Asymptotics for nonlocal evolution equations

In this talk, we will discuss the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal di usion problems. We will consider equations like

ut(x, t) = ∫R J(x, y)(u(y, t) − u(x, t))dy + f(u)(x, t),

and a nonlocal analogous to the p-Laplacian

ut(x, t) = ∫R J(x, y)|u(y) − u(x)|^p−2 (u(y, t) − u(x, t))dy.

With these energy methods, we can deal with nonlocal problems that do not necessarily involve a convolution, that is, J(x+y) = J(x-y), equations that have been treated before using di erent techniques.

Joint work with Julio Rossi.