Nonlinear evolution equations that are non-local in space and time

Abstract

We deal with a nonlocal nonlinear evolution problem of the form∬Rn×RJ(x−y,t−s)|v‾(y,s)−v(x,t)|p−2(v‾(y,s)−v(x,t))dyds=0 for (x,t)∈Rn×[0,∞). Here p≥2, J:Rn+1→R is a nonnegative kernel, compactly supported inside the set {(x,t)∈Rn+1:t≥0} with ∬Rn×RJ(x,t)dxdt=1 and v‾ stands for an extension of a given initial value f, that is,v‾(x,t)={v(x,t)t≥0,f(x,t)t<0. For this problem we prove existence and uniqueness of a solution. In addition, we show that the solutions approximate viscosity solutions to the local nonlinear PDE ‖∇u‖p−2ut=Δpu when the kernel is rescaled in a suitable way.