Mathematics

Aim of the course is to provide the basic tools concerning the abstract approach to main (linear and semilinear) evolution equations: Schrodinger, wave, Klein-Gordon, heat - equations.

• Bochner integral, Pettis and Bochner theorems.
• Elements on unbounded operators: closed, dissipative and maximal dissipative operators.
• Fundamental theorems of semigroup theory.
• Hille theorem, infinitesimal generators, Cauchy problem for abstract equations, Duhamel formula.
• Hille-Yosida, Lumer-Phillips and Stone theorems, construction of (semi)groups associated to heat, wave, Klein Gordon and Schrödinger equations.
• Semilinear abstract problem, local solution, extension, global solution, continuous dependence on data, conservations laws.
• Special properties of Heat semigroup.
• Example of a nonlinear problem for Klein-Gordon equation. 
• Elements of interpolation theory (Three Lines and Riesz-Thorin theorem) and application to the Cauchy problem for nonlinear Schrödinger and Wave equation via contraction mapping theorem.
• Conservation laws.
• Global solutions, continuous dependence on data. 

The course is intended to give to students the main and basic tools used in Functional analysis to treat the most important evolution equations: Heat, Schrodinger, Wave and Klein-Gordon equations.