Degenerate Quasilinear Parabolic Systems Research Articles

The Cauchy problem for a nonlinear degenerate parabolic system in non-divergence form

We deal with degenerate quasilinear parabolic systems in the non-divergence form under positive initial conditions. An asymptotic behavior of self-similar solutions in the case of slow diffusion is established. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. In addition, the asymptotic representation of the solution is obtained.

Hölder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems

In this paper, we establish Hölder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to Hörmander type vector fields by deriving a parabolic Poincaré inequality and the higher integrability of gradients of weak solutions.

Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system

The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$ u_t = v^p \left( {\Delta u + a\int_\Omega {udx} } \right),v_t = u^q \left( {\Delta v + b\int_\Omega {vdx} } \right) $$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution (u, v) to this problem. Moreover, a necessary and sufficient condition for the non- global existence of the solution is obtained. Under some further conditions on the initial data, we get criteria for the finite time blow-up of the solution.

Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source

This paper deals with a class of localized and degenerate quasilinear parabolic systems $$u_t=f(u)(\Delta u+av(x_0,t)),\qquad v_t=g(v)(\Delta v+bu(x_0,t))$$ with homogeneous Dirichlet boundary conditions. Local existence of positive classical solutions is proven by using the method of regularization. Global existence and blow-up criteria are also obtained. Moreover, the authors prove that under certain conditions, the solutions have global blow-up property.

Blow-up rate and profile for a class of quasilinear parabolic system

This paper deals with positive solutions to a class of nonlocal and degenerate quasilinear parabolic system with null Dirichlet boundary conditions. The blow-up rate and blow-up profile are gained if the parameters and the initial data satisfy some conditions.

A class of nonlocal and degenerate quasilinear parabolic system not in divergence form

In this paper, we investigate the positive solution of the nonlocal and degenerate quasilinear parabolic system not in divergence form u t = f ( u ) ( Δ u + a ∫ Ω v d x ) , v t = g ( v ) ( Δ v + b ∫ Ω u d x ) with null Dirichlet boundary conditions. The blow-up and global existence criteria are obtained. Furthermore, we prove that under certain conditions the solutions have global blow-up.

A degenerate parabolic system for three-phase flows in porous media

A classical model for three-phase capillary immiscible flows in a porous medium is considered. Capillarity pressure functions are found, with a corresponding diffusion-capillarity tensor being triangular. The model is reduced to a degenerate quasilin-ear parabolic system. A global existence theorem is proved under some hypotheses on the model data.

Nonlinear transmission problems for quasilinear diffusion systems

We study degenerate quasilinear parabolic systems in two different domains, which are connected by a nonlinear transmission condition at their interface. For a large class of models, including those modeling pollution aggression on stones and chemotactic movements of bacteria, we prove global existence, uniqueness and stability of the solutions.

Blow-up solution for the degeneratequasilinear parabolic system with nonlinear boundary condition

In this paper, we discuss the global existence and blow-up of the solution to the quasilinear parabolic system with nonlinear boundary condition. We also discuss the blow-up set of the blow-up solution and obtain the blow-up rate of the blow-up solution and prove the existence of the self-similar blow-up solution.

Global existence and decay properties of solutions for some degenerate quasilinear parabolic systems modelling chemotaxis

The following degenerate parabolic system modelling chemotaxis is considered. ( KS ) τ u t = ∇ · ( ∇ u m - u ∇ v ) , x ∈ R N , t > 0 , τ v t = Δ v - v + u , x ∈ R N , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , where τ = 0 or 1. We show here that the system of ( KS ) τ with m > 1 has a time global weak solution ( u , v ) with a uniform bound in time when ( u 0 , v 0 ) is a nonnegative function and in L 1 ∩ L ∞ ( R N ) × L 1 ∩ H 1 ∩ W 1 , ∞ ( R N ) , u 0 m ∈ H 1 R N ) . The decay properties of the solution with small initial data are also discussed.

Local boundedness for doubly degenerate quasi-linear parabolic systems

We establish the local boundedness of weak solutions for the degenerate quasilinear parabolic systems ∂u i ∂t − ∑ j=1 n ∂ ∂x j |u| α ∂u ∂x p−2 ∂u i ∂x j =C|u| α−1 ∂u ∂x p−1 u i, i=1,…,N; N≥1 .

On the large time behavior of solutions for some degenerate quasilinear parabolic systems

On the large time behavior of solutions for some degenerate quasilinear parabolic systems

On the support properties of solutions for some degenerate quasilinear parabolic systems

On the support properties of solutions for some degenerate quasilinear parabolic systems