On the evolution pp-Laplacian with nonlocal memory

We study the homogeneous Dirichlet problem for the evolution pp-Laplacian with the nonlocal memory term equation(0.1)ut−Δpu=∫0tg(t−s)Δpu(x,s)ds+Θ(x,t,u)+f(x,t)in  Q=Ω×(0,T),  where Ω⊂RnΩ⊂Rn is a bounded domain, ΘΘ, gg and ff are given functions. It is proved that for max{1,2nn+2}2p>2 and sΘ(x,t,s)≤0sΘ(x,t,s)≤0 the disturbances from the data propagate with finite speed and the “waiting time” effect is possible. We present simple explicit solutions that show the failure of the maximum and comparison principles for the solutions of equation (0.1).