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Abstract
We study the properties of space localization of weak solutions of the equation
which appears in the mathematical description of filtration of an ideal barotropic gas in a porous medium. The functions and
are assumed to satisfy the nonstandard growth conditions:
,
,
,
,
with some positive constants
and measurable bounded functions
,
,
. It is shown that if
,
, and
,
meet certain regularity requirements, then every weak solution possesses the property of finite speed of propagation of disturbances from the initial data. In the case that
in a ball
and
in
, the solutions display the waiting time property: if
with a positive exponent
, depending on
and
, and a sufficiently small
, then there exists
such that
in
.
Acknowledgements
The first author was partially supported by the Research [grant number 15-11-20019] of the Russian Science Foundation (Russia). The second author acknowledges the support of the Research Project MTM2010-18427, MICINN (Spain). Both authors were partially supported by the Research [grant number CAPES-PVE-88887.059583/2014-00], [grant number 88881.0303888/2013-01] (Brazil).
Notes
No potential conflict of interest was reported by the authors.
1 Notice an abuse of notation: from now on we follow suit and denote by the given exponent of nonlinearity instead of the unknown pressure
.