Degenerate Parabolic Systems Research Articles

Global existence for a class of highly degenerate parabolic systems

We consider a system of differential equations describing diffusion in polymers. By means of techniques from semigroup theory we prove that it possesses a unique global smooth solution, thus settling some questions, which came up in numerical investigations of this model. From the mathematical point of view the interest in this problem lies in the fact, that it is a degenerate parabolic (2 × 2)-system, whose “diffusion matrix” has rank one only.

Numerical simulation for degenerate diffusions

Numerical treatment of certain diffusion problems via indirect (analytical) methods requires solving boundary-value problems for degenerate parabolic equations or systems. Degeneracy may arise from coordinate singularity (polar coordinates) or other more intrinsic reasons. Examples of degenerate diffusions can be found in wave propagation in random media, genetics and mathematical theory of evolution. Methods based on suitable finite difference schemes either suggested by the structure of the problems (symmetry, formally obtained limiting-equations, etc.) or by the conservation of certain quantities, as it happens for their continuous analogue, are introduced to compute the solutions with accuracy and stability.

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

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

Hölder estimates for second-order quasilinear degenerate parabolic systems

Interior and boundary Holder estimates are established for bounded generalized solutions of quasilinear degenerate parabolic systems with unbounded coefficients.

Boundary estimates for solutions of nonlinear degenerate parabolic systems.

On etablit la regularite partout jusqu'a la frontiere pour des solutions faibles de systemes paraboliques du type ∂u (l) /∂t−div(|Du| p−2 Du (l) =f (l) (x,t,u,Du) dans Ω T ; l=1,2,…,m; u:Ω T →R m , m∈N, p>max{1;2N/(N+2)}. Ω est un ouvert borne de R N , N≥2, 0<T<∞, Ω T ≡Ω×(0,T]

ON SOME NONLINEAR SYSTEMS DESCRIBING THE DYNAMICS OF COMPETING BIOLOGICAL SPECIES

The Cauchy problem is considered for quasilinear degenerate parabolic systems of second order with generally nonsmooth lowest terms. Such systems describe the joint evolution of several biological species consuming the same food. The existence of a generalized solution of this problem is proved, and its behavior for large values of time is studied. In particular, the conditions are found for the survival of individual species and for their extinction in a finite time as well as for boundless proliferation and for spatial localization.Bibliography: 19 titles.

Hölder continuity of the gradient of solutions of nonlinear degenerate parabolic systems

Hölder continuity of the gradient of solutions of nonlinear degenerate parabolic systems

Addendum to “Holder estimates for nonlinear degenerate parabolic Systems”.

Article Addendum to “Holder estimates for nonlinear degenerate parabolic Systems”. was published on January 1, 1985 in the journal Journal für die reine und angewandte Mathematik (volume 1985, issue 363).

Cauchy problem for a class of degenerate parabolic systems

Cauchy problem for a class of degenerate parabolic systems

Construction of the fundamental solution for degenerate parabolic systems and its application to construction of a parametrix of □_b

On considere un systeme parabolique degenere d'operateurs pseudodifferentiels: [d/dt+p(x,D)] E(t)=0, t>0, x∈R n , E(O)=I ou p(x, ξ) est une matrice k×k. On cherche E(t) dans une certaine classe d'operateurs pseudodifferentiels

PERIODIC BOUNDARY PROBLEMS FOR SEMI LINEAR DEGENERATE EVOLUTION SYSTEMS OF HIGHER ORDER

Abstract It is the purpose of this paper to consider some semilinear evolution systems of partial differential equations, which are found in physics, chemical reactions and biology. The existence, uniqueness and regularity of solutions for periodic boundary problems in global are proved for following degenerate parabolic and hyperbolic systems of higher order ∂ u ∂ t = ∑ m - 1 M ( - 1 ) m + 1 A m ( x ) ∂ 2 m u ∂ x 2 m + ∑ r = 0 R B r ( t ) ∂ 2 r + 1 u ∂ x 2 r + 1 + f ( u ) , where the matrices A(t), m = 1, …, M are nonnegative definite and the matrices B(t), r = 1 · ·, M are symmetric. One can show that the nonlinear Schrodinger equation or system, which may be treated as a real degenerate parabolic system, the Sine-Gordon equation, Klein-Gordon equation, some simultaneous equations of Schrodinger equation and wave equation etc. may be changed into the above type.

Regularity of solutions of nonlinear degenerate parabolic systems.

Soit le systeme parabolique δu i /δt-div(|⊇u| p−2 ⊇u i )=0 (≤i≤m), avec p>1, ou u i =u i (x,t), ⊇=grad x et x∈Ω⊂R N ouvert. Pour toute solution de ce systeme avec max{1, 2N/N+2}<p<∞ ⊇u est continu dans Ω T

ON THE PERIODIC BOUNDARY PROBLEMS FOR SEMILINEAR DEGENERATE EVOLUTION SYSTEMS OF HIGHER ORDER

It is the purpose of this paper to consider some semilinear evolution systems of partial differential equations, which are found in physics, chemical reactions and biology. The existence, uniqueness and regularity of solutions for periodic boundary problems in global are proved for following degenerate parabolic and hyperbolic systems of higher order ∂u∂t = ∑m-1M(-1)m+1Am(x)∂2mu∂x2m + ∑r=0RBr(t)∂2r+1u∂x2r+1+f(u),where the matrices A(t), m = 1, …, M are nonnegative definite and the matrices B(t), r = 1 · ·, M are symmetric. One can show that the nonlinear Schrödinger equation or system, which may be treated as a real degenerate parabolic system, the Sine-Gordon equation, Klein-Gordon equation, some simultaneous equations of Schrodinger equation and wave equation etc. may be changed into the above type.