Space and time localization in the flow of 2 immiscible fluids through a porous-medium - energy methods applied to systems

Abstract

Since the beginnings of the 1980s some energy methods have been introduced as an alternative to comparison principles in order to prove space and time localization of solutions of suitable nonlinear parabolic or elliptic equations. The study of nonhomogeneous equations (i.e. with prescribed right-hand terms) was considered by the authors [(*) Recent advances in nonlinear elliptic and parabolic problems, Proc. Int. Conf., Nancy/France 1988, Pitman Res. Notes Math. Ser. 208, 3-14 (1989; Zbl 0696.35090), and Dokl. Akad. Nauk SSSR 303, No.3, 524-529 (1988; Zbl 0684.35025)] proving new results on the retention of the free boundary (separating the region of the domain where the solution vanishes). In this work we wish to explain how to extend the results obtained in (*) for scalar equations to the case of systems of equations (even of different types).