Abstract

In this paper we study the Existence and Uniqueness of solutions for the following Cauchy problem:A2u″(t)+A1u′(t)+A(t)u(t)+M(u(t))=f(t),   t∈(0,T)       (1)u(0)=u0;   A2u′(0)=A212u1;                      where A1 and A2 are bounded linear operators in a Hilbert space H, {A(t)}0≤t≤T is a family of self-adjoint operators, M is a non-linear map on H and f is a function from (0,T) with values in H.As an application of problem (1) we consider the following Cauchy problem:k2(x)u″+k1(x)u′+A(t)u+u3=f(t)   in   Q,              (2)u(0)=u0;   k2(x)u′(0)=k2(x)12u1                     where Q is a cylindrical domain in ℝ4; k1 and k2 are bounded functions defined in an open bounded set Ω⊂ℝ3,A(t)=−∑i,j=1n∂∂xj(aij(x,t)∂∂xi);where aij and a′ij=∂∂tuij are bounded functions on Ω and f is a function from (0,T) with values in L2(Ω).