Abstract

In this paper the following Cauchy problem, in a Hilbert space H, is considered: (I+λA)u″+A2u+[α+M(|A12u|2)]Au=fu(0)=u0u′(0)=u1M and f are given functions, A an operator in H, satisfying convenient hypothesis, λ≥0 and α is a real number.For u0 in the domain of A and u1 in the domain of A12, if λ>0, and u1 in H, when λ=0, a theorem of existence and uniqueness of weak solution is proved.