Higher order evolution inequalities with nonlinear convolution terms

Abstract

We are concerned with the study of existence and nonexistence of weak solutions to ∂ku∂tk+(−Δ)mu≥(K∗|u|p)|u|qinRN×R+,∂iu∂ti(x,0)=ui(x)inRN,0≤i≤k−1, where N,k,m≥1 are positive integers, p,q>0 and ui∈Lloc1(RN) for 0≤i≤k−1. We assume that K is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, K∗|u|p denotes the standard convolution operation between K(|x|) and |u|p. We obtain necessary conditions on N,m,k,p and q such that the above problem has solutions. Our analysis emphasizes the role played by the sign of ∂k−1u∂tk−1.