On a Φ-Laplacian Parabolic Equation with Singular and Convective Reaction

The aim of the talk is to analyze an evolutionary Φ- Laplacian problem, with singular and convective reactions: u_t -A u= f+g .The differential operator A considered is an elliptic operator, driven by a Young function Φ, while the reaction terms are Carathéodory functions obeying suitable growth conditions. The problem possesses three features of interest: • the operator A can be non-homogeneous, and with unbalanced growth; • the reaction term f is singular (i.e., it behaves like u^{−γ} with γ ∈ (0, 1)) and f(x, ·) can be non-monotone); • the reaction term g is convective (i.e., it depends on ∇u). We will first introduce the functional setting of the problem, by recalling the most relevant function spaces involved and their basic properties. Secondly, the main issues concerning both singular and convective terms are highlighted, together with the sub-solution and freezing techniques. Finally, we will briefly sketch the proof of our existence result, based on a priori estimates and a semi-discretization (in time) procedure. The seminar will have an introductory fashion, under the trend proposed by the cycle ASK (Analysis Student Kernel), for young analysis researchers, at the University of Bologna (https://sites.google.com/view/askbologna/home?authuser=1).