THE UNCONDITIONAL CONVERGENT DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR QUASILINEAR PARABOLIC SYSTEMS WITH TWO DIMENSIONS

In the present work we are going to solve the boundary value problem for the quasilinear parabolic systems of partial differential equations with two space dimensions by the finite difference method with intrinsic parallelism. Some fundamental behaviors of general finite difference schemes with intrinsic parallelism for the mentioned problems are studied. By the method of a priori estimation of the discrete solutions of the nonlinear difference systems, and the interpolation formulas of the various norms of the discrete functions and the fixed-point technique in finite dimensional Euclidean space, the existence of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the convergence of the discrete vector solutions of these difference schemes to the unique generalized solution of the original quasilinear parabolic problem is proved.

A Quasilinear Parabolic System with Nonlocal Sources and Weighted Nonlocal Boundary Conditions

In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In addition, the blow-up rate estimate of non-global solutions for a class of weight functions is also obtained, which is found to be independent of nonlinear diffusion parameters m and n.

THE FIRST BOUNDARY VALUE PROBLEM FOR STRONGLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS IN ANISOTROPIC SOBOLEV SPACES

This article concerns the existence of weak solutions of the first boundary value problem for a kind of strongly degenerate quasilinear parabolic equation in the anisotropic Sobolev Space. With the theory of anisotropic Sobolev spaces an existence result is proved.

Existence and Nonexistence of the Global Solution on the Quasilinear Parabolic Equation

The paper studies the existence, the exponential decay and the nonexistence of global solution for a class of quasilinear parabolic equations.

Existence and Nonexistence of Global Solution for Quasilinear Parabolic Equation

This paper deals with the existence and nonexistence of global positive solutions of the following quasilinear parabolic equations:u t=1m△u m-u n, x∈Ω,t>0 1m·u mv=u p,x∈Ω,t>0 u(x,0)=u 0(x)>0,x∈Ω-where Ω∈R N is a bounded domain with smooth boundary Ω,m,n,p are positive constants, γ is the outward normal vector. The necessary and sufficient conditions for the global existence of solutions are obtained.

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.

FINITE SPEED OF PROPAGATION OF SOLUTIONS FOR SOME QUASILINEAR HIGHER ORDER PARABOLIC EQUATIONS WITH DOUBLY STRONG DEGENERATION

Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.

On the Solvability of Degenerate Quasilinear Parabolic Equations of Second Order

In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.

On Doubly Degenerate Quasilinear Parabolic Equations of Higher Order

We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order,which can degenerate,on a part of the boundary,on a segment in the interior of the domain and in time.

SOURCE－TYPE SOLUTIONS OF A QUASILINEAR DEGENERATE PARABOLIC EQUATION WITH ABSORPTION

The existence and nonexistence of non-trivial solutions for the Cauchy problem of the formare studied.

Existence of Solutions for Degenerate Quasilinear Parabolic Equationsof Higher Order

In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.

Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ＇（.） is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].

A NEWTON MULTIGRID METHOD FOR QUASILINEAR PARABOLIC EQUATIONS

A combination of the classical Newton Method and the multigrid method, i.e., a Newton multigrid method is given for solving quasilinear parabolic equations discretized by finite elements. The convergence of the algorithm is obtained for only one step Newton iteration per level. The asymptotically computational cost for quasilinear parabolic problems is O（NNk） similar to multigrid method for linear parabolic problems.

On Double Degenerate Quasilinear Parabolic Variational Inequalities

We deal with the existence of weak solutions of double degenerate quasilinear parabolic inequalities with a Signorini-Dirichlet-Neumann type mixed boundary condition, which may degenerate in certain subset of the boundary or on a segment in the interior of the domain and in time. The main tools in our study are the maximM monotone property of the derivative operator with zero-initial valued conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.

BLOW-UP OF SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS

This paper investigates the qualitative properties of solutions to certain quasilinear parabolic equations. Under appropriate conditions, we obtain that the solution either exists globally or blows up in finite time by making use of the energy method and subsolution techniques. We find out that the behavior of solution heavily depends on the sign and the growth rate of the nonlinear reaction term and the nonlinear flux through boundary at infinity.

The Existence of Interfaces for General Porous Medium Equations with Strong Degeneracy

The aim of this paper is to discuss the Cauchy problem for quasilinear degenerate parabolic equations of the formwhere φ∈C1(R1) is a strictly monotonically increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ'(-) is permitted to have zero measure. In particular, the existence of interfaces of solutions is obtained.

Classification of certain qualitative properties of solutions for the quasilinear parabolic equations

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.

ON THE DIFFUSION PHENOMENON OF QUASILINEAR HYPERBOLIC WAVES

The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlinear parabolic equation v_t-div(a(v)v)=0,in the sense that the norm||u(.,t)-v(.,t)||_(L∞(R^n))of the difference u-v decays faster than that of either u or v.This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves,first observed by Hsiao,L.and Liu Taiping(see[1,2]).

Properties of positive solutions to a nonlinear parabolic problem

This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vázquez and the comparison principle, we deduce that the blow-up occurs only on the boundary ?Ω. In addition, for a bounded Lipschitz domain Ω, we establish the blow-up rate estimates for the positive solution to this problem with a = 0.

Non-Newton Filtration Equation with Nonconstant Medium Void and Critical Sobolev Exponent

In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy initial value; we also describe the asymptotic behavior of global solutions with high energy initial value.