Solvability of a mixed problem for the nonlinear Schrödinger equation

Existence and uniqueness theorems are established for a generalized solution of a mixed problem for the nonlinear Schrödinger equation in the presence of dissipation in the space $L_\infty(0,T;\overset\circ W{}^1_2(G))$ and $L_\infty(0,T;\overset\circ W{}^1_2(G)\cap W^2_2(G))$.
The method of proving uniqueness of a solution is based on the assumption of the existence and boundedness in $t\in[0,T]$ of the integral of a solution $\int_G\exp(\varkappa|u|^p)\,dx$ for some $\varkappa>0$, where $p$ is the degree of nonlinearity in the equation.
Bibliography: 16 titles.