| Contents |
| In this talk we will present some nonlocal evolution problems that involve operators of the type:
$$\mathcal{L}u(x)= \int_{\mathbf{R}^d} J(x-y)(u(y)
- u(x)) \, dy
$$
We analyze the asymptotic behaviour of the solutions of the following nonlocal convection-diffusion equation
$$u_t=J\ast u-u+G\ast u^2 -u^2.$$
The results are mainly obtained by scaling arguments and a new compactness argument that is adapted to nonlocal evolution problems.
The compactness tool is the following one:
Let $\Omega\subset \mathbf{R}^d$ be an open set. Let $\rho:\mathbf{R}^d\rightarrow \mathbf{R}$ be a nonnegative smooth radial function with compact support, non identically zero, and $\rho_n(x)=n^d\rho(nx)$. Let $\{f_n\}_{n\geq 1}$ be a sequence of functions in $L^p((0,T)\times \Omega)$ such that
$$\int _0^T \int _{\Omega} |f_n|^p \leq \ {M}
$$
and
$$
n^p\int _0^T\int _{\Omega}\int _{\Omega} \rho_n(x-y) |f_n(t,x)-f_n(t,y)|^pdxdydt\leq {M}.
$$
If $\{f_n\}_{n\geq 1}$ is weakly convergent in $L^p((0,T)\times \Omega)$ to $f$ then $f\in L^p((0,T),W^{1,p}(\Omega))$ for $p>1$ and $f\in L^1((0,T),BV(\Omega))$ for $p=1$.
Let $p>1$. Assuming that $\Omega$ is a smooth bounded domain in $\mathbf{R}^d$, $\rho(x)\geq \rho(y)$ if $|x|\leq |y|$ and that
$$
\|\partial_t f_n\|_{L^p((0,T),W^{-1,p}(\Omega))}\leq M
$$
then $\{f_n\}_{n\geq 1}$ is relatively compact in $L^p((0,T)\times \Omega)$. |
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