I am only looking for an intuitive answer, so I won't provide any specific equations.
In this model: firm A and firm B compete on prices. Firm A is always better off choosing a smaller price than firm B. So firm A behaves similar to the firms in the standard Bertrand model.
Firm B is different. If firm A chooses a large enough price (say, $p_A > c$ for some boundary $c$), then firm B also starts to behave like firm A and wants to undercut the price of firm A. However, if $p_A < c$, then firm B doesn't care about firm $A$ and simply sets its monopoly price.
In this model, I cannot find a pure-strategy Nash equilibrium, because for any price chosen by the second firm B, firm A will find it optimal to undercut firm B's price. So there is no "unique" best-response by firm A.
Does there intuitively exist some sort of mixed-strategy equilibrium? If so, how might it be characterized? Who will be mixing, and over what range?