This paper is concerned with the existence and uniqueness of nonnegative classical solutions to the initial-boundary value problems for the pseudo-parabolic equation with strongly nonlinear sources. Furthermore, we di...This paper is concerned with the existence and uniqueness of nonnegative classical solutions to the initial-boundary value problems for the pseudo-parabolic equation with strongly nonlinear sources. Furthermore, we discuss the asymptotic behavior of solutions as the viscosity coefficient k tends to zero.展开更多
In this paper, the modification of double Laplace decomposition method is pro- posed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples ar...In this paper, the modification of double Laplace decomposition method is pro- posed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.展开更多
In this paper,the existence,the uniqueness,the asymptotic behavior and the non-existence of the global generalized solutions of the initial boundary value problems for the non-linear pseudo-parabolic equation ut-αuxx...In this paper,the existence,the uniqueness,the asymptotic behavior and the non-existence of the global generalized solutions of the initial boundary value problems for the non-linear pseudo-parabolic equation ut-αuxx-βuxxt=F(u)-βF (u)xx are proved,where α,β 0 are constants,F(s) is a given function.展开更多
The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the exis...The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the existence and uniqueness are investigated for the weak solution of the nonhomogeneous initial-boundary value problem.The Nitschebased projection method is adopted to impose the boundary conditions in a weak way.The interpolation operator is used to deal with the nonlinear term.The Crank-Nicolson scheme is employed to discretize the temporal variable.There are two main features of the proposed scheme:(i)the internal degrees of freedom are avoided no matter what type of mesh is utilized,and(ii)the Jacobian is simple to calculate when Newton’s iteration method is applied to solve the fully discrete scheme.The error estimates are established for the discrete schemes and the theoretical results are illustrated through some numerical examples.展开更多
The inverse problem for a class of nonlinear evolution equations of dispersive type type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equippi...The inverse problem for a class of nonlinear evolution equations of dispersive type type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution it,as given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin.展开更多
In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreov...In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreover,we prove the blow-up of solution and estimate the upper bound of the blow-up time for J(u0)≤0.Finally,we prove the finite time blow-up and estimate the upper bound of blow-up time for J(u0)>0.展开更多
In this paper, we study the initial-boundary value problem for the semilinear pseudoparabolic equations ut —△xut —△xu =|u|^p-1u, where X =(X1, X2,..., Xm) is a system of real smooth vector fields which satisfy the...In this paper, we study the initial-boundary value problem for the semilinear pseudoparabolic equations ut —△xut —△xu =|u|^p-1u, where X =(X1, X2,..., Xm) is a system of real smooth vector fields which satisfy the Hormander's condition, and △x =∑j=1^m Xj^2 is a finitely degenera te elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.展开更多
We investigate the initial boundary value problem of the pseudo-parabolic equation ut-/△ut-/△u=φuu+|u|p-1u,whereφu is the Newtonian potential,which was studied by Zhu et al.(Appl.Math.Comput.,329(2018)38-51),and t...We investigate the initial boundary value problem of the pseudo-parabolic equation ut-/△ut-/△u=φuu+|u|p-1u,whereφu is the Newtonian potential,which was studied by Zhu et al.(Appl.Math.Comput.,329(2018)38-51),and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels.We in this note determine the upper and lower bounds for the blow-up time.While estimating the upper bound of blow-up time,we also find a sufficient condition of the solution blowingup in finite time at arbitrary initial energy level.Moreover,we also refine the upper bounds for the blow-up time under the negative initial energy.展开更多
We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term.We establish a lower bound for the blow-up time if blow-up does occur.Also both the upper bo...We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term.We establish a lower bound for the blow-up time if blow-up does occur.Also both the upper bound for T and blow up rate of the solution are given when J(u 0)<0.Moreover,we establish the blow up result for arbitrary initial energy and the upper bound for T.As a product,we refine the lifespan when J(u 0)<0.展开更多
In this paper, we prove the existence of nonnegative solutions to the initial boundary value problems for the pseudo-parabolic type equation with weakly nonlin- ear sources. Further, we discuss the asymptotic behavior...In this paper, we prove the existence of nonnegative solutions to the initial boundary value problems for the pseudo-parabolic type equation with weakly nonlin- ear sources. Further, we discuss the asymptotic behavior of the solutions as the viscous coefficient k tends to zero.展开更多
In this paper,we study the initial boundary value problem of pseudo-parabolic p-Laplacian type equation,which be use to model some important physical and biological phenomena.By using the potential well method,we obta...In this paper,we study the initial boundary value problem of pseudo-parabolic p-Laplacian type equation,which be use to model some important physical and biological phenomena.By using the potential well method,we obtain the global existence,asymptotic behavior and blow up results of weak solution with subcritical initial energy.Then we also extend these results to the critical initial energy.展开更多
基金The NSFC,CPSF,SRFDP and 973 Program(2010CB808002)
文摘This paper is concerned with the existence and uniqueness of nonnegative classical solutions to the initial-boundary value problems for the pseudo-parabolic equation with strongly nonlinear sources. Furthermore, we discuss the asymptotic behavior of solutions as the viscosity coefficient k tends to zero.
文摘In this paper, the modification of double Laplace decomposition method is pro- posed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.
基金Supported by the National Natural Science Foundation of China(10671182)
文摘In this paper,the existence,the uniqueness,the asymptotic behavior and the non-existence of the global generalized solutions of the initial boundary value problems for the non-linear pseudo-parabolic equation ut-αuxx-βuxxt=F(u)-βF (u)xx are proved,where α,β 0 are constants,F(s) is a given function.
基金supported by the National Natural Science Foundation of China(Grant No.12071100)by the Fundamental Research Funds for the Central Universities(Grant No.2022FRFK060019).
文摘The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the existence and uniqueness are investigated for the weak solution of the nonhomogeneous initial-boundary value problem.The Nitschebased projection method is adopted to impose the boundary conditions in a weak way.The interpolation operator is used to deal with the nonlinear term.The Crank-Nicolson scheme is employed to discretize the temporal variable.There are two main features of the proposed scheme:(i)the internal degrees of freedom are avoided no matter what type of mesh is utilized,and(ii)the Jacobian is simple to calculate when Newton’s iteration method is applied to solve the fully discrete scheme.The error estimates are established for the discrete schemes and the theoretical results are illustrated through some numerical examples.
文摘The inverse problem for a class of nonlinear evolution equations of dispersive type type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution it,as given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin.
基金Supported by National Natural Science Foundation of China(Grant No.11271141).
文摘In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreover,we prove the blow-up of solution and estimate the upper bound of the blow-up time for J(u0)≤0.Finally,we prove the finite time blow-up and estimate the upper bound of blow-up time for J(u0)>0.
基金Supported by National Natural Science Foundation of China(Grants Nos.11631011 and 11626251)
文摘In this paper, we study the initial-boundary value problem for the semilinear pseudoparabolic equations ut —△xut —△xu =|u|^p-1u, where X =(X1, X2,..., Xm) is a system of real smooth vector fields which satisfy the Hormander's condition, and △x =∑j=1^m Xj^2 is a finitely degenera te elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.
基金Supported by the Doctoral Scientific Research Starting Foundation of Guizhou Normal University of China,2018(No.GZNUD[2018]34).
文摘We investigate the initial boundary value problem of the pseudo-parabolic equation ut-/△ut-/△u=φuu+|u|p-1u,whereφu is the Newtonian potential,which was studied by Zhu et al.(Appl.Math.Comput.,329(2018)38-51),and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels.We in this note determine the upper and lower bounds for the blow-up time.While estimating the upper bound of blow-up time,we also find a sufficient condition of the solution blowingup in finite time at arbitrary initial energy level.Moreover,we also refine the upper bounds for the blow-up time under the negative initial energy.
基金NSFC(Nos.11811145,12071364)the Fundamental Research Funds for the Central Universities(WUT:2020IA003)Key Scientific Research Foundation of the Higher Education Institutions of Henan Province,China(No.19A110004)。
文摘We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term.We establish a lower bound for the blow-up time if blow-up does occur.Also both the upper bound for T and blow up rate of the solution are given when J(u 0)<0.Moreover,we establish the blow up result for arbitrary initial energy and the upper bound for T.As a product,we refine the lifespan when J(u 0)<0.
文摘In this paper, we prove the existence of nonnegative solutions to the initial boundary value problems for the pseudo-parabolic type equation with weakly nonlin- ear sources. Further, we discuss the asymptotic behavior of the solutions as the viscous coefficient k tends to zero.
基金Supported by the National Natural Science Foundation of China(Grant No.11271141).
文摘In this paper,we study the initial boundary value problem of pseudo-parabolic p-Laplacian type equation,which be use to model some important physical and biological phenomena.By using the potential well method,we obtain the global existence,asymptotic behavior and blow up results of weak solution with subcritical initial energy.Then we also extend these results to the critical initial energy.