Finite time extinction for the 1D stochastic porous medium equation with transport noise

We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $$\smash {O(t^\frac{1}{2})}$$ O ( t 1 2 ) whereas the support of solutions to the deterministic PME grows only with rate $$\smash {O(t^{\frac{1}{m{+}1}})}$$ O ( t 1 m + 1 ) . The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.

Dificultades en el registro y transmisión de un arte fugaz.

Desde la segunda mitad del siglo XX las intervenciones artísticas se independizan de los hartados y constantes componentes estéticos, miméticos y matéricos. La imposibilidad de custodiar la obra como objeto demanda nuevos mecanismos que registren un arte tan cambiante como caprichoso. Con los avances de la tecnología, la fotografía digital y las innovaciones acontecidas en el campo del arte, los artistas comenzarán a dejar constancia de sus intervenciones haciendo uso de cámaras fotográficas y pequeñas grabaciones para inmortalizar sus obras. Destinamos esta investigación a trabajar, a través de una representación de autores emergentes, las dificultades en el registro de obras artísticas caracterizadas por su intangibilidad. El desafío está servido, cualquier persona o entidad vinculada a lo artístico se enfrenta a la difícil tarea de registrar para conservar la inmortalidad del acontecimiento artístico. Estas innovadoras y fugaces categorías llevan implícita la condición participativa del espectador, el espacio como continente y contenido de la obra, el tiempo, la extinción de los materiales, la acción, el desplazamiento, la pérdida de la unicidad del arte, así como su acelerada mortalidad. Condiciones y aspectos que hacen del arte un “arte-acontecimiento”, predispuesto por la condición espaciotemporal a su caducidad, modificación o desaparición. Since the second half of 20th century, artistic interventions have become independent of the jaded and constant aesthetic, mimetic and mathematical components. The impossibility of guarding the work as an object demands new mechanisms that register an art as changing as it is capricious. With the advances in technology, digital photography and innovations in the field of art, artists will begin to record their interventions, using cameras and small recordings to immortalize their works. We dedicate this research to work through a representation of emerging authors, difficulties in the registration of artistic works characterized by their intangibility. The challenge is served, any person or entity linked to the artistic, faces the difficult task of registering to preserve the immortality of the artistic event. These innovative and fleeting categories implicitly involve the participatory condition of the viewer, space as a continent and content of the work, time, extinction of materials, action, displacement, loss of the uniqueness of art, as well as its accelerated dematerialization Conditions and aspects that make art an “art-event”, predisposed by the space-time condition to its expiration, modification or disappearance.

Extinction for a Singular Diffusion Equation with Strong Gradient Absorption Revisited

When {2N/(N+1)<p<2} and {0<q<p/2}, non-negative solutions to the singular diffusion equation with gradient absorption\partial_{t}u-\Delta_{p}u+|\nabla u|^{q}=0\quad\text{in }(0,\infty)\times% \mathbb{R}^{N}vanish after a finite time. This phenomenon is usually referred to as finite-time extinction and takes place provided the initial condition {u_{0}} decays sufficiently rapidly as {|x|\to\infty}. On the one hand, the optimal decay of {u_{0}} at infinity guaranteeing the occurrence of finite-time extinction is identified. On the other hand, assuming further that {p-1<q<p/2}, optimal extinction rates near the extinction time are derived.

On the Roughness of Quasinilpotency Property of One-parameter Semigroups

Let S := {S(t)}t≥0 be a C0-semigroup of quasinilpotent operators (i.e., σ(S(t)) = {0} for eacht> 0). In dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical ûnite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a C0-semigroup is preserved under the perturbations of its infinitesimal generator.