Existence of weak solutions for abstract hyperbolic-parabolic equations

In this paper we study the Existence and Uniqueness of solutions for the following Cauchy problem: A 2 u ″ ( t ) + A 1 u ′ ( t ) + A ( t ) u ( t ) + M ( u ( t ) ) = f ( t ) , t ∈ ( 0 , T ) ( 1 ) u ( 0 ) = u 0 ; A 2 u ′ ( 0 ) = A 2 1 2 u 1 ; where A 1 and A 2 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: k 2 ( x ) u ″ + k 1 ( x ) u ′ + A ( t ) u + u 3 = f ( t ) in Q , ( 2 ) u ( 0 ) = u 0 ; k 2 ( x ) u ′ ( 0 ) = k 2 ( x ) 1 2 u 1 where Q is a cylindrical domain in ℝ 4 ; k 1 and k 2 are bounded functions defined in an open bounded set Ω ⊂ ℝ 3 , A ( t ) = − ∑ i , j = 1 n ∂ ∂ x j ( a i j ( x , t ) ∂ ∂ x i ) ; where a i j and a ′ i j = ∂ ∂ t u i j are bounded functions on Ω and f is a function from ( 0 , T ) with values in L 2 ( Ω ) .