The Method of Characteristics

This chapter is dedicated to the method of characteristics. We explicitly solve first order equations of the type \(\displaystyle \partial _t u+a(t,x) \partial _x u=h(t,x,u). \) Moreover, we show that if the coefficient a depends on the unknown u \(\displaystyle \partial _t u+a(t,x, u) \partial _x u=h(t,x,u) \) the catastrophe of the gradient may occur, namely characteristics cross and there is the creation of discontinuities in finite time even if the initial datum is analytic.