Systems of semilinear evolution inequalities with temporal fractional derivative on the Heisenberg group

We investigate nonexistence results of nontrivial solutions of fractional differential inequalities of the form ( FS q m ) : { D 0 / t q x i − Δ H ( λ i x i ) ≥ | η | α i + 1 | x i + 1 | β i + 1 , ( η , t ) ∈ H N × ] 0 , + ∞ [ , 1 ≤ i ≤ m , x m + 1 = x 1 , where D 0 / t q is the time-fractional derivative of order q ∈ ( 1 , 2 ) in the sense of Caputo, Δ H is the Laplacian in the ( 2 N + 1 ) -dimensional Heisenberg group H N , | η | is the distance from η in H N to the origin, m ≥ 2 , α m + 1 = α 1 , β m + 1 = β 1 , and λ i ∈ L ∞ ( H N × ] 0 , + ∞ [ ) , 1 ≤ i ≤ m . The main results are concerned with Q ≡ 2 N + 2 , less than the critical exponents that depend on q , α i , and β i , 1 ≤ i ≤ m . For q = 2 , we deduce the results given by El Hamidi and Kirane (Abstr. Appl. Anal. 2004(2):155-164, 2004 ) and El Hamidi and Obeid (J. Math. Anal. Appl. 208(1):77-90, 2003 ) from the hyperbolic systems. For m = 1 , we study the scalar case ( FI q ) : D 0 / t q x − Δ H ( λ x ) ≥ | η | α | x | β , where β > 1 , α are real parameters. In the last case, for q = 2 , we return to the approach of Pohozaev and Véron (Manuscr. Math. 102:85-99, 2000 ) from the hyperbolic inequalities.