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The relation between the porous medium and the eikonal equations in several space dimensions.

Pierre-Louis Lions, Panagiotis E. Souganidis, Juan Luis Vázquez (1987)

Revista Matemática Iberoamericana

We study the relation between the porous medium equation ut = Δ(um), m > 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.

The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation

Neil Watson (2000)

Annales Polonici Mathematici

Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip X = ℝ n × ] 0 , a [ . We prove a representation theorem for an arbitrary C 2 , 1 function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with C ∞ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of...

The solution operator for a partial differential equation with delay

Gabriella Di Blasio, Karl Kunisch, Eugenio Sinestrari (1983)

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

Viene dimostrata l’esistenza e l’unicità globale della soluzione di un’equazione funzionale in uno spazio di Hilbert e si caratterizza il generatore infinitesimale del semigruppo ad essa associato. Il risultato è applicato ad equazioni integrodifferenziali a derivate parziali di tipo parabolico in cui compaiono argomenti con ritardo (discreto e continuo) nelle derivate spaziali di ordine massimo.

The solutions of the quasilinear Keller-Segel system with the volume filling effect do not blow up whenever the Lyapunov functional is bounded from below

Tomasz Cieślak (2006)

Banach Center Publications

In [2] we proved two kinds of mechanisms of preventing the blow up in a quasilinear non-uniformly parabolic Keller-Segel systems. One of them was a priori boundedness from below of the Lyapunov functional. In fact, we were able to present a condition under which the Lyapunov functional is bounded from below and a solution exists globally. In the present paper we prove that whenever the Lyapunov functional is bounded from below the solution exists globally.

The speed of propagation for KPP type problems. I: Periodic framework

Henry Berestycki, François Hamel, Nikolai Nadirashvili (2005)

Journal of the European Mathematical Society

This paper is devoted to some nonlinear propagation phenomena in periodic and more general domains, for reaction-diffusion equations with Kolmogorov–Petrovsky–Piskunov (KPP) type nonlinearities. The case of periodic domains with periodic underlying excitable media is a follow-up of the article [7]. It is proved that the minimal speed of pulsating fronts is given by a variational formula involving linear eigenvalue problems. Some consequences concerning the influence of the geometry of the domain,...

Currently displaying 101 – 120 of 187