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.

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.

EXISTENCE OF WEAK SOLUTIONS FOR A DEGENERATE GENERALIZED BURGERS EQUATION WITH LARGE INITIAL DATA

It is obtained the existence of the weak solution for a degenerate generalized Burgers equation under the restriction u0 ∈ L∞. The main method is to add viscosity perturbation and obtain some estimates in L1 norm. Meanwhile it is obtained the solution is exponential decay when the initial data has compact support.

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.

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.

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.

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 Hormander's condition, and △x=m∑j=1Xj^2is 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.

EXISTENCE AND DECAY OF WEAK SOLUTIONS TO DEGENERATE DAVEY-STEWARTSON EQUATIONS

In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.

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.

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.

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.

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 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）.

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.

INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ELLIPTIC EQUATIONS

Let X = （X1, ···, Xm） be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum （ ） form j=1 to m Xj Xj.