Qualitative Study of Nonlinear Parabolic Equations: an introduction
Jesús Ildefonso Díaz Díaz · Extracta Mathematicae · 2001
Before making explicit the structural assumptions on the data b,A, f, h and u0 let us mention some important special examples. Perhaps the simpler example isthe linear heat equation ut −∆u = f. (1) So, b(s) = s,A(x, u, ξ) = ξ and g ≡ 0. This is a typical example of linear partial differential equation of parabolic type usually considered in undergraduate courses (see, e.g., John [31]). A modern treatment starts by introducing the notion of weak solution or by its reformulation as an abstract Cauchy problem on a Banach space du dt (t) + Au(t) = f(t), u(0) = u0, ∗Partially sponsored by the DGES (Spain), project REN2000-0766.