Oscillations for an Equation Arising in Groundwater Flow with the Relaxation Time

Abstract

Groundwater flow problems are mostly formulated by means of massbalance equation combined with Darcy's law. In this way, the flow is governed by a parabolic equation. To prevent inaccuracies which may result from this formulation, the Cattaneo approach can be utilized. The paper presents groundwater flow equation adopting the Cattaneo approach. In both 2D and 3D cases, the equation is of hyperbolic type and contains a constant known as relaxation time. The article focuses further on energy solutions defined on unbounded time interval. It is shown that under certain conditions, such solutions are oscillatory. The conditions sufficient to ensure the oscillatory solutions are derived. An upper bound for the oscillatory time is proved to be independent of the particular solution.

Keywords:

• modified Darcy's law
• Cattaneo approach
• relaxation time
• oscillatory solutions of an autonomous problem
• oscillatory time

AMS Subject Classification::

• 35B05
• 35L20
• 76S05
• 86A05