We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow...We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.展开更多
This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depen...This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.展开更多
In this paper, we consider a strongly-coupled parabolic system with initial boundary values. Under the appropriate conditions, using Gagliard-Nirenberg inequality, Poincare inequality, Gronwall inequality and imbeddin...In this paper, we consider a strongly-coupled parabolic system with initial boundary values. Under the appropriate conditions, using Gagliard-Nirenberg inequality, Poincare inequality, Gronwall inequality and imbedding theorem, we obtain the global existence of solutions.展开更多
The initial boundary value problem for the fourth-order wave equation u_(tt)+△~2u+u=|u|^(p-1)u is considered.The existence and uniqueness of global weak solutions is obtained by using the Galerkin method and the conc...The initial boundary value problem for the fourth-order wave equation u_(tt)+△~2u+u=|u|^(p-1)u is considered.The existence and uniqueness of global weak solutions is obtained by using the Galerkin method and the concept of stable set due to Sattinger.展开更多
In this paper we study the decay estimate of global solutions to the initial-boundary value problem for double degenerate nonlinear parabolic equation by using a dif-ference inequality.
This paper is concerned with the existence of global solutions to the Cauchy problem of a hyperbolic-parabolic coupled system with large initial data. To this end, we first construct its local solutions by the standar...This paper is concerned with the existence of global solutions to the Cauchy problem of a hyperbolic-parabolic coupled system with large initial data. To this end, we first construct its local solutions by the standard iteration technique, then we deduce the basic energy estimate by constructing a convex entropy-entropy flux pair to this system. Moreover, the L∞-estimates and H^2-estimates of solutions are obtained through some delicate estimates. In our results, we do not ask the far fields of the initial data to be equal and the initial data can be arbitrarily large which generalize the result obtained in [7].展开更多
The paper is concerned with some chemotaxis model au/at=DV（u↓△ln（u/w））＋u（a-bu）,aw/at--f（u, w）. To study the behavior of the solution, some function transformations are intro- duced, and the main tool is sup...The paper is concerned with some chemotaxis model au/at=DV（u↓△ln（u/w））＋u（a-bu）,aw/at--f（u, w）. To study the behavior of the solution, some function transformations are intro- duced, and the main tool is sup-sub-solution method. The result shows that, whether the solution blows up in finite time depends on the parameters and the initial data. As the chemical substance w has linear growth, f（u,w）=βu-δw, where β〉0, δ〉0, and α＋δ〉0, wherein the solution exists globally. However, as w possesses ex- ponential growth, f（u,w）=（βu-δ）w, wherein both u and w share the same blowup point and time if the solution blows up in finite time.展开更多
Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are pro...Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.展开更多
In this paper,we apply the method given in the paper“Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors”(Mathematische Annalen,2022,382:1031–1046)to study the Cauchy problem...In this paper,we apply the method given in the paper“Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors”(Mathematische Annalen,2022,382:1031–1046)to study the Cauchy problem for a one dimensional inhomogeneous hydrodynamic model of two-carrier types for semiconductors with the velocity relaxation.展开更多
We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for shor...We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.展开更多
This note studies the global solutions of a semilineear reaction diffusion system which comes from an exothermic themical reaction.This is a complement of paper[1]and gives a positive answer to the question mentioned...This note studies the global solutions of a semilineear reaction diffusion system which comes from an exothermic themical reaction.This is a complement of paper[1]and gives a positive answer to the question mentioned by paper[2].展开更多
This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment ...This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.展开更多
This paper is concerned with an initial boundary value problem for the planar magnetohydrodynamic compressible flow with temperature dependent heat conductivity in a half-line.In particular,the transverse magnetic fie...This paper is concerned with an initial boundary value problem for the planar magnetohydrodynamic compressible flow with temperature dependent heat conductivity in a half-line.In particular,the transverse magnetic field is assumed to satisfy the Neumann boundary condition,which was first investigated by Kazhikhov in 1987.We establish the global existence of the unique strong solutions to the MHD equations without any smallness conditions on the initial data.More precisely,our result can be regarded as a natural generalization of Kazhikov’s result for applying the constant heat-conductivity in bounded domains to the degenerate case in unbounded domains.展开更多
In this paper,we consider the Cauchy problem of the d-dimensional damping incompressible magnetohydrodynamics system without dissipation.Precisely,this system includes a velocity damped term and a magnetic damped term...In this paper,we consider the Cauchy problem of the d-dimensional damping incompressible magnetohydrodynamics system without dissipation.Precisely,this system includes a velocity damped term and a magnetic damped term.We establish the existence and uniqueness of global solutions to this damped system in the critical Besov spaces by means of the Fourier frequency localization and Bony paraproduct decomposition.展开更多
This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space R^(3). For regular initial data with m0E0 is suitable small, where m0 and E0 represent the upper bound of initial density ...This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space R^(3). For regular initial data with m0E0 is suitable small, where m0 and E0 represent the upper bound of initial density and initial energy, we prove that if ρ0 ∈ Lγ ∩ H3 with γ ∈ (1, 6), then the problem possesses a unique global classical solution on R^(3) × [0, T] with any T ∈ (0, ∞). It’s worth noting that both the vacuum states and possible random largeness of initial energy are allowed.展开更多
In this paper,we study the higher-order semilinear parabolic system{ut+(-△)^mu=a|v|^p-1v,(t,x)∈R^1+×R^N,vt+(-△)^mv=b|u|^q-1u,(t,x)∈R^1+×R^N,u(0,x)=φ(x),v(0,x)=ψ(x),x∈R^N,where m,p,q〉1,a,b∈R.We prove...In this paper,we study the higher-order semilinear parabolic system{ut+(-△)^mu=a|v|^p-1v,(t,x)∈R^1+×R^N,vt+(-△)^mv=b|u|^q-1u,(t,x)∈R^1+×R^N,u(0,x)=φ(x),v(0,x)=ψ(x),x∈R^N,where m,p,q〉1,a,b∈R.We prove that the global existence of mild solutions for small initial data with respect to certain norms.Some of these solutions are proved to be asymptotically self-similar.展开更多
In this paper,the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries(nonlocal mKdV for short)equation under the initial data u0∈H^(3)(R)∩H^...In this paper,the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries(nonlocal mKdV for short)equation under the initial data u0∈H^(3)(R)∩H^(1,1)(R)with the L^(1)(R)small-norm assumption.A Lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem.The map from initial potential to reflection coefficient is obtained in direct scattering transform.The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator.The bijective relation naturally yields the existence of global solutions in a Sobolev space H^(3)(R)∩H^(1,1)(R)to the Cauchy problem.展开更多
The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied.The existence of global solutions for this problem is proved by using potential well ...The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied.The existence of global solutions for this problem is proved by using potential well method,and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function.Meanwhile,blow-up of solutions in the unstable set is also obtained.展开更多
In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.
This paper considers the Cauchy problem of pseudo-parabolic equation with inhomogeneous terms u_(t)=△u+k△u_(t)+w(x)u^(P)(x,t).In[1],Li et al.gave the critical Fujita exponent,second critical exponent and the life sp...This paper considers the Cauchy problem of pseudo-parabolic equation with inhomogeneous terms u_(t)=△u+k△u_(t)+w(x)u^(P)(x,t).In[1],Li et al.gave the critical Fujita exponent,second critical exponent and the life span for blow-up solutions under w(x)=|x|^(σ)with>0.We further generalize the weight function w(x)~|x|^(σ)for-2<σ<0,and discuss the global and non-global solutions to obtain the critical Fujita exponent.展开更多
基金supported by the National Natural Science Foundation of China (11101102)Ph.D. Programs Foundation of Ministry of Education of China (20102304120022)+3 种基金the Support Plan for the Young College Academic Backbone of Heilongjiang Province (1252G020)the Natural Science Foundation of Heilongjiang Province (A201014)Science and Technology Research Project of Department of Education of Heilongjiang Province (12521401)Foundational Science Foundation of Harbin Engineering University and Fundamental Research Funds for the Central Universities (HEUCF20131101)
文摘We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.
基金supported by the Key Project of the NSFC(12131010)the NSFC(11771155,12271032)+1 种基金the NSF of Guangdong Province(2021A1515010249,2021A1515010303)supported by the NSFC(11971179,12371205)。
文摘This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.
文摘In this paper, we consider a strongly-coupled parabolic system with initial boundary values. Under the appropriate conditions, using Gagliard-Nirenberg inequality, Poincare inequality, Gronwall inequality and imbedding theorem, we obtain the global existence of solutions.
基金the National Natural Science Foundation of China(10301026)the Research Foundation of Chengdu University of Information Technology(CRF200702)
文摘The initial boundary value problem for the fourth-order wave equation u_(tt)+△~2u+u=|u|^(p-1)u is considered.The existence and uniqueness of global weak solutions is obtained by using the Galerkin method and the concept of stable set due to Sattinger.
基金Supported by the NNSF of China(10441002)Supported by NNSF of Henan Province(200510466011)
文摘In this paper we study the decay estimate of global solutions to the initial-boundary value problem for double degenerate nonlinear parabolic equation by using a dif-ference inequality.
基金Huijiang Zhao was supported by the National Natural Science Foundation of China (10871151)Changjiang Zhu was supported by the National Natural Science Foundation of China (10625105 and 10431060)the Program for New Century Excellent Talentsin University (NCET-04-0745)
文摘This paper is concerned with the existence of global solutions to the Cauchy problem of a hyperbolic-parabolic coupled system with large initial data. To this end, we first construct its local solutions by the standard iteration technique, then we deduce the basic energy estimate by constructing a convex entropy-entropy flux pair to this system. Moreover, the L∞-estimates and H^2-estimates of solutions are obtained through some delicate estimates. In our results, we do not ask the far fields of the initial data to be equal and the initial data can be arbitrarily large which generalize the result obtained in [7].
基金Supported by the National Natural Science Foundation of China(10471108)
文摘The paper is concerned with some chemotaxis model au/at=DV（u↓△ln（u/w））＋u（a-bu）,aw/at--f（u, w）. To study the behavior of the solution, some function transformations are intro- duced, and the main tool is sup-sub-solution method. The result shows that, whether the solution blows up in finite time depends on the parameters and the initial data. As the chemical substance w has linear growth, f（u,w）=βu-δw, where β〉0, δ〉0, and α＋δ〉0, wherein the solution exists globally. However, as w possesses ex- ponential growth, f（u,w）=（βu-δ）w, wherein both u and w share the same blowup point and time if the solution blows up in finite time.
基金Project supported by the National Natural Science Foundation of China (Nos. 10371073 and 10572156) the Natural Science Foundation of Henan Province of China (No.0611050500)
文摘Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.
基金supported by Zhejiang Province NSFC(LY20A010023 and LY22A010015)the NSFC(12071106)of China+1 种基金supported by the Natural Science Foundation of Jiangsu Province(BK20211293)the“Qing-Lan Engineering”Foundation of Jiangsu Higher Education Institutions。
文摘In this paper,we apply the method given in the paper“Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors”(Mathematische Annalen,2022,382:1031–1046)to study the Cauchy problem for a one dimensional inhomogeneous hydrodynamic model of two-carrier types for semiconductors with the velocity relaxation.
基金Zhai was partially supported by the Guangdong Provincial Natural Science Foundation (2022A1515011977)the Science and Technology Program of Shenzhen(20200806104726001)+1 种基金Zhong was partially supported by the NNSF of China (11901474, 12071359)the Exceptional Young Talents Project of Chongqing Talent (cstc2021ycjh-bgzxm0153)。
文摘We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.
文摘This note studies the global solutions of a semilineear reaction diffusion system which comes from an exothermic themical reaction.This is a complement of paper[1]and gives a positive answer to the question mentioned by paper[2].
文摘This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.
基金supported by the National Natural Science Foundation of China(12401279,12371219)the Academic and Technical Leaders Training Plan of Jiangxi Province(20212BCJ23027).
文摘This paper is concerned with an initial boundary value problem for the planar magnetohydrodynamic compressible flow with temperature dependent heat conductivity in a half-line.In particular,the transverse magnetic field is assumed to satisfy the Neumann boundary condition,which was first investigated by Kazhikhov in 1987.We establish the global existence of the unique strong solutions to the MHD equations without any smallness conditions on the initial data.More precisely,our result can be regarded as a natural generalization of Kazhikov’s result for applying the constant heat-conductivity in bounded domains to the degenerate case in unbounded domains.
基金supported partially by the National Natural Science Foundation of China (No. 12371230)the Natural Science Foundation of Guangdong Province (No.2019B151502041)+4 种基金supported partially by the Natural Science Foundation of Guangdong Province (No.2016A030313390)the SCAU Fund for High-level University Buildingsupported partially by the R&D project of Pazhou Lab (Huangpu)(No.2023K0601)the National Key Research and Development Program of China (No.2020YFA0712500)the Shenzhen Science and Technology Program (No.CJGJZD20210408091403008)
文摘In this paper,we consider the Cauchy problem of the d-dimensional damping incompressible magnetohydrodynamics system without dissipation.Precisely,this system includes a velocity damped term and a magnetic damped term.We establish the existence and uniqueness of global solutions to this damped system in the critical Besov spaces by means of the Fourier frequency localization and Bony paraproduct decomposition.
基金supported by the Natural Science Foundation of Shandong Province of China(ZR2024MA033ZR2021QA049).
文摘This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space R^(3). For regular initial data with m0E0 is suitable small, where m0 and E0 represent the upper bound of initial density and initial energy, we prove that if ρ0 ∈ Lγ ∩ H3 with γ ∈ (1, 6), then the problem possesses a unique global classical solution on R^(3) × [0, T] with any T ∈ (0, ∞). It’s worth noting that both the vacuum states and possible random largeness of initial energy are allowed.
基金This work was supported by the National Natural Science Foundation of China 10701024 and the Natural Science Foundation of Tianjin of China(08JYBJC12100).
文摘In this paper,we study the higher-order semilinear parabolic system{ut+(-△)^mu=a|v|^p-1v,(t,x)∈R^1+×R^N,vt+(-△)^mv=b|u|^q-1u,(t,x)∈R^1+×R^N,u(0,x)=φ(x),v(0,x)=ψ(x),x∈R^N,where m,p,q〉1,a,b∈R.We prove that the global existence of mild solutions for small initial data with respect to certain norms.Some of these solutions are proved to be asymptotically self-similar.
基金supported by the National Natural Science Foundation of China(No.12271104)。
文摘In this paper,the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries(nonlocal mKdV for short)equation under the initial data u0∈H^(3)(R)∩H^(1,1)(R)with the L^(1)(R)small-norm assumption.A Lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem.The map from initial potential to reflection coefficient is obtained in direct scattering transform.The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator.The bijective relation naturally yields the existence of global solutions in a Sobolev space H^(3)(R)∩H^(1,1)(R)to the Cauchy problem.
基金This research was supported by the Natural Science Foundation of Zhejiang Province(Grant No.LY17A010009).
文摘The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied.The existence of global solutions for this problem is proved by using potential well method,and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function.Meanwhile,blow-up of solutions in the unstable set is also obtained.
文摘In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.
基金Supported by the National Natural Science Foundation of China(Grant No.11501438).
文摘This paper considers the Cauchy problem of pseudo-parabolic equation with inhomogeneous terms u_(t)=△u+k△u_(t)+w(x)u^(P)(x,t).In[1],Li et al.gave the critical Fujita exponent,second critical exponent and the life span for blow-up solutions under w(x)=|x|^(σ)with>0.We further generalize the weight function w(x)~|x|^(σ)for-2<σ<0,and discuss the global and non-global solutions to obtain the critical Fujita exponent.
