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.

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.

A REMARK OF THE HLDER ESTIMATE OFSOLUTIONS OF SOME DEGENERATE PARABOLICEQUATIONS

In this paper, the Cauchy problem of the degenerate parabolic equationsis studied for some cases, and the explicit Holder estimates of the solution u with respectto x is given.

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.

GLOBAL EXISTENCE,EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS

In this paper,we study the initial-boundary value problem for the semilinear parabolic equations ut-△Xu=|u|p-1u,where X=（X1,X2,…,Xm） is a system of real smooth vector fields which satisfy the H?rmander’s condition,and △X=∑j=1m Xj2 is a finitely degenerate elliptic operator.Using potential well method,we first prove the invariance of some sets and vacuum isolating of solutions.Finally,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,and also we discuss the asymptotic behavior of the global solutions.

CAUCHY PROBLEM FOR A DEGENERATE PARABOLIC EQUATION WITH NON-DIVERGENCE FORM

In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.

REGULARITY OF RANDOM ATTRACTORS FOR A STOCHASTIC DEGENERATE PARABOLIC EQUATIONS DRIVEN BY MULTIPLICATIVE NOISE

We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.

EXISTENCE AND NONEXISTENCE OF GLOBAL POSITIVE SOLUTIONS FOR DEGENERATE PARABOLIC EQUATIONS IN EXTERIOR DOMAINS

This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = （σ ＋ m ）n / （ n-σ- 2 ） is its critical exponent provided max{-1, [ （1- m ）n- 2] / （ n ＋ l ） } 〈 σ 〈 n- 2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Futhermore, we demonstrate that if max（1, σ ＋ m） 〈 p 〈 pc, then every positive solution of the equations blows up in finite time; whereas for p 〉 pc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n〈σ＋2.

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 of Local Solutions to a Class of Doubly Degenerate Parabolic Equations

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate pLaplace equation （p 〉 2）ut-div（b（x,t,u）｜↓△u｜^p-2↓△u）=f（x,u,t）The initial-boundary value problem in a bounded domain of R^N is considered under mixed boundary conditions. The existence of local-in-time weak solutions is obtained.

EXISTENCE AND UNIQUENESS OF RENORMALIZED SOLUTIONS FOR A CLASS OF DEGENERATE PARABOLIC EQUATIONS

This article discusses the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equations b（u）t - div（a（u, u）） = H（u）（f ＋ divg）.

Existence of Solutions to a Class of Nonlinear Degenerate Parabolic Equations Not in Divergence Form

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.

Homogenization of a nonlinear degenerate parabolic equation

The homogenization of one kind of nonlinear parabolic equation is studied. The weak convergence and corrector results are obtained by combining carefully the compactness method and two-scale convergence method in the homogenization theory.

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.

THE CAUCHY PROBLEM FOR A CLASS OF DOUBLY DEGENERATE NONLINEAR PARABOLIC EQUATION

This article studies the Cauchy problem for a class of doubly nonlinear degenerate parabolic equations au/at = div（A（｜△↓B（u）｜）△↓B（u））. Under certain conditions, the author considers its regularized problem and establishes some estimates. On the basis of the estimates, the existence and uniqueness of the generalized solutions in BV space are proved.

Blow-up rate estimate for degenerate parabolic equation with nonlinear gradient term

Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.

A NONLINEAR DEGENERATE PARABOLIC EQUATION WITH NON-LOCAL SOURCE

The global existence and finite time blow up of the positive solution for a nonlinear degenerate parabolic equation with non- local source are studied

Initial－boundary Value Problem for a Class of Quasilinear Degenerate Parabolic Equations

in this work,we study the quasilinear initial-boundary value problem , where is a system of real smooth vector fields which is defined on an open domain M of R'', and satisfies the Hormanderls condition,.Assume that is non characteristic for the system X,,..',Xm. Under some hypothesis for the boundary of domain and the elliptic structure condition for nonlinear coerfficients Aij, Bj, C,(i, j= 1, ..', m), we have proved that the existence and regularity of solution for aboveinitialboudary value problems.

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.