SOLVING COUPLED PSEUDO-PARABOLIC EQUATION USING A MODIFIED DOUBLE LAPLACE DECOMPOSITION METHOD

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.

A Pseudo-parabolic Type Equation with Nonlinear Sources

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.

Initial Boundary Value Problems for a Class of Non-linear Pseudo-parabolic Equation

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.

ERROR ANALYSIS OF THE SECOND-ORDER SERENDIPITY VIRTUAL ELEMENT METHOD FOR SEMILINEAR PSEUDO-PARABOLIC EQUATIONS ON CURVED DOMAINS

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.

Fujita phenomena in nonlinear pseudo-parabolic system

This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.

Global Existence and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Pseudo-parabolic Equations

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.

Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

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.

Difference schemes for fully nonlinear pseudo-parabolic systems with two space dimensions

The first boundary value problem for the fully nonlinear pseudoparabolic systems of partial differential equations with two space dimensions by the finite difference method is studied. The existence and uniqueness of the discrete vector solutions for the difference systems are established by the fixed point technique. The stability and convergence of the discrete vector solutions of the difference schemes to the vector solutions of the original boundary problem of the fully nonlinear pseudo-parabolic system are obtained by way of a priori estimation. Here the unique smooth vector solution of the original problems for the fully nonlinear pseudo-parabolic system is assumed. Moreover, by the method used here, it can be proved that analogous results hold for fully nonlinear pseudo-parabolic system with three space dimensions, and improve the known results in the case of one space dimension.

Regularization of the Inverse Problem for Time Fractional Pseudo-parabolic Equation with Non-local in Time Conditions

This paper is devoted to identifying an unknown source for a time-fractional diffusion equation in a general bounded domain.First,we prove the problem is non-well posed and the stability of the source function.Second,by using the Modified Fractional Landweber method,we present regularization solutions and show the convergence rate between regularization solutions and sought solution are given under a priori and a posteriori choice rules of the regularization parameter,respectively.Finally,we present an illustrative numerical example to test the results of our theory.

Blow-Up Phenomena for Some Pseudo-Parabolic Equations with Nonlocal Term

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.

FINITE DIFFERENCE SCHEMES OF THE NONLINEAR PSEUDO-PARABOLIC SYSTEM

In this paper, we deal with the finite difference method for the initial boundary value problem of the nonlinear pseudo-parabolic system in the rectangular domain , where u(x,t)=(u1(x,t),u2(x,t)…,um(x .t)). F(x,t,u,um,…ux2M)are m-dimensional vector functions,and A(x,t,u,nx,…ux 2M)is an m×m positive definite matrix. The existence and uniqueness of solution for the finite difference system are proved by the fixed-point theory. Stability, convergence and error estimates are derived.

A Pseudo-Parabolic Type Equation with Weakly Nonlinear Sources

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.

具有奇异势和一般非线性的伪抛物方程解的局部存在性和爆破

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.

具有对数非线性项的伪抛物型p-Laplacian型方程的初边值问题

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.

GLOBAL SOLUTION OF THE INVERSE PROBLEM FOR ACLASS OF NONLINEAR EVOLUTION EQUATIONSOF DISPERSIVE TYPE

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.

One-dimensional Viscous Diffusion Equation of Higher Order with Gradient Dependent Potentials and Sources

In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ（s） and sources A（s）. We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ（s） and A（s） are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T 〉 0, the solution will never blow up during the period 0 〈 t 〈 T, so long as thc viscous coefficient is large enough.