Nonexistence of global solutions for a fractional problems with a nonlinearity of the Fisher type

This paper deals with the Cauchy problem for a nonlinear hyperbolic equation D1+α 0|t u+D 0|tu+ (− ) γ 2 u = h(t, x) | u |p| 1 − u |q, posed in Q = R+ × R, where pi, qi > 1, −1 < α < 1, 0 < β < 2, 0 < γ ≤ 2, and β < 1+α with given initial position and velocity u(x, 0) = u0(x), ut (x, 0) = u1(x), and the Cauchy problem for a nonlinear hyperbolic system with initial data   D 1+α1 0|t u+D1 0|tu+ (− ) γ1 2 u = h1(t, x) | v |p1 | 1 − v |q1, (t, x) ∈ Q D 1+α2 0|t v +D2 0|t v + (− ) γ2 2 v = h2(t, x) | u |p2 | 1 − u |q2, (t, x) ∈ Q u(x, 0) = u0(x) ≥ 0, ut (x, 0) = u1(x) ≥ 0, x ∈ R v(x, 0) = v0(x) ≥ 0, vt (x, 0) = v1(x) ≥ 0, x ∈ R where −1 < αi < 1, 0 < βi < 2, 0 < γi ≤ 2, and βi < 1 + αi. Di (i = 1, 2) denote the time-derivative of arbitrary order αi in the sense of Caputo. We find a critical exponent of Fujita type in the case of the particular values of the fractional order and the separate terms pi, qi(i = 1, 2) and N. AMS subject classification: 26A33, 34K30, 35R10, 47D06.