Higher order evolution inequalities with nonlinear convolution terms

We are concerned with the study of existence and nonexistence of weak solutions to {partial derivative(k)u/partial derivative t(k) + (-Delta)(m) u >= (K * vertical bar u vertical bar(p))vertical bar u vertical bar(q) in R-N x R+, partial derivative(i)u/partial derivative t(i)(x, 0) = u(i)(x) in R-N, 0 <= i <= k - 1, where N, k, m >= 1 are positive integers, p, q > 0 and u(i) is an element of L-loc(1)(R-N) 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 (*) vertical bar u vertical bar(p) denotes the standard convolution operation between K(vertical bar x vertical bar) and vertical bar u vertical bar(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 partial derivative(k-1)u/partial derivative t(k-1). (C) 2022 The Author(s). Published by Elsevier Ltd.

Automated summary

This article investigates a problem concerning the existence and nonexistence of weak solutions, providing necessary conditions for the problem to have solutions and emphasizing the role played by the sign of partial derivatives.