On certain boundary value problems which contain a small parameter (Q1333688)

Let \((0,b)\equiv V_ t\) be a bounded interval of the real axis \(t\); \(\mathbb{H}_ t= L_ 2(V_ t)\). Let also \(\mathbb{H}'\) be a separable Hilbert space. In the space \(\mathbb{H}= \mathbb{H}_ t\otimes \mathbb{H}'\) we consider the second order differential-operator equation (which contains a small parameter \(\varepsilon>0\)) with respect to ``time \(t\)'' of the form \[ L_ \varepsilon u_ \varepsilon\equiv \varepsilon {d^ 2 u_ \varepsilon\over dt^ 2}+ 2A_ 1 {du_ \varepsilon\over dt}- A_ 2 u_ \varepsilon= f.\tag{L}\(_ \varepsilon\) \] Here \(A_ j: \mathbb{H}'\to \mathbb{H}'\) are unbounded operators of specific structure. In addition, there is assumed that for every \(t\in [0,b]\) the functions \(u_ \varepsilon(t)\) take their values in \(\mathbb{H}'\), and \(f(t)\) is a given element in the Hilbert space \(\mathbb{H}\). Failure to demand the spectrum of operators \(A_ j\) to obey standard requirements results in conditions of other form, i.e., non-local, boundary (by \(t\)) conditions \[ \mu_ 1 u_ \varepsilon(t)\bigl|_{t=0}- u_ \varepsilon(t)\bigr|_{t= b}= 0,\quad \mu_ 2D_ t u_ \varepsilon(t)\bigl|_{t=0}- D_ t u_ \varepsilon(t)\bigr|_{t= b}= 0,\leqno{(\Gamma_ \varepsilon)} \] which we must set for correctly stated solvability of equation \((\text{L}_ \varepsilon)\) in \(\mathbb{H}\). Here \(u_ j\) are complex parameters, and \(0\leq |\mu_ j| \leq+\infty\). Our aim is to elucidate under what choice of parameters \(\mu_ j\) and under what decompositions of points of spectrum of operators \(A_ j\) on \(\mathbb{C}\) the problem \((\text{L}_ \varepsilon)\)-\((\Gamma_ \varepsilon)\) and the corresponding non-perturbed problem \((\text{L}_ 0)\)-\((\Gamma_ 0)\) will be correctly solvable? On the other hand, we want to show the presence of functions of bilateral boundary layer type in the asymptotics of the solution \(u_ \varepsilon\) of problem \((\text{L}_ \varepsilon)\)-\((\Gamma_ \varepsilon)\). In addition, the conditions, guaranteeing that \(u_ \varepsilon\) tends to \(u_ 0\) with \(\varepsilon\to 0\), as well as this converging characteristic, are of our interest, too. On some examples we illustrate the distinguishing peculiarities of the singularly perturbed non-local problem in comparison with the classical Cauchy problems. In particular, it will be established that in the Cauchy problem for a special form of equation \((\text{L}_ \varepsilon)\) the boundary layer does not exist at all.