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Análise / Equações Diferenciais Parciais

Título
Asymptotic behaviour for nonlocal diffusion-convection equations
Expositor
Liviu Ignat

Institute of Mathematics of the Romanian Academy - IMAR, Romania
Data
Terça-feira, 8 de janeiro de 2019, 15:30
Local
Sala 232
Resumo

In this talk we analyze the long time behaviour of the solutions of the equation
\[
u_{t}(t,x) + \mathcal{L}u(t,x)+(f(u))_x=0, \quad t>0,\quad x\in \mathbf{R},
\]
where  $\mathcal{L}$ is a nonlocal operator and  $f(s)=|s|^{q-1}s/q$ with $q>1$. When $\mathcal{L}=(-\Delta)^{\alpha/2}$ we prove that in the one-dimensional case, for $1<q<\alpha<2$ 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 relays on tricky inequalities to guarantee an Oleinik type inequality.

Joint work with Diana Stan. This presentation is partially supported by CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE- 2016-0035.