TY - JOUR

T1 - On the absence of global solutions for quantum versions of Schrödinger equations and systems

AU - Jleli, Mohamed

AU - Kirane, Mokhtar

AU - Samet, Bessem

N1 - Funding Information:
The first and third authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP-1436-034 .
Funding Information:
The first and third authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP-1436-034.
Publisher Copyright:
© 2018 Elsevier Ltd

PY - 2019/2/1

Y1 - 2019/2/1

N2 - 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 0q 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 01, 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.

AB - 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 0q 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 01, 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.

KW - Global weak solution

KW - Nonexistence

KW - Quantum calculus

KW - Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=85055537281&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2018.10.010

DO - 10.1016/j.camwa.2018.10.010

M3 - Article

AN - SCOPUS:85055537281

SN - 0898-1221

VL - 77

SP - 740

EP - 751

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 3

ER -