Geodesic
Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 2, pp. 245-255

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The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered: $$ \rho(x)\frac{\partial u}{\partial t}=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+u^p. $$ The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained. 
@article{ZVMMF_2007_47_2_a7,
     author = {A. V. Martynenko and A. F. Tedeev},
     title = {Cauchy problem for a~quasilinear parabolic equation with a~source term and an inhomogeneous density},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {245--255},
     publisher = {mathdoc},
     volume = {47},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_2_a7/}
}
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A. V. Martynenko; A. F. Tedeev. Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 2, pp. 245-255. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_2_a7/