Über das homogene Dirichlet‐Problem bei nichtlinearen partiellen Differentialgleichungen vom Typ der Boussinesq‐Gleichung

摘要

AbstractSemilinear equations of Boussinesq type, e.g.utt+uxx−uxxxx+ (u2)xx= 0,utt+uxx−uxxxx+uxuxx= 0, or certain equations containing the squared wave operator, e.g.uxxtt−uk= 0,kϵNk≥ 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 Sobole