**3. Asymptotics of Solutions in the Case** *b <* **0 and** *d <* **0**

Initially, we consider asymptotics of solution to Equation (2) with initial conditions from *S*+. In the first step (on the segment *t* ∈ [0, *T*]), function *u*(*t* − *T*) is greater or equal than *pR*, which is why in this segment Equation (2) has the form of (4). Therefore, for *t* ∈ [0, *T*], the solution of Equation (2) has the form of (5).

In this case, *<sup>d</sup>* < 0, so we obtain *<sup>u</sup>*(*t*) < *pL* for *<sup>t</sup>* ∈ [*δ*, *<sup>T</sup>* + *<sup>δ</sup>*], where *<sup>δ</sup>* = *<sup>O</sup>*(*λ*−1) at *λ* → +∞; therefore, Equation (2) has the form of (6) in the segment *t* ∈ [*T* + *δ*, 2*T* + *δ*]. As in the previous case, in the time segment *t* ∈ [*T*, *T* + *δ*], the solution *u*(*t*) depends on the values of the function *f* , but this dependence is smaller than the leading term of the asymptotics of the solution, and this leading term of the asymptotics of the solution coincides with the leading term of the asymptotics of the solution to the Cauchy problem (6),

$$
\mu(T) = p\_R e^{-T} + \lambda d(1 - e^{-T}).
$$

Hence, it follows that the solution of Equation (2) with initial conditions from set *S*+ has the form

$$u(t) = \lambda d(1 - e^{-T})e^{-(t-T)} + \lambda b(1 - e^{-(t-T)}) + o(\lambda). \tag{17}$$

Since *b* < 0 and *d* < 0, expression (17) is less then *pL* for all *t* ∈ [*T*, +∞). This is why Equation (2) has the form of (6) for all *t* ∈ [*T* + *δ*, +∞), and Formula (17) holds for all *t* ≥ *T*.

Now, we study asymptotics of the solution to Equation (2) with the initial conditions from *S*−.

On the segment *t* ∈ [0, *T*], the function *u*(*t* − *T*) is less than or equal to *pL*, which is why on this segment Equation (2) has the form of (6), and its solution has the form of (7).

As *b* < 0, we obtain *u*(*t*) < *pL*, and Equation (2) has the form of (6); until then, *u*(*t*) > *pL*. However, expression (7) is less than *pL* for all *t* > 0. This is why solution has the form of (7) for all *t* ∈ [0, +∞).

Therefore, in the case that *b* < 0 and *d* < 0, all solutions with initial conditions from sets *S*<sup>+</sup> and *S*<sup>−</sup> tend to the constant *λb* at *t* → +∞.
