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Semigroup theory and applications

Course Type: 
PhD Course
Academic Year: 

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.

April 26,28 - Room A-138; May 3, 5 - Room A-132
Next Lectures: 

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