Volume 3, Issue 3
Some Problems of Nonlinear Schrodinger Equations with the Effect of Dissipation

Guo Doling, Wang Lireng

DOI:

J. Part. Diff. Eq.,3(1990),pp.1-23

Published online: 1990-03

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  • Abstract

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.

  • Keywords

effect of dissipation global generalized solution global smooth solution asymptotic behavior blow up Sobolev inequality strong differential function optimal constant

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@Article{JPDE-3-1, author = {Guo Doling, Wang Lireng}, title = {Some Problems of Nonlinear Schrodinger Equations with the Effect of Dissipation}, journal = {Journal of Partial Differential Equations}, year = {1990}, volume = {3}, number = {3}, pages = {1--23}, abstract = { 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.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5802.html} }
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.
Guo Doling, Wang Lireng. (1970). Some Problems of Nonlinear Schrodinger Equations with the Effect of Dissipation. Journal of Partial Differential Equations. 3 (3). 1-23. doi:
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