This article is cited in 15 papers

The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data

A. V. Martynenko^a, An. F. Tedeev^b, V. N. Shramenko<sup>c

a</sup> Lugansk Taras Shevchenko National University
^b Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine
^c National Technical University of Ukraine "Kiev Polytechnic Institute"

Abstract: 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.

UDC: 517.946

MSC: 35K65, 35K15, 35B44

Received: 31.01.2011

DOI: 10.4213/im6847

 English version: Izvestiya: Mathematics, 2012, 76:3, 563–580

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