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Decay estimates for solutions of a class of parabolic problems arising in filtration through porous media

G. A. Philippin, S. Vernier-Piro (2001)

Bollettino dell'Unione Matematica Italiana

In questo lavoro si studia un problema di valori al contorno parabolico non lineare che si incontra nello studio dell'infiltrazione di un gas in un mezzo poroso. Si stabiliscono condizioni sui dati che determinano un comportamento di tipo esponenziale decrescente nel tempo per la soluzione e il suo gradiente. Si costruiscono inoltre stime esplicite.

Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes

Christian Heinemann, Christiane Kraus (2014)

Mathematica Bohemica

This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent...

Difference methods for parabolic functional differential problems of the Neumann type

K. Kropielnicka (2007)

Annales Polonici Mathematici

Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions...

Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity q 2

Luisa Fattorusso (2004)

Commentationes Mathematicae Universitatis Carolinae

Let Ω be a bounded open subset of n , let X = ( x , t ) be a point of n × N . In the cylinder Q = Ω × ( - T , 0 ) , T > 0 , we deduce the local differentiability result u L 2 ( - a , 0 , H 2 ( B ( σ ) , N ) ) H 1 ( - a , 0 , L 2 ( B ( σ ) , N ) ) for the solutions u of the class L q ( - T , 0 , H 1 , q ( Ω , N ) ) C 0 , λ ( Q ¯ , N ) ( 0 < λ < 1 , N integer 1 ) of the nonlinear parabolic system - i = 1 n D i a i ( X , u , D u ) + u t = B 0 ( X , u , D u ) with quadratic growth and nonlinearity q 2 . This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions u belonging to W 1 , q C 0 , λ .

Diffusion and cross-diffusion in pattern formation

Wei-Ming Ni (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We discuss the stability and instability properties of steady state solutions to single equations, shadow systems, as well as 2 × 2 systems. Our basic observation is that the more complicated the pattern are, the more unstable they tend to be.

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

Oto Havle, Vít Dolejší, Miloslav Feistauer (2010)

Applications of Mathematics

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation...

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