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MSC 2010: 44A35, 35L20, 35J05, 35J25In this paper are found explicit solutions of four nonlocal boundary value
problems for Laplace, heat and wave equations, with Bitsadze-Samarskii
constraints based on non-classical one-dimensional convolutions. In fact,
each explicit solution may be considered as a way for effective summation
of a solution in the form of nonharmonic Fourier sine-expansion. Each
explicit solution, may be used for numerical calculation of the solutions too.
On présente dans cet exposé des résultats récents de Merle et Raphael sur l’analyse des solutions explosives de l’équation de Schrödinger critique. On s’intéresse en particulier à leur preuve du fait que les solutions d’énergie négative (dont on savait qu’elles explosaient par l’argument du viriel) et dont la norme est proche de celle de l’état fondamental, explosent au régime du “log log”et que ce comportement est stable.
Estudiamos la existencia de soluciones del sistema elíptico no lineal Δu + |∇u| = p(|x|)f(v), Δv + |∇v| = q(|x|)g(u) en Ω que explotan en el borde. Aquí Ω es un dominio acotado de RN o el espacio total. Las nolinealidades f y g son funciones continuas positivas mientras que los potenciales p y q son funciones continuas que satisfacen apropiadas condiciones de crecimiento en el infinito. Demostramos que las soluciones explosivas en el borde dejan de existir si f y g son sublineales. Esto se tiene...
We consider an evolution equation similar to that introduced by Vese in [Comm.
Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution
converges in large time to the convex envelope of the initial datum. We give a stochastic
control representation for the solution from which we deduce, under quite general
assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
We consider an evolution equation similar to that introduced by Vese in [Comm.
Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution
converges in large time to the convex envelope of the initial datum. We give a stochastic
control representation for the solution from which we deduce, under quite general
assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
We show that the entropy method, that has been used successfully in order to prove exponential convergence towards equilibrium with explicit constants in many contexts, among which reaction-diffusion systems coming out of reversible chemistry, can also be used when one considers a reaction-diffusion system corresponding to an irreversible mechanism of dissociation/recombination, for which no natural entropy is available.
We show that the entropy method, that has been used successfully in order
to prove exponential convergence towards equilibrium with explicit constants in many contexts,
among which reaction-diffusion systems coming out of reversible chemistry, can also be used
when one considers a reaction-diffusion system corresponding to an irreversible mechanism of
dissociation/recombination, for which no natural entropy is available.
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