Some Regularizations of Semilinear Evolution Equations M. Ahsan, G. Moro¸ sanu Central European University, Department of Mathematics and its Applications, Nador u. 9, H-1051 Budapest, Hungary ABSTRACT We consider in a real Hilbert space the Cauchy problem (P 0 ): u ′ (t)+ Au(t)+ Bu(t)= f (t), 0 ≤ t ≤ T ; u(0) = u 0 , where −A is the generator of a C 0 -semigroup of linear contractions and B is a smooth nonlinear operator. Next we associate with (P 0 ) the problem (P ε 1 ): −εu ′′ (t)+u ′ (t)+Au(t)+Bu(t)= f (t), 0 ≤ t ≤ T ; u(0) = u 0 ,u(T )= u 1 , where ε> 0 is a small parameter. Existence, uniqueness and higher regularity for both (P 0 ) and (P ε 1 ) are investigated and an asymptotic expansion for the solution of problem (P ε 1 ) is established, showing the presence of a boundary layer near t = T . Some open problems are formulated, such as the case of the boundary condition u ′ (T )= u T in (P ε 1 )(instead of the above one), higher order asymptotic expansions. Keywords: semilinear evolution equation, elliptic-like regularization, higher reg- ularity of solutions, singular perturbation, asymptotic expansion 1. INTRODUCTION Let H be a real Hilbert space with scalar product (·, ·) and the induced norm ‖·‖. Denote by (P 0 ) the following Cauchy problem  u ′ (t)+ Au(t)+ Bu(t)= f (t), 0 ≤ t ≤ T, (E) u(0) = u 0 , (IC ) (P 0 ) where T> 0 is a given time instant, u 0 ∈ H is a given initial state, f : [0,T ] → H , A : D(A) ⊂ H → H is a linear operator, such that Further author information: (Send correspondence to G. Moro¸ sanu.) G. Moro¸ sanu: E-mail: morosanug@ceu.hu M. Ahsan: E-mail: ahsanmaths@gmail.com