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The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation

Nicole Schadewaldt (2011)

Commentationes Mathematicae Universitatis Carolinae

We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation u t = [ sign ( u x ) log | u x | ] x on Q T = [ 0 , T ] × [ 0 , l ] . For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at u x = 0 . We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us...

The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are hölder continuous with respect to space variables

Davide Guidetti (1996)

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

We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution u which is bounded with values in C 2 + θ ⁢ Ω ¯ (0 < < 1), with ∂ t ⁡ u bounded with values in C θ ⁢ Ω ¯ .

The periodic unfolding method for a class of parabolic problems with imperfect interfaces

Zhanying Yang (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we use the adapted periodic unfolding method to study the homogenization and corrector problems for the parabolic problem in a two-component composite with ε-periodic connected inclusions. The condition imposed on the interface is that the jump of the solution is proportional to the conormal derivative via a function of order εγ with γ ≤ −1. We give the homogenization results which include those obtained by Jose in [Rev. Roum. Math. Pures Appl. 54 (2009) 189–222]. We also get the...

The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation

Juan Luis Vázquez (2004)

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

We review the main mathematical questions posed in blow-up problems for reaction-diffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.

The quasilinear parabolic kirchhoff equation

Łukasz Dawidowski (2017)

Open Mathematics

In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.

The regularity of the positive part of functions in L 2 ( I ; H 1 ( Ω ) ) ∩ H 1 ( I ; H 1 ( Ω ) * ) with applications to parabolic equations

Daniel Wachsmuth (2016)

Commentationes Mathematicae Universitatis Carolinae

Let u ∈ L 2 ( I ; H 1 ( Ω ) ) with ∂ t u ∈ L 2 ( I ; H 1 ( Ω ) * ) be given. Then we show by means of a counter-example that the positive part u + of u has less regularity, in particular it holds ∂ t u + ∉ L 1 ( I ; H 1 ( Ω ) * ) in general. Nevertheless, u + satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

Currently displaying 81 – 100 of 187