Geodesic
The heat equation in a medium with nonuniformly distributed nonlinear heat sources or absorbers
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1983), pp. 20-24

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For a heat conduction equation with non-linear sources whose action intensifies at infinity, it is proved that the Cauchy problem has no locally bounded solution in any time interval. In the case of non-linear heat absorption decaying at infinity, necessary and sufficient conditions are found for instantaneous compactification of the support of the solution for the Cauchy problem. 
@article{VMUMM_1983_3_a3,
     author = {A. S. Kalashnikov},
     title = {The heat equation in a medium with nonuniformly distributed nonlinear heat sources or absorbers},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {20--24},
     publisher = {mathdoc},
     number = {3},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_1983_3_a3/}
}
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A. S. Kalashnikov. The heat equation in a medium with nonuniformly distributed nonlinear heat sources or absorbers. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 3 (1983), pp. 20-24. http://geodesic.mathdoc.fr/item/VMUMM_1983_3_a3/