Abstract:

Besides the classical questions of existence and regularity of partial differential equations (PDEs), energy (or Lyapunov) functions can also be used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Matano (1988) constructed a Lyapunov function for quasilinear non-degenerate parabolic equations with gradient dependency. We modify Matano's method to construct an energy formula for fully nonlinear parabolic PDEs of non-degenerate or degenerate type with Dirichlet or nonlinear Robin boundary conditions. This is based on joint explorations with B. Fiedler and E. Beatriz.