TY - JOUR
T1 - Some Problems of Nonlinear Schrodinger Equations with the Effect of Dissipation
AU - Guo Doling, Wang Lireng
JO - Journal of Partial Differential Equations
VL - 3
SP - 1
EP - 23
PY - 1990
DA - 1990/03
SN - 3
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5802.html
KW - effect of dissipation
KW - global generalized solution
KW - global smooth solution
KW - asymptotic behavior
KW - blow up
KW - Sobolev inequality
KW - strong differential function
KW - optimal constant
AB - We first consider the initial value problem of nonlinear Schrödinger equation with the effect of dissipation, and prove the existence of global generalized solution and smooth solution as some conditions respectively. Secondly, we disscuss the asymptotic behavior of solution of mixed problem in bounded domain for above equation. Thirdly, we find the “blow up” phenomenon of the solution of mixed problem for equation iu_t = Δu + βf(|u|²)u - i\frac{ϒ(t)}{2}u, \quad x ∈ Ω ⊂ R³, t > 0 i. e. there exists T_0 > 0 such that lim^{t→Γ_0} || ∇u || ²_{L_t(Ω)} = ∞. The main means are a prior estimates on fractional degree Sobolev space, related properties of operator's semigroup and some integral identities.