Elliptic-like regularization of semilinear evolution equations

Consider in a real Hilbert space the Cauchy problem ( P 0 ): u ′ ( t ) + A u ( t ) + B u ( 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. We associate with ( P 0 ) the following problem: ( P 1 ε ): − ε u ″ ( t ) + u ′ ( t ) + A u ( t ) + B u ( 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 1 ε ) are investigated and an asymptotic expansion for the solution of problem ( P 1 ε ) is established, showing the presence of a boundary layer near t = T .