Weighted weak formulation for a nonlinear degenerate parabolic equation arising in chemotaxis or porous media

Abstract

This paper is devoted to the mathematical analysis of a degenerate nonlinear parabolic equation. This kind of equations stems either from the modeling of a compressible two phase flow in porous media or from the modeling of a chemotaxis-fluid process. In the degenerate equation, the strong nonlinearities are technically difficult to be controlled by the degenerate dissipative term because the equation itself presents degenerate terms of order 0 and of order 1. In the case of the sub-quadratic degeneracy of the dissipative term at one point, a weak and classical formulation is possible for the expected solutions. However, in the case of the degeneracy of the dissipative term at two points, we obtain solutions in a weaker sense compared to the one of the classical formulation. Therefore, a degenerate weighted formulation is introduced taking into account the degeneracy of the dissipative term.