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The solution of anisotropic sixth‐order Schrödinger equation
Author(s) -
Su Hailing,
Guo Cuihua
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6009
Subject(s) - sobolev space , mathematics , uniqueness , anisotropy , initial value problem , schrödinger equation , order (exchange) , mathematical analysis , banach space , space (punctuation) , banach fixed point theorem , physics , quantum mechanics , finance , economics , linguistics , philosophy
This paper studies the local existence of solutions in Sobolev space for anisotropic sixth‐order Schrödinger‐type equation i u t + Δ u + ∑ i = 1 d ( a ∂x i4 u + b ∂x i6 u ) + c | u | α u = 0 , x ∈ R n , t ∈ R , 1 ≤ d < n , under the initial conditions u ( x ,0)= φ ( x ), x ∈ R n . In particular, when n =2 and d =1, we consider the global existence of solutions in Sobolev space for anisotropic sixth‐order Schrödinger equation. By using the Banach fixed point theorem, we obtain the existence, the uniqueness, the continuous dependence of the solution on the initial value, and the decay estimate of the global solution about such problems in anisotropic Sobolev spacesHy →s 1 , ρHz →s 2 , r .