| dc.description.abstract | This project concerns Evolution Equations in Banach spaces and lies at the interface between
Functional Analysis, Dynamical Systems, Modeling Theory and Natural Sciences.
Evolution Equations include Partial Di erential Equations (PDEs) with time t as one of the
independent variables and arise from many elds of Mathematics as well as Physics, Mechanics
and Material Sciences (e.g., Systems of Conservation Law from Dynamics, Navier-Stokes and Eu-
ler equations from Fluid Mechanics, Di usion equations from Heat transfer and Natural Sciences,
Klein-Gordon and Schrödinger equations from Quantum Mechanics, Cahn-Hilliard equations and
Porous media equations from Material Sciences, Evolution equations with memory from Pharma-
cokinetics).
In this project, we present the fundamental theory of abstract Evolution Equations by using the
semigroup approach (which arises naturally from well-posed Cauchy problems: Theorem 2.2.6) and
Fixed-point methods. More precisely, rst we review the basic notions of Functional Analysis and
Di erential Analysis, secondly we study the theory of semigroups of bounded linear operators, and
thirdly we consider Linear Evolution Equations (with emphasis on the di erence between the nite
dimensional and the in nite dimensional case, that is due to domain restrictions) and moreover we
give existence results (in appropriate sense) for Semilinear Evolution Equations of the form
du
= Au + f (t, u),
dt
t>0 ;
u(0) = u0
where A is a linear operator that generates a C0 -semigroup and f satis es certain conditions. As
applications we start with the evolution equation ∂t u + ∂x u = 0 in R and then after we show the
existence of solutions to some Homogeneous Heat Equations, classical Wave equations, nonlinear
Heat Equation, and to some nonlinear Wave equation. | en_US |
