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.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

A-high-order Accuraqcy Implicit Difference Scheme for Solving the Equation of Parabolic Type

In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(Δt 4+Δx 4) It can be easily solved by double sweeping method.

Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

Perfectly Matched Layer for an Elastic Parabolic Equation Model in Ocean Acoustics

The perfectly matched layer(PML) is an effective technique for truncating unbounded domains with minimal spurious reflections. A fluid parabolic equation(PE) model applying PML technique was previously used to analyze the sound propagation problem in a range-dependent waveguide(Lu and Zhu, 2007). However, Lu and Zhu only considered a standard fluid PE to demonstrate the capability of the PML and did not take improved one-way models into consideration. They applied a [1/1] Padé approximant to the parabolic equation. The higher-order PEs are more accurate than standard ones when a very large angle propagation is considered. As for range-dependent problems, the techniques to handle the vertical interface between adjacent regions are mainly energy conserving and single-scattering. In this paper, the PML technique is generalized to the higher order elastic PE, as is to the higher order fluid PE. The correction of energy conserving is used in range-dependent waveguides. Simulation is made in both acoustic cases and seismo-acoustic cases. Range-independent and range-dependent waveguides are both adopted to test the accuracy and efficiency of this method. The numerical results illustrate that a PML is much more effective than an artificial absorbing layer(ABL) both in acoustic and seismo-acoustic sound propagation modeling.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.

DISAPPEARANCE OF INTERFACES FOR DEGENERATE PARABOLIC EQUATIONS WITH VARIABLE DENSITY AND ABSORPTION

In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite time under some assumptions on the density functions.

Calculation of Wave Radiation Stress in Combination with Parabolic Mild Slope Equation

A new method for the calculation of wave radiation stress is proposed by linking the expressions for wave radiation stress with the variables in the parabolic mild slope equation. The governing equations are solved numerically by the finite difference method. Numerical results show that the new method is accurate enough, can be efficiently solved with little programming effort, and can be applied to the calculation of wave radiation stress for large coastal areas.

Superconvergence of Finite Element Approximations to Parabolic and Hyperbolic Integro-Differential Equations

The object of this paper is to investigate the superconvergence properties of finite element approximations to parabolic and hyperbolic integro-differential equations. The quasi projection technique introduced earlier by Douglas et al. is developed to derive the O(h2r) order knot superconvergence in the case of a single space variable, and to show the optimal order negative norm estimates in the case of several space variables.

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.

THE HOMOGENEOUS DIRICHLET PROBLEM FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC-HYPERBOLIC EQUATION WITH L^p INITIAL VALUE

The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.

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.

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.

Parabolic Approximation of the Weakly Nonlinear Mild Slope Equation with Bottom Friction

This paper presents a refined parabolic approximation model of the mild slope equation to simulate the combination of water wave refraction and diffraction in the large coastal region. The bottom friction and weakly nonlinear term are included in the model. The difference equation is established with the Crank-Nicolson scheme. The numerical test shows that some numerical prediction results will be inaccurate in complicated topography without considering weak nonlinearity; the bottom friction will make wave height damping and it can not be neglected for calculation of wave field in large areas.

PERIODIC OPTIMAL CONTROL PROBLEMS GOVERNED BY SEMILINEAR PARABOLIC EQUATIONS WITH IMPULSE CONTROL

This paper is concerned with periodic optimal control problems governed by semi- linear parabolic differential equations with impulse control. Pontryagin＇s maximum principle is derived. The proofs rely on a unique continuation estimate at one time for a linear parabolic equation.

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR QUASILINEAR PARABOLIC DIFFERENTIAL EQUATION

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.

Dependence of the Blow-up Time with Respect to Parameters for Semilinear Parabolic Equations with Nonlinear Memory

In this paper we discuss the bounds for the modulus of continuity of the blow-up time with respect to three parameters of λ, h, and p respectively for the initial boundary value problem of the semilinear parabolic equation.

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.

Second-order two-scale analysis and numerical algorithms for the hyperbolic–parabolic equations with rapidly oscillating coefficients

We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.

A Note on Homogenization of Parabolic Equation in Perforated Domains

We are concerned with a class of parabolic equations in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes.By using the periodic unfolding method in perforated domains, we obtain the homogenization results under the conditions slightly weaker than those in the corresponding case considered by Nandakumaran and Rajesh（Nandakumaran A K, Rajesh M. Homogenization of a parabolic equation in perforated domain with Neumann boundary condition. Proc. Indian Acad. Sci.（Math. Sci.）, 2002, 112（1）： 195–207）. Moreover,these results generalize those obtained by Donato and Nabil（Donato P, Nabil A. Homogenization and correctors for the heat equation in perforated domains. Ricerche di Matematica L. 2001, 50： 115–144）.