Orateur
Liviu Ignat
(IMAR, Bucarest, Roumanie)
Description
In this talk, we analyze the long time behaviour of the solutions of the equation
$ u_t(t,x)+(-\Delta) ^{\alpha/2}u(t,x)+(f(u))_x=0,\ t\in (0,\infty),\ x\in\mathbf{R},$
where $\alpha\in (0,2)$ and $f(s)=|s|^{q-1}s/q$ with $q\in (1,\infty)$. We present some prvious results on the asymptotic expansion of the solutions when the time goes to infinity. We prove that in the one-dimensional case, for $q\in (1,\alpha)$ the asymptotic behaviour is given by the entropy solution of the conservation law $u_t(t,x)+(f(u))_x=0$, $u(0)=M\delta_0$ where $M$ is the mass of the initial data. The proof relies on tricky inequalities to guarantee an Oleinik type inequality $(u^{q-1})_x\leq 1/t$.
This is a joint work with Diana Stan. This presentation is partially supported by CNCS-UEFISCDI No. PN-III-P4- ID-PCE-2016-0035.
Auteur principal
Liviu Ignat
(IMAR, Bucarest, Roumanie)
Co-auteur
Diana Stan
(Basque Center for Applied Mathematics, Bilabao, Espagne)
