Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source

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.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SINGULAR PARABOLIC EQUATION

In this paper,we study the initial-boundary value problem for a class of singular parabolic equations.Under some conditions,we obtain the existence and asymptotic behavior of solutions to the problem by parabolic regularization method and the sub-super solutions method.As a byproduct,we prove the existence of solutions to some problems with gradient terms,which blow up on the boundary.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING

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.

EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/｜x｜s｜u｜q-2u,（x,t）∈Ω×（0,∞）,u（x,0）=f（x）,x∈Ω,u（x,t）=0,（x,t）∈Ω×（0,∞）,（P）}where-△pu ≡-div（｜▽u｜p-2▽u）,1〈p〈N,0〈s≤p,p≤q≤p＊（s） = N-s/N-pp,Ω is a bounded domain in RN such that 0∈Ω with a C1 boundaryΩ,f≥0 satisfying some convenient regularity assumptions.The analysis reveals that the existence of solutions for（P） depends on p,q,s in general,and on the relation between λ and the best constant in the Sobolev-Hardy inequality.

ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.

Asymptotic Behavior for A Class of Non-autonomous Nonclassical Parabolic Equations with Delay on Unbounded Domain

In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.

THE EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR EVOLUTION EQUATIONS AND APPLICATIONS TO P. D. E.

The main purpose of this paper is to discuss the existence and asymptotic behavior of solutions for [GRAPHICS] and for which the sufficient conditions of asymptotic behavior are obtained and the restriction for the existence is reduced.

A SINGULAR DIRICHLET PROBLEM FOR THE MONGE-AMPÈRE TYPE EQUATION

We consider the singular Dirichlet problem for the Monge-Ampère type equation{\rm det}\D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2},\u<0,\x\in\Omega,\u|_{\partial\Omega}=0,whereΩis a strictly convex and bounded smooth domain inℝn,q∈[0,n+1),g∈C∞(0,∞)is positive and strictly decreasing in(0,∞)with\lim\limits_{s\rightarrow 0^+}g(s)=\infty,and b∈C∞(Ω)is positive inΩ.We obtain the existence,nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g.Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

Nonlinear singularly perturbed problems of ultra parabolic equations

A class of nonlinear singularly perturbed problem of ultra parabolic equations are considered. Using the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.

Global existence and decay of solutions for the generalized bad Boussinesq equation

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.

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.

Uniform Blow-up Behavior for Degenerate and Singular Parab olic Equations

This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blowup exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases.

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.

NONLINEAR REACTION DIFFUSION PROBLEMS WITH ULTRA PARABOLIC LIMITING EQUATIONS

In this paper the nonlinear reaction diffusion problems with ultraparabolic equations are considered. By using comparison theorem, the existence, uniqueness and asymptotic behavior of solution for the problem are studied.

THE BLOW-UP PROPERTIES FOR A DEGENERATE SEMILINEAR PARABOL IC EQUATION WITH NONL OCAL SOURCE

This paper deals with the blow up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u t-(x αu x) x=∫ a 0f(u) d x in (0,a)×(0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow up in finite time of positive solutions are obtained. It is also proved that the blow up set is almost the whole domain. This differs from the local case. Furthermore, the blow up rate is precisely determined for the special case: f(u)=u p,p>1.

SOME RESULTS ON A DEGENERATE AND SINGULAR DIFFUSION EQUATION

In this article, a degenerate and singular diffusion problem is studied. The existence of solutions is established by parabolic regularization. Some properties of solutions, for instance, asymptotic behavior, are also discussed.

ASYMPTOTIC BEHAVIOR AND EXISTENCE OF POSITIVE SOLUTIONS FOR A NEUTRAL DIFFERENCE EQUATION

In this paper, we consider the neutral difference equation△(x n-cx n-m )+p nx n-k =0, n=N, N+1, N+2, …,where c and p n are real numbers, k, m are positive integers with m<k, and △ denotes the forward difference operator: △ u n=u n+1 -u n. By using the Krasnoselskii fixed theorem, we obtain some sufficient conditions under which such an equation has a bounded and eventually positive solution which tends to zero as n→∞.

Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Two Parameters and Boundary Perturbation

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with two parameters and boundary perturbation were considered.Under suitable conditions,the existence,uniqueness and asymptotic behavior of solutions for the initial boundary value problems were studied.An example was also given to illustrate our main results.

LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF DIFFERENCE SCHEME FOR A SEMILINEAR PARABOLIC EQUATION (Ⅱ)

In this paper we prove the solution of explicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the relevant nonlinear stationary problem as t→∞. For nonlinear parabolic problem, we obtain the long time asymptotic behavior of its discrete solution which is analogous to that of its continuous solution. For simplicity, we discuss one-dimensional problem.

LONG TIME ASYMPTOTIC BEHAVIOR OF SOLUTION OF DIFFERENCE SCHEME FOR A SEMILINEAR PARABOLIC EQUATION (I)

In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as t -&gt infinity. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.