New applications of monotonicity methods to a class of non-monotone parabolic quasilinear sub-homogeneous problems

The main goal of this survey is to show how some monotonicity methods related with the subdifferential of suitable convex functions lead to new and unexpected results showing the continuous and monotone dependence of solutions with the respect to the data (and coefficients) of the problem. In this way, this paper offers ‘a common roof’ to several methods and results concerning monotone and non-monotone frameworks.Besides to present here some new results, this paper offers also a peculiar review to some topics which attracted the attention of many specialists in elliptic and parabolic nonlinear partial differential equations in the last years under the important influence of Häım Brezis .To be more precise,the model problem under consideration concerns to positive solutions of a class of doubly nonlinear diffusion parabolic equations with some sub-homogeneous non-monotone forcing terms