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

Firstly, we consider the asymptotics of the solution to Equation (2) with initial conditions from *S*+.

In the first step (in the segment *t* ∈ [0, *T*]), the function *u*(*t* − *T*) is greater than or equal to *pR*, which is why on this segment, Equation (2) has the form of (4), and the solution to Equation (2) has the form of (5). As in Section 2, we identify that Expression (5) is greater than *pR* for all *t* > 0; therefore, Equation (2) has the form of (4) for all *t* > 0. This is why the solution of (2) with initial conditions from *S*+ does not depend on the values of *f* and *b* and has the form of (5) for all *t* > 0.

Similarly, the solution of Equation (2) with initial conditions from *S*<sup>−</sup> does not depend on the values of *f* and *d* and for all *t* > 0 has the form of (7).

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