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We extend, to parabolic equations of the KPP type in periodic media, a result of Bramson which asserts that, in the case of a spatially homogeneous reaction rate, the time lag between the position of an initially compactly supported solution and that of a traveling wave grows logarithmically in time.

The $p$ -Laplace eigenvalue problem as $p \to \infty$ in a Finsler metric

M. Belloni, Bernhard Kawohl, P. Juutinen (2006)

Journal of the European Mathematical Society

We consider the $p$ -Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite $p$ and investigate the limit problem as $p \to \infty$ .

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

D. Erhard, F. den Hollander, G. Maillard (2014)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we study the parabolic Anderson equation $\partial u (x, t) / \partial t = κ 𝛥 u (x, t) + ξ (x, t) u (x, t)$ , $x \in ℤ^{d}$ , $t \geq 0$ , where the $u$ -field and the $ξ$ -field are $ℝ$ -valued, $κ \in [0, \infty)$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $ξ$ -field plays the role of adynamic random environmentthat drives the equation. The initial condition $u (x, 0) = u_{0} (x)$ , $x \in ℤ^{d}$ , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump...

The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation

D. G. Aronson, J. L. Vazquez (1987)

Annales de l'I.H.P. Analyse non linéaire

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 quasineutral limit problem in semiconductors sciences

Ling Hsiao (2004)

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

The mathematical analysis on various mathematical models arisen in semiconductor science has attracted a lot of attention in both applied mathematics and semiconductor physics. It is important to understand the relations between the various models which are different kind of nonlinear system of P.D.Es. The emphasis of this paper is on the relation between the drift-diffusion model and the diffusion equation. This is given by a quasineutral limit from the DD model to the diffusion equation.

The regularity of weak and very weak solutions of the Poisson equation on polygonal domains with mixed boundary conditions (part II)

Adam Kubica (2005)

Applicationes Mathematicae

We examine the regularity of weak and very weak solutions of the Poisson equation on polygonal domains with data in L². We consider mixed Dirichlet, Neumann and Robin boundary conditions. We also describe the singular part of weak and very weak solutions.

The regularized asymptotics in the singularly perturbed parabolic problem with angular boundary layers.

Omuraliev, A.S. (2007)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

The second eigenvalue of the $p$ -Laplacian as $p$ goes to 1.

Parini, Enea (2010)

International Journal of Differential Equations

The topological derivative of the Dirichlet integral under formation of a thin ligament.

Nazarov, S.A., Sokolowski, Jan (2004)

Sibirskij Matematicheskij Zhurnal

Théorie de la diffusion pour l'équation de Schrödinger

Pierre Cartier (1978/1979)

Séminaire Bourbaki

Time asymptotic description of an abstract Cauchy problem solution and application to transport equation

Boulbeba Abdelmoumen, Omar Jedidi, Aref Jeribi (2014)

Applications of Mathematics

In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in $L_{1}$ -space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.

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The lifespan of spherically symmetric solutions of the compressible Euler equations outside an impermeable sphere

The light in the neighborhood of a caustic

The limiting equation for Neumann Laplacians on shrinking domains.

The linearization principle for some evolution problems arising in chemical kinetics.

The logarithmic delay of KPP fronts in a periodic medium