Geodesic
The Cauchy problem for a~degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data
Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 563-580

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Given a degenerate parabolic equation of the form $\rho(x) u_t=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+\rho(x)u^p$ with a source and inhomogeneous density, we consider the Cauchy problem with an initial function slowly tending to zero as $|x| \to \infty$. We find conditions for the global-in-time existence or non-existence of solutions of this problem. These conditions depend essentially on the behaviour of the initial data as $|x|\to \infty$. In the case of global solubility we obtain a sharp estimate of the solution for large values of time.
Keywords: inhomogeneous density, degenerate parabolic equation, blow-up, slowly decaying initial function. 
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     author = {A. V. Martynenko and An. F. Tedeev and V. N. Shramenko},
     title = {The {Cauchy} problem for a~degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data},
     journal = {Izvestiya. Mathematics },
     pages = {563--580},
     publisher = {mathdoc},
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     number = {3},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a5/}
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A. V. Martynenko; An. F. Tedeev; V. N. Shramenko. The Cauchy problem for a~degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data. Izvestiya. Mathematics , Tome 76 (2012) no. 3, pp. 563-580. http://geodesic.mathdoc.fr/item/IM2_2012_76_3_a5/