Fujita type results for a parabolic inequality with a non-linear convolution term on the Heisenberg group

The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group { u t – ∆ H u ≥ ( K ∗ H | u | p ) | u | q , η ∈ H n , t > 0 , u ( η , 0 )= u 0 ( η ), η ∈ H n , where n ≥1, p , q >0, u 0 ∈ L loc 1 ( H n ) , ∆ H is the Heisenberg Laplacian, and K:(0,∞)→(0,∞) is a continuous function satisfying K ( | · | H ) ∈ L loc 1 ( H n ) which decreases in a vicinity of infinity. In addition, ∗ H denotes the convolution operation in H n . Our approach is based on the non-linear capacity method.