Geodesic
The Cauchy problem in classes of increasing functions for the~equation of filtration with convection
Sbornik. Mathematics, Tome 186 (1995) no. 6, pp. 803-825

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We consider the Cauchy problem with a non-negative continuous initial function for the equation $$ u_t=(u^m)_{xx}+c(u^n)_x, $$ where $m>1$, $m\geqslant n\geqslant 1$ and $c$ is a positive constant. We prove a number of existence and uniqueness theorems for generalized solutions increasing at infinity for this Cauchy problem; we also investigate the behaviour of these solutions for large values of the time. 
@article{SM_1995_186_6_a2,
     author = {A. L. Gladkov},
     title = {The {Cauchy} problem in classes of increasing functions for the~equation of filtration with convection},
     journal = {Sbornik. Mathematics},
     pages = {803--825},
     publisher = {mathdoc},
     volume = {186},
     number = {6},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_6_a2/}
}
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A. L. Gladkov. The Cauchy problem in classes of increasing functions for the~equation of filtration with convection. Sbornik. Mathematics, Tome 186 (1995) no. 6, pp. 803-825. http://geodesic.mathdoc.fr/item/SM_1995_186_6_a2/