Abstract
We establish conditions that ensure the absence of global solutions to the nonlinear hyperbolic equation with a time–space fractional damping:
u
tt
-
Δ
u
+
(
-
Δ
)
β
/
2
D
+
α
u
=
|
u
|
p
,
where (−
Δ)
β/2
, 1
⩽
β
⩽
2 stands for the
β/2 fractional power of the Laplacien and
D
+
α
is the Riemann–Liouville’s time fractional derivative
[10]. Our results include nonexistence results as well as necessary conditions for the local and global solvability. The method used is based on a duality argument with an appropriate choice of the test function and a scaling argument.