L'autore dà una condizione necessaria e sufficiente per la risolubilità formale degli operatori differenziali a coefficienti costanti, lineari, , in termini di prolungamento, come distribuzioni, delle , tali che .
We provide a mathematical proof of the existence of traveling vortex rings solutions to the Gross–Pitaevskii (GP) equation in dimension . We also extend the asymptotic analysis of the free field Ginzburg–Landau equation to a larger class of equations, including the Ginzburg–Landau equation for superconductivity as well as the traveling wave equation for GP. In particular we rigorously derive a curvature equation for the concentration set (i.e. line vortices if ).
A tempered distribution which is an exact solution of the wave equation with a concentrated moving source on the right-hand side, is obtained in the paper by means of the Cagniard - de Hoop method.
The existence of a traveling wave with special properties to modified KdV and BKdV equations is proved. Nonlinear terms in the equations are defined by means of a function f of an unknown u satisfying some conditions.
A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic pre-pattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is...
In this paper, we use -convergence techniques to study the following variational problemwhere , with , and is a bounded domain of , . We obtain a -convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem . Finally, a second order expansion in -convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.