Premium
Über das homogene Dirichlet‐Problem bei nichtlinearen partiellen Differentialgleichungen vom Typ der Boussinesq‐Gleichung
Author(s) -
Warnecke G.,
Hsiao G. C.
Publication year - 1987
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.1670090133
Subject(s) - mathematics , sobolev space , dirichlet problem , hilbert space , mathematical physics , mathematical analysis , bounded function , boundary value problem , pure mathematics
Semilinear equations of Boussinesq type, e.g. u tt + u xx − u xxxx + ( u 2 ) xx = 0, u tt + u xx − u xxxx + u x u xx = 0, or certain equations containing the squared wave operator, e.g. u xxtt − u k = 0, k ϵ N k ≥ 2, are studied. A generalized boundary value problem on bounded domains can be treated using Hilbert space methods. The linear parts of these equations are not elliptic, the latter not even hypoelliptic. A mountain pass lemma is used to prove the existence of nontrivial weak solutions. These solutions are obtained in anisotropic Sobolev spaces.