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

Abstract

We, first, consider the quantum version of the nonlinear Schrödinger equation iqDq|tu(t,x)+Δu(qt,x)=λ|u(qt,x)|p,t>0,x∈RN,where 0<q<1, iq is the principal value of iq, Dq|t is the q-derivative with respect to t, Δ is the Laplacian operator in RN, 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 iqDq|tu(t,x)+Δu(qt,x)=λ|v(qt,x)|p,t>0,x∈RN,iqDq|tv(t,x)+Δv(qt,x)=λ|u(qt,x)|m,t>0,x∈RN,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.