On the localization of solutions of doubly nonlinear parabolic equations with nonstandard growth in filtration theory

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 .

Keywords:

• doubly nonlinear parabolic equation
• variable nonlinearity
• space localization
• waiting time

AMS Subject Classifications:

• 35K65
• 35K67
• 35B36

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 .