Higher order evolution inequalities with nonlinear convolution terms. (August 2022) - Filippucci, Roberta; Ghergu, Marius
Abstract: We are concerned with the study of existence and nonexistence of weak solutions to ∂ k u ∂ t k + ( − Δ ) m u ≥ ( K ∗ | u | p ) | u | q in R N × R +, ∂ i u ∂ 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 ∈ 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 ∗ | 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 − 1 u ∂ t k − 1 .