# Evolution Equations and Applications

Thesis

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 Differential Equations (PDEs) with time t as one of the independent variables and arise from many fields of Mathematics as well as Physics, Mechanics and Material Sciences (e.g., Systems of Conservation Law from Dynamics, Navier-Stokes and Euler equations from Fluid Mechanics, Diffusion 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, first we review the basic notions of Functional Analysis and Differential Analysis, secondly we study the theory of semigroups of bounded linear operators, and thirdly we consider Linear Evolution Equations (with emphasis on the difference between the finite dimensional and the infinite 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), t> 0 ; u(0) = u 0 dt where A is a linear operator that generates a C 0 -semigroup and f satisfies 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.