Subject: EQUAZIONI DI EVOLUZIONE (A.A. 2021/2022)
Unit Equazioni di evoluzione
Related or Additional Studies (lesson)
The course gives a basic knowledge on the weak formulation of non-linear evolution equations, with particular emphasis on reaction-diffusion equations and damped wave equations. When the problem is well posed and the solution operator satisfies suitable properties, the student will be able to set the model in the framework of infinite dimensional dynamical systems. Whenever possible, the longtime behavior of the system will be described by global attractors.
Prerequisites are Calculus 1 and 2, Theory of Banach and Hilbert spaces, basic theory of Lebesgue spaces. The knowledge of Sobolev spaces is recommended but not mandatory.
The main topics are: Banach spaces valued functions; evolution problems for (non-linear) partial differential equations as Cauchy problems in infinite dimensional dynamical spaces. Global analysis of the longterm behavior of dissipative evolution equations: attractors.
Teaching consists usually in lectures at the (possibly virtual) blackboard: there subjects are throughly developed and duly commented. Student questions and comments are welcome and encouraged. Although not mandatory, attendance is strongly recommended. The course is held in Italian. All teaching technical and practical information, as well as the teaching material, will be uploaded to the Dolly platform. The student is invited to register and consult this platform regularly. (*) due to the COVID19 health situation, the methods of delivery of the lessons could undergo changes and be carried out remotely on a platform to be indicated, however favoring the virtual synchronous mode.
The student is required to present a 45 minutes talk devoted to course topics or in-depth analysis of close subjects. During the oral exam, questions on the whole course contents may be asked. The exams might be either in presence or on online, depending on the COVID19 epidemic’s evolution. The score, out of thirty, will be known at the end of the exam.
1. Knowledge and understanding
Through lectures and individual study, knowledge of the main concepts in Evolution Equations with particular emphasis on nonlinear parabolic and hyperbolic equations.
2. Applying knowledge and understanding
Through classroom exercises, support activities and individual work, ability to model and solve mathematical problems by analytical tools.
3. Making judgements
Attitude towards a methodological approach that leads to verify claims and methods through rigorous arguments.
Ability to self-assessment of their skills and abilities.
4. Comunication skills
Ability to deal with a dialectical discourse in a timely and consistent way, arguing with precision.
Lasting acquisition of mathematical knowledge to be used at any time of their cultural journey.
5. Learning skills
Attitude towards a methodological approach that leads to an improvement in the method of study in order to deepen learning capacity.
H. Brezis, Analisi funzionale. Teoria e applicazioni. Liguori
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, NY, 2011
J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer