Partial Differential Equations and the Existence and Uniqueness Theorem for the Cauchy Problem, linear case

Abstract

If Ω of R2 is a open region, I an open interval and the linear equation of first order non homogeneous in its most general forma (x,y)ux + b(x,y)uy = c(x,y) and the initial condition u(σ(t),ρ(t))=f(t) that a, b and c are C1 class in Ω that contains the smoth curve γ=(σ(t),ρ(t)), t∈I, initial curve of problem. This type of problem is called a Cauchy Problem. To solve this Cauchy problem is fundamental the concept of characteristic curves of the equation, since these represent the starting point in the search for solution to the problem. Also, we will see that the way in which the characteristic curves intersect the initial curve (σ(t),ρ(t)) given determines whether the problem will have unique solution, endless solutions or if the solution does not exist. We will see the Existence and Uniqueness Theorem that gives us the necessary conditions for the existence and uniqueness of solution of Cauchy Problem cited. It is important to note that the Existence and Uniqueness Theorem guarantees just local results, since the behavior of the characteristic curves away from the initial curve can become to complex.