This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utiliz...This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.展开更多
In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the glob...In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the global existence of the solution to this system without any small condition on the initial data.展开更多
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
The authors prove the local existence and uniqueness of weak solution of a hyperbolic-parabolic system and establish the global existence of the weak solution for this system for the spatial dimension n = 1.
This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut= △u^l + u^p1v^q1 and vt = △v ^m + u^p2 v^q2 with homogeneous Dirichlet boundary conditions....This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut= △u^l + u^p1v^q1 and vt = △v ^m + u^p2 v^q2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1 - (l -p1)(m- q2), the initial data, and the domain Ω.展开更多
In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically...In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically to zero as (1 + t)-(1/7) when t approaches to infinity, provided the initial data are sufficiently small and regular.展开更多
We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical s...We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.展开更多
In this paper we consider the Elastic membrane equation:with memory term and nonlinear boundary damping: Under some appropriate assumptions on the relaxation function h and with certain initial data, the global exis...In this paper we consider the Elastic membrane equation:with memory term and nonlinear boundary damping: Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions :and a general decay for the energy are established using the multiplier technique. Also, 'we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.展开更多
By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result o...By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.展开更多
This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multipli...This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.展开更多
This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term --u/(1+t)λpu, where λ≥ 0 ...This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term --u/(1+t)λpu, where λ≥ 0 and μ 〉 0 are constants. We have showed that, for all λ ≥ 0 andμ 〉 0 the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial- boundary value problem in the half space R+^d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 〈 1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.展开更多
Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension. It is proved that the global existence holds when a conv...Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension. It is proved that the global existence holds when a convenient null condition is satisfied by nonlinearities.展开更多
We prove the existence of global solutions to the initial-boundary-value problem on the half space R+ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence ...We prove the existence of global solutions to the initial-boundary-value problem on the half space R+ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.展开更多
According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in...According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.展开更多
In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local exist...In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.展开更多
In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f(u)xx,x∈Ω,t〉0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t≥0, where Ω...In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f(u)xx,x∈Ω,t〉0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t≥0, where Ω = (0, 1). First, we obtain the existence of local Wkp solutions. Then, we prove that, if f(s) ∈ Ck+1(R) is nondecreasing, f(0) = 0 and |f(u)| ≤ C1|u|∫0uf(s)ds + C2, u0(x),u1(x) ∈ WkP(Ω) ∩ W01,P(Ω), k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u(x, t) ∈ W2,∞(0, T; WkP(Ω) ∩ W01,P(Ω)). Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.展开更多
In this paper,the Cauchy problem of the classical Keller-Segel chemotaxis model with initial data of large mass is considered.By the Green’s function method and scaling technique,we prove that if ku0k 1nL1 ku0k1−1n L...In this paper,the Cauchy problem of the classical Keller-Segel chemotaxis model with initial data of large mass is considered.By the Green’s function method and scaling technique,we prove that if ku0k 1nL1 ku0k1−1n L∞andκk∇v0kLp are less than some constant,then the problem always admits a unique global classical solution,even when the initial mass ku0kL1 is large.Moreover,the decay rates of the classical solution are also obtained.展开更多
Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an e...Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an extension of certain results for ordinary differential equations.展开更多
This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fuji...This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel'dovich-Kompaneetz-Barenblatt profile展开更多
A global existence theorem is established for an initial-boundary value prob- lem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonl...A global existence theorem is established for an initial-boundary value prob- lem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolic-parabolic sys- tem. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate a-priori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a different set of a priori estimates is generated which allows for taking the limit as the artificial viscosity parameter converges to zero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the initial-boundary value problem for the original, nonlinear, hyperbolic-parabolic system.展开更多
文摘This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.
文摘In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the global existence of the solution to this system without any small condition on the initial data.
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.
文摘The authors prove the local existence and uniqueness of weak solution of a hyperbolic-parabolic system and establish the global existence of the weak solution for this system for the spatial dimension n = 1.
基金This work is supported in part by NNSF of China (10571126)in part by Program for New Century Excellent Talents in University.
文摘This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut= △u^l + u^p1v^q1 and vt = △v ^m + u^p2 v^q2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1 - (l -p1)(m- q2), the initial data, and the domain Ω.
文摘In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically to zero as (1 + t)-(1/7) when t approaches to infinity, provided the initial data are sufficiently small and regular.
基金supported by the National Natural Science Foundation of China(11301172,11226170)China Postdoctoral Science Foundation funded project(2012M511640)Hunan Provincial Natural Science Foundation of China(13JJ4095)
文摘We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.
文摘In this paper we consider the Elastic membrane equation:with memory term and nonlinear boundary damping: Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions :and a general decay for the energy are established using the multiplier technique. Also, 'we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.
文摘By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.
基金Sponsored by the NNSF of China(11031003,11271066,11326158)a grant of Shanghai Education Commission(13ZZ048)Chinese Universities Scientific Fund(CUSF-DH-D-2013068)
文摘This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.
文摘This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term --u/(1+t)λpu, where λ≥ 0 and μ 〉 0 are constants. We have showed that, for all λ ≥ 0 andμ 〉 0 the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial- boundary value problem in the half space R+^d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 〈 1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.
文摘Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension. It is proved that the global existence holds when a convenient null condition is satisfied by nonlinearities.
文摘We prove the existence of global solutions to the initial-boundary-value problem on the half space R+ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.
基金supported by National Natural Science Foundation of China(11126336 and 11201324)New Teachers’Fund for Doctor Stations,Ministry of Education(20115134120001)+1 种基金Fok Ying Tuny Education Foundation(141114)Youth Fund of Sichuan Province(2013JQ0027)
文摘According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.
基金supported by a grant from the National High Technology Researchand and Development Program of China (863 Program) (2009AA044501)by NSFC (10776035+2 种基金10771085)by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Educationby the 985 program of Jilin University
文摘In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.
基金supported by National Natural Science Foundation of China(10871055,10926149)Natural Science Foundation of Heilongjiang Province (A2007-02+2 种基金A200810)Science and Technology Foundation of Education Office of Heilongjiang Province(11541276)Foundational Science Founda-tion of Harbin Engineering University
文摘In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f(u)xx,x∈Ω,t〉0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(0,t)=u(1,t)=0,t≥0, where Ω = (0, 1). First, we obtain the existence of local Wkp solutions. Then, we prove that, if f(s) ∈ Ck+1(R) is nondecreasing, f(0) = 0 and |f(u)| ≤ C1|u|∫0uf(s)ds + C2, u0(x),u1(x) ∈ WkP(Ω) ∩ W01,P(Ω), k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u(x, t) ∈ W2,∞(0, T; WkP(Ω) ∩ W01,P(Ω)). Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.
基金Supported by the Natural Science Foundation of Jiangsu Province(BK20160856)the National Natural Science Foundation of China(11801137)。
文摘In this paper,the Cauchy problem of the classical Keller-Segel chemotaxis model with initial data of large mass is considered.By the Green’s function method and scaling technique,we prove that if ku0k 1nL1 ku0k1−1n L∞andκk∇v0kLp are less than some constant,then the problem always admits a unique global classical solution,even when the initial mass ku0kL1 is large.Moreover,the decay rates of the classical solution are also obtained.
文摘Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an extension of certain results for ordinary differential equations.
基金supported in part by NSF of China (11071266)in part by NSF project of CQ CSTC (2010BB9218)partially supported by the Educational Science Foundation of Chongqing(KJ101303) China
文摘This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel'dovich-Kompaneetz-Barenblatt profile
文摘A global existence theorem is established for an initial-boundary value prob- lem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolic-parabolic sys- tem. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate a-priori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a different set of a priori estimates is generated which allows for taking the limit as the artificial viscosity parameter converges to zero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the initial-boundary value problem for the original, nonlinear, hyperbolic-parabolic system.