Blow-up in a reactive-diffusive medium with a moving heat source

A concentrated source of heat is allowed to move in one-dimension through a reactive-diffusive medium. For rather general motion, it is shown that if the speed remains sufficiently high then a blow-up can be avoided; whereas, if the speed remains sufficiently low then a blow-up must eventually occur. Other criteria involving the displacement and velocity of the heat source are found to guarantee a blow-up. Special examples of (i) rapid deceleration from one velocity level to a lower level and (ii) a simple periodic motion are examined.