In order to make a full use of the Hint 3, the following considerations might be useful.
First, suppose that the wire leftwards from a point Q is in the low-resistivity state, and rightwards – in the high-resistivity state. This means that far from Q (where we can neglect the heat transfer along the wire), the temperature of the wire T(x) is almost constant, i.e. independent of the coordinate x along the wire; let the temperatures to the left and to the right be respectively T_1, and T_2. A more complicated temperature profile will be formed in the immediate neighbourhood of Q. However, we can use the fact that the heat transfer equation determining the temperature profile T(x) is a linear inhomogeneous differential to show that T(x)-T(0), is an odd function (assuming that x=0 at Q). Hence, we can immediately express T(0) in terms of T_1 and T_2.
Second, notice that at the moment when the hottest point of the wire P is undergoing a phase transition, two fast process are triggered: widening of the high-resistivity region around P, and decreasing of the current strength.
Please submit the solution to this problem via e-mail to email@example.com. The next and final hint for the Problem 1 will be published at 13:00 GMT, 29th January 2023, together with the next updates of the intermediate results.