This course deals with some type evolution equations arising in applied sciences, from the point of view of the existence, uniqueness, and properties of the solutions. These equations involve maximal monotone operators and subdifferentials of convex and lower semicontinuous functions.
Expected learning outcomes
The course provides analytical tools to investigate evolution equations related to applied problems, possibly involving strong nonlinearities (e.g. due to the presence of multivalued operators). The skills refer to the theory of abstract evolution equations, approximation (and dicretization) techniques, a priori estimates on the (approximated) solutions, and passage to the limit (by compactness and semicontinuity arguments).
-Linear unbounded operators. The Hille-Yosida theorem for the linear case. -Elements of Sobolev spaces and generalized derivatives. -Conjugate convex functions. -Maximal monotone operators. Subdifferential operators. -Variational method (Lions method, in Hilbert triplets) for equations of the form u'+Au=f. Application to the heat equation. The Faedo-Galerkin approximation. -Strong and weak solvability of the equation u'+Au=f (A maximal monotone or subdifferential operator). The Hille-Yosida theorem in the nonlinear case. -Applications to the Allen-Cahn equation and to the semilinear damped wave equation