Nonexistence of Solutions of Higher-order Nonlinear Non-gauge Schrodinger Equation

A nonexistence result is proved of the space higher-order nonlinear Schrodinger equation i partial derivative(t)u - (-Delta)(m) (vertical bar u vertical bar(n-1) u) = lambda vertical bar u vertical bar(p), x is an element of R-N, t > 0, where m > 1, n > 1 and p > n. Our method of proof rests on a judicious choice of the test function in the weak formulation of the equation. Then, we obtain an upper bound of the life span of solutions. Furthermore, the necessary conditions for the existence of local or global solutions are provided. Next, we extend our results to the 2 x 2 - system i partial derivative(t) u - (-Delta)(m)u = lambda vertical bar v vertical bar(p), x is an element of R-N, t > 0, i partial derivative(t) v - (-Delta)(m)v = delta vertical bar v vertical bar(q), x is an element of R-N, t > 0, where m > 1, p, q > 1, and lambda, delta is an element of C\{0}.

Automated summary

In this paper, a nonexistence result is proved for the higher-order nonlinear Schrodinger equation. The upper bound of the lifespan of solutions and the necessary conditions for the existence of local or global solutions are obtained. Moreover, the results are extended to a 2 x 2 system.