Some degenerate and quasilinear parabolic systems not in divergence form

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u_t=u^p(Δu+av), v_t=v^q(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ₁², where λ₁ is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

Homogenization of degenerate nonlinear parabolic equation in divergence form

In this paper the homogenization of degenerate nonlinear parabolic equations

where a(t,y,λ) is periodic in (t,y), is studied via a weighted compensated compactness result.     

Asymptotic behavior of a degenerate nonlocal parabolic equation

In this paper, we consider the asymptotic behavior for the degenerate nonlocal parabolic equation

     

Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f(u)=u^p,0<p1, we gained blow-up rate estimate.     

Blow-up for a degenerate parabolic equation with a nonlocal source

In this paper, we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation

     

Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources

This paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions.     

Regularity of viscosity solutions of a degenerate parabolic equation

We study the Cauchy problem for the nonlinear degenerate parabolic equation of second order

and present regularity results for the viscosity solutions.

     

Large time behavior of solutions to a class of doubly nonlinear parabolic equations

We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation u _t = div(u ^m−1|Du| ^p−2 Du) − u ^q with an initial condition u(x, 0) = u ₀(x). Here the exponents m, p and q satisfy m + p ⩾ 3, p > 1 and q > m + p − 2. The paper was supported by NSF of China (10571144), NSF for youth of Fujian province in China (2005J037) and NSF of Jimei University in China.     

一类双重退化抛物方程重整化解的稳定性

讨论了一类有双重退化性的抛物方程的Cauchy问题,并基于Kruzhkov技术,证明了重整化解的稳定性.     

Blow-up and global existence for a nonlocal degenerate parabolic system

This paper investigates the blow-up and global existence of nonnegative solutions of the system

     

Global and nonglobal weak solutions to a degenerate parabolic system

This paper deals with a degenerate parabolic system coupled via general reaction terms of power type. Global weak solutions are obtained by means of energy estimates and the De Giorgi's technique. In particular, the criterion for global nonexistence of weak solutions is proved by introducing suitable weak sub-solutions together with a weak comparison principle. In summary, the critical exponent for weak solutions of the degenerate parabolic system is determined.     

Asymptotic behavior and blow-up of solutions for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity $u_t?{△}_{X}u=u\log ?|u|$, where $X=(X_1,X_2,?,X_m)$ is an infinitely degenerate system of vector fields, and ${△}_X:={\sum }_{j=1}^m{X}_j^2$ is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the logarithmic Sobolev inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and we also discuss the asymptotic behavior of the solutions.     

Hölder estimates of solutions to a degenerate diffusion equation

This paper is concerned with the Hölder estimates of weak solutions of the Cauchy problem for the general degenerate parabolic equations

with the initial data , where the diffusion function can be a constant on a nonzero measure set, such as the equations of two-phase Stefan type. Some explicit Hölder exponents of the composition function with respect to the space variables are obtained by using the maximum principle.

     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05     

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.     

Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values

In this paper, we investigate the behavior of the positive solution of the following Cauchy problem

ut−div(|∇um^|p−2∇um)=uq     

Blow-up and global existence to a degenerate reaction-diffusion equation with nonlinear memory

In this paper, we consider a degenerate reaction-diffusion equation