Abstract
We, first, consider the nonlinear Schrodinger equation
i(alpha) (C)(0)D(t)(alpha)u + Delta u = lambda vertical bar u vertical bar(p) + mu a(x) . Delta vertical bar u vertical bar(q), t > 0, x is an element of R-N,
where 0 < alpha < 1, i(alpha) is the principal value of i(alpha), D-C(0)t(alpha) is the Caputo fractional derivative of order alpha, lambda is an element of C\{0}, mu is an element of C, p > q > 1, u(t, x) is a complex-valued function, and a : R-N -> R-N is a given vector function. We provide sufficient conditions for the nonexistence of global weak solution under suitable initial data. Next, we extend our study to the system of nonlinear coupled equations
i(alpha) D-C(0)t(alpha) u + Delta u = lambda vertical bar v vertical bar(p) + mu a(x) . del vertical bar v vertical bar(q), t > 0, x is an element of R-N,
i(beta) D-C(0)t(beta) v + Delta v = lambda vertical bar u vertical bar(kappa) + mu b(x) . del vertical bar u vertical bar(sigma), t > 0, x is an element of R-N,
where 0 < beta <= alpha < 1, lambda is an element of C\{0}, mu is an element of C, p > q > 1, kappa > sigma > 1, and a, b : R-N -> R-N are two given vector functions. Our approach is based on the test function method.