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This is a basic course about evolutionar partial differential equations. We will deal with parabolic and linear hyperbolic equations of the second order.
Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (11.09.2013)
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Exercises: the students are supposed to solve homeworks (written). The deadlines for each set of homework will be announced at least one week in advance. To obtain credits, students must obtain at least 50% of the maximal number of points form each homework. According to POS, Art. 8, Par. 2 it is not possible to repeat this.
The credit from the exercices is required to participate at the exam.
Exam: based mostly on the material discussed at the lecture and in the written homeworks. It will have the written form, in some cases can be followed by the oral part.
Most recommended sources are the book PDE's by L.C. Evans and Lecture notes which will be available on the web page. Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (18.02.2024)
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L. C. Evans: Partial Differential Equations, AMS, 2010. E. Zeidler: Nonlinear Functional Analysis and its Applications II/A, (Chapters 23 and 24), Springer, 1990. Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (11.09.2013)
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The credit from the exercises is required to be allowed to participate at the exam.
The exam will have written part (based on problems studied at the exercises and during the lecture) and it may be followed by additional oral part to clarify the written part. The exercises will be sent by email or be available on the webpage, the lectures are either covered by the material of the book PDE's aby L.C. Evans or by the Lecture notes. All this required theoretical material will be covered at the lectures. Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (18.02.2024)
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Sobolev spaces: embedding theorems, trace theorems. (with proofs) Nonlinear scalar elliptic equations of second order: weak formulation, uniqueness and existence theory, monotone operators, regularity, minimum and maximum principles. Introduction to calculus of variations: fundamental theorem of calculus of variations, weak lower semicontinuity of convex functionals, relation to the elliptic equation (Sobolev-) Bochner spaces: continuous and compact (Aubin-Lions theorem) embeddings. (with proofs) Semigroup theory: Hille-Yosida theorem, application to linear parabolic and hyperbolic equations. Nonlinear parabolic equations of second order: existence, uniqueness and regularity of solution. Last update: Kaplický Petr, doc. Mgr., Ph.D. (07.01.2019)
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