Degenerate Parabolic Systems Research Articles

Krein–Rutman type property and exponential separation of a noncompact operator

We investigate exponentially separated property for a noncompact linear operator T T on a Banach space. We obtain the relationship between exponentially separated property and the well-known Krein–Rutman type property for a noncompact operator. Under the assumption of an essential spectral gap, we prove that any u u -bounded operator T T with a reproducing cone admits the exponentially separated property and, hence, is of Krein–Rutman type automatically. We also establish an amenable sufficient condition for the exponentially separated property of some degenerate linear parabolic systems.

On Algebraic condition for null controllability of some coupled degenerate systems

In this paper we will generalize the Kalman rank condition for the null controllability to $n$-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and $m$-controls. For that we prove a global Carleman estimate for the solutions of a scalar $2n$-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

Calderón–Zygmund theory for degenerate parabolic systems involving a generalized p-Laplacian type

We study a local Calderon–Zygmund theory for degenerate parabolic systems involving a generalized p-Laplacian to obtain the interior integrability of the gradient.

Lp Estimates for Weak Solutions to Nonlinear Degenerate Parabolic Systems

This paper is devoted to the Lp estimates for weak solutions to nonlinear degenerate parabolic systems related to Hörmander’s vector fields. The reverse Hölder inequalities for degenerate parabolic system under the controllable growth conditions and natural growth conditions are established, respectively, and an important multiplicative inequality is proved; finally, we obtain the Lp estimates for the weak solutions by combining the results of Gianazza and the Caccioppoli inequality.

A Priori Error Analysis for the Galerkin Finite Element Semi-discretization of a Parabolic System with Non-Lipschitzian Nonlinearity

This paper deals with the numerical approximation of certain degenerate parabolic systems arising from flow problems in porous media with slow adsorption. The characteristic difficulty of these problems comes from their monotone but non-Lipschitzian nonlinearity. For a model problem of this type, optimal-order pointwise error estimates are derived for the spatial semi-discretization by the finite element Galerkin method. The proof is based on linearization through a parabolic duality argument in L∞(L∞) spaces and corresponding sharp L1 estimates for regularized parabolic Green functions.

Study of properties of solutions for quasilinear parabolic systems

In this paper, we concern with degenerate and quasilinear parabolic systems not in divergence form with null Dirichlet boundary conditions and positive initial conditions. The local existence and uniqueness of classical solution are proved. Moreover, It will be proved that all solutions exist globally with homogeneous Dirichlet boundary condition.

Optimal behavior of the support of the solutions to a class of degenerate parabolic systems

In this paper we deal with a class of degenerate/singular quasilinear parabolic systems. We study a Cauchy problem in \mathbb R^N with an initial datum in L^1 . Sharp L^{\infty} estimates are proved. In the degenerate case, assuming that the initial datum has compact support, we prove the optimal speed of propagation of the support.

Partial regularity and blow-up asymptotics of weak solutions to degenerate parabolic systems of porous medium type

In the present paper, we deal with the degenerate parabolic system of porous medium type with non-linear diffusion m and interaction q in \({\rm{I\!R}}^N\) in the critical case of \({q =m+\frac{2}{N}}\). We establish the \({\varepsilon}\) -regularity theorem for weak solutions. As an application, the structure of asymptotics of blow-up solution is clarified. In particular, we show that the solution behaves like the δ-function at the blow-up points. Moreover, we prove that the number of blow-up points is finite, which can be controlled in terms of the mass of initial data. We also give a sharp constant for the \({\varepsilon}\) -regularity theorem.

Blow-up analysis in degenerate parabolic systems coupled via norm-type reactions

This paper deals with a class of degenerate parabolic equations coupled via nonlinear reactions in -norm type, subject to null Dirichlet boundary conditions. Firstly, we give the existence and uniqueness of local classical solutions and the comparison principle. Secondly, we determine the critical exponents for the blow-up solutions. Thirdly, all of the uniform blow-up profiles are obtained for simultaneous blow-up solutions.

On Regularity of the Time Derivative for Degenerate Parabolic Systems

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) $p$-Laplace system and models for non-Newtonian fluids of powerlaw or Carreau type. By the use of special weak different quotients adapted to the variational structure we bound fractional derivatives of $u_t$ in time and space directions. Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic $p$-Laplace equation.

On a weak attractor of a class of PDEs with degenerate diffusion and chemotaxis

In this article we deal with a class of degenerate parabolic systems that encompasses two different effects: porous medium and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We prove estimates related to the existence of the global attractor under certain `balance conditions' on the order of the porous medium degeneracy and the growth of the chemotactic function.

Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate

Abstract In this paper, we study inverse problems for multi-dimensional linear degenerate parabolic equations and strongly coupled systems. In particular we discuss the Lipschitz type stability results for the inverse source problems which determine a source term by boundary data on an appropriate sub-boundary and the data on any fixed time. Our arguments are based on the Carleman estimate. Here we prove and use the Carleman estimate with the x-independent weight function for linear degenerate parabolic equations and systems.

Very weak solutions of degenerate parabolic systems with non-standard [formula omitted]-growth

We study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p(x,t)-Laplacian system, ∂tu−div(|Du|p(x,t)−2Du)=div(|F|p(x,t)−2F). Under natural assumptions on the exponent function p:Ω×(0,T)→[2,∞), we prove that any very weak solution u:Ω×(0,T)→RN with |Du|p(⋅)(1−ε)∈L1 belongs to the natural energy space, i.e. |Du|p(⋅)∈Lloc1, provided ε>0 is small enough.

Qualitative properties of a cooperative degenerate Lotka-Volterra model

Abstract This paper considers a kind of degenerate parabolic systems. First, we consider the initial boundary value problem of the two-species degenerate parabolic cooperative system. By using the method of a parabolic regularization and energy estimate, we establish the existence of the weak solution of the problem. Then we establish the comparison principle and discuss the uniqueness and the uniform bound. At last, we consider the periodic boundary value problem of the system. By constructing a pair of ordered upper and lower solutions, we establish the existence of nontrivial nonnegative periodic solutions.

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.

New perturbation methods for nonlinear parabolic problems

We develop methods aimed at deriving regularity results for solutions to nonlinear degenerate parabolic equations and systems via local perturbation; as a consequence we obtain, in a unified way, Lipschitz continuity of solutions under weak parabolicity assumptions, and gradient continuity results in borderline cases. Nonlinear Schauder estimates as those of Misawa (2002) [29] are recovered and extended to more general settings.

Local boundedness of local weak solutions for doubly degenerate parabolic systems with singular weights

In this paper, we prove that local weak solutions for doubly degenerate parabolic systems with singular weights are locally bounded.

Degenerate parabolic systems of vector order Kolmogorov-type equations

We define a new class of degenerate \(\overrightarrow {2b}\)-parabolic systems of Kolmogorov-type equations with coefficients depending only on time. We also construct a fundamental matrix of solutions to a Cauchy problem for systems of this class and study its main properties.

The regularity of general parabolic systems with degenerate diffusion

The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\partial_t u - \mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.

On the well posedness of a class of PDEs including porous medium and chemotaxis effect

In this article we deal with a class of degenerate parabolic systems which exhibit two phenomena: porous medium effects and chemotaxis ones. Such classes of equations arise in the modelling (on the mesoscale) of biomass spreading mechanisms via chemotaxis. Under certain "balance conditions'' on the order of the porous medium degeneracy and the growth of the chemotactic functions, we prove well posedness.