On the behaviour of Colombeau's generalized functions at a point: Applications to semilinear systems

Abstract

Abstract. We define the quasiasymptotics at zero (or at any other point) of a Colombeau's generalized function, element of script G sign, and show that this definition restrected on a Schwartz's distribution embedded into script G sign gives the well-known notion of the quasiasymptotics at zero and, in a special case, the value at zero. We analyze the quasiasymptotics of a Cauchy problem for a strictly semilinear hyperbolic system and show that under suitable assumptions on the non-linear term, the behaviour of the solution is determined by the behaviour of initial data.