600,19JC1420101)。
文摘For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.
文摘In this paper the authors prove the existence and uniqueness of global classical solutions to some kinds of typical boundary-value problems and typical free-boundary problems for quasilinear hyperbolic systems.
基金Project supported by the Special Funds forMajor State Basic Research Projects ofChina.
文摘For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems, this paper establishes the local exact boundary controllability with boundary controls only acting on one end. As an application, the authors show the local exact boundary controllability for a kind of nonlinear vibrating string problem.
