Elliptic-like regularization of semilinear evolution equations

Abstract

Consider in a real Hilbert space the Cauchy problem ($P_0$): $u^{\prime }\left(t\right)+Au\left(t\right)+Bu\left(t\right)=f\left(t\right),0\le t\le T;u\left(0\right)=u_0$, 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: ($P_1^{\epsilon }$): $-\epsilon u^{″}\left(t\right)+u^{\prime }\left(t\right)+Au\left(t\right)+Bu\left(t\right)=f\left(t\right),0\le t\le T;u\left(0\right)=u_0,u\left(T\right)=u_1$, 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 $t=T$.