Elliptic-like regularization of semilinear evolution equations

[摘要] Consider in a real Hilbert space the Cauchy problem (P-0): u' (t)+Au +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. We associate with (P-0) the following problem: (P-1(epsilon): -epsilon u ''(t) + u'(t) + Au(t) + Bu(t) = f (t), 0 <= t <= T; u(0) = u(0), u(T) = u(1), where epsilon > 0 is a small parameter. Existence, uniqueness and higher regularity for both (P-0) and (P-1(epsilon)) are investigated and an asymptotic expansion for the solution of problem (P-1(epsilon)) is established, showing the presence of a boundary layer near t = T. (C) 2012 Elsevier Inc. All rights reserved.

[发布日期] 2012-12-15

[关键词] Semilinear evolution equation;Elliptic-like regularization;Higher regularity of solutions;Singular perturbation;Asymptotic expansion