Seminario de EDPs y Afines (IMERL)

Diffusion models appear in multiple sciences such as biology, physics or even economics. They come up naturally as a broad class of natural processes, like the transport of matter due to random molecular motions. The most common diffusion operator in dynamical systems is the laplacian $\Delta$, which is derived from Fick's laws and leads us to the local diffusion model $u_t=\Delta u$. It is called local diffusion because this model only considers the possibility of the particles moving very short distances in each instant of time. However, there can be more than that. There are other phenomena, such as the propagation of a pathogen, in which the particles could jump long distances in each instant of time thanks to various means of transport. We call this non-local diffusion and it is modeled by different operators, such as those of the type $J*u-u$, where the kernel $J$ is a density function of the probability of any jump happening. The most famous non-local operator is the fractional laplacian $(-\Delta)^s$, which has by itself spawned a whole array of literature.

The phenomenon of quenching in a dynamical system consists of the explosion of the velocity of the solution while the solution itself remains bounded. It was first assessed by Hideo Kawarada in 1974 for the equation $u_t=u_{x x}+(1-u)^{-1}$, where it happens whenever the solution reaches the value $u=1$. The phenomenon of quenching appears naturally in physical models such as the nonlinear heat conduction in solid hydrogen, or the Arrhenius Law in combustion theory. It is for this reason that quenching has been the subject of numerous studies since Kawarada's paper.

The aim of this talk is to speak about our study of a system of equations with weakly coupled singular absorption terms and a non-local diffusion operator of the type convolution with a smooth kernel and the quenching phenomena that arises. First we will offer a suitable introduction to the non-local diffusion operator and the quenching phenomenon so that the talk can be followed by anyone interested but without prior knowledge on these topics. Then we will show our results about the system, which tackle the appearance of stationary solutions, the quenching rates of both components, the possibility of both components presenting quenching at the same time and the added difficulties this problem presents with respect to the single equation with non-local diffusion.

We consider the problem
$$ -\left(a+b\left([u]_s^p\right)^{p-1}\right)(-\Delta)_p^s u(x)+V(x)|u(x)|^{p-2} u(x)=f(x, u(x)), $$
with \(|u(x)| \rightarrow 0\),as \(|x| \rightarrow+\infty,\) where \(s \in(0,1), 1<p<+\infty, a>0, b>0, N \geq 2,\) \(1<s p<N<+\infty, V \in C\left(\mathbb{R}^N\right)\) verifies \(\theta=\inf _{\mathbb{R}^N} V>0\), and \(f \in C\left(\mathbb{R}^N \times \mathbb{R}\right).\) The operator \((-\Delta)_p^s\) denotes the fractional \(p\)-Laplacian, and the seminorm \([u]_{s, p}^p\) is associated with the fractional Sobolev space \(W^{s, p}\left(\mathbb{R}^N\right)\). Our approach combines variational methods and critical point theory to establish the existence of nontrivial solutions. Under appropriate conditions on \(V\) and \(f\), we prove the existence of a ground-state solution. Moreover, by assuming an additional oddness condition on the nonlinearity, we obtain the multiplicity using the genus theory.

En esta conferencia nos centraremos en la dinámica de Langevin en un dominio acotado, con reflexión especular en el contorno. Presentamos una construcción explícita del proceso de difusión asociado e identificamos su generador infinitesimal. A continuación, establecemos una serie de supuestos que garantizan la irreducibilidad, la aperiodicidad y la ergodicidad exponencial. También estableceremos la relación con las Ecuaciones Diferenciales Parciales hipoelípticas asociadas.