On the absence of global solutions for quantum versions of Schrödinger equations and systems

We, first, consider the quantum version of the nonlinear Schrödinger equation i^qD_q|tu(t,x)+Δu(qt,x)=λ|u(qt,x)|^p,t>0,x∈R^N,where 0<q<1, i^q is the principal value of i^q, D_q|t is the q-derivative with respect to t, Δ is the Laplacian operator in R^N, λ∈ℂ∖{0}, p>1, and u(t,x) is a complex-valued function. Sufficient conditions for the nonexistence of global weak solution to the considered equation are obtained under suitable initial data. Next, we study the system of nonlinear coupled equations i^qD_q|tu(t,x)+Δu(qt,x)=λ|v(qt,x)|^p,t>0,x∈R^N,i^qD_q|tv(t,x)+Δv(qt,x)=λ|u(qt,x)|^m,t>0,x∈R^N,where 0<q<1, λ∈ℂ∖{0}, p>1, m>1, and u(t,x),v(t,x) are complex-valued functions. The used approach is based on an extension of the test function method to quantum calculus.