Convergence estimates for some abstract linear second order differential equations with two small parameters (Q2814648)

The authors consider in a Hilbert space \(H\) the linear differential equation NEWLINENEWLINE\[NEWLINE\varepsilon u^{\prime \prime}(t) + \delta u^{\prime}(t) + Au(t)=f(t),\quad 0<t<TNEWLINE\]NEWLINE NEWLINEwith initial conditions at \(t=0\) on \(u\) and \(u^{\prime}\), where \(\varepsilon\) and \(\delta\) are positive small parameters, and \(A\) is a linear, self-adjoint, strongly positive operator from \(D(A)=V\) to \(H\), with \(V\) a Hilbert subspace of \(H\), densely and continuously embedded in \(H\). The behavior of \(u=u(t;\varepsilon, \delta)\) is studied when either \(\varepsilon \rightarrow 0\) and \(\delta \geq \delta_0>0\) or when both parameters \(\varepsilon\) and \(\delta\) tend to zero. As expected \(u\) has a singular behavior near \(t=0\). The authors describe the boundary layer functions in both cases and establish sharp approximation estimates.