Schedule
Tuesday 12h30-14h (room 4.35) & Friday 14h30-16h30 (room 5.18)
Office Hours
After the Friday session or contact me directly.
Main Bibliography
A. Haraux, T. Cazenave, An introduction to semilinear evolution equations, Oxford University Press, 1998
L. C. Evans, Partial Differential Equations, AMS, 1998
S. Zheng, Nonlinear Evolution Equations, Chapman&Hall, 2004
E. Godlewski, P. Raviart, Hyperbolic Systems of Conservation Laws, 1991
Syllabus
Some tools from functional analysis: Sobolev spaces, Bochner integral, vector-valued Lp spaces
Semigroup theory: m-dissipative operators. Characterization in the Hilbertian case. Construction of the associate contraction semigroup. The Hille Yosida theorem. Smoothing effect. Perturbation of m-dissipative operators.
Semilinear theory: the inhomogeneous equation. Strong integral solution vs classical solution. Semilinear equations with locally Lipschitz nonlinearity. The semilinear heat equation: well-posedness, global existence, blow-up, asymptotic behavior. The semilinear Klein-Gordon equation: well-posedness, global existence, blow-up.
Compactness methods for linear and nonlinear problems: energy methods, Galerkin approximations. Monotone operators.
Scalar conservation laws.
Evaluation
Three written homework (50%) + Final project (50%)
Students may be called for a short oral examination on their homework. The final project constitutes a written report (20p. max) and an oral presentation.
Exercise list
The exercise list will be updated throughout the semester.