ABSTRACT
This chapter presents some methods of using weighted integrals in the qualitative theory of P.D.E.s. It begins by presenting the weight function approach to uniqueness on bounded and unbounded domains. The chapter introduces the weight function approach to the stability of the Cauchy problem for the Korteweg-Burgers-de Vries equation. The aim of the weight function approach to uniqueness on unbounded domains is to obtain uniqueness without prescribing, at infinity, conditions on the solutions and on its spatial derivatives which are “too strong”. The introduction of weight functions depending both on space variables and time, can be very useful. One might term the method the method of weighted cross-sectional integrals. It is believed that this is the first example in the literature of such a method.
