Weak asymptotic methods for scalar equations and systems

In this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslov asymptotic analysis. We obtain a sequence of functions which tend to satisfy the equation(s) in the weak sense in the space variable and in the strong sense in the time variable. To this end we reduce the partial differential equations to a family of ordinary differential equations in a classical Banach space. For scalar equations we prove that the initial value problem is well posed in the L1L1 sense for the approximate solutions we construct. Then we prove that this method gives back the widely accepted solutions when they are known. For systems we obtain existence in the general case and uniqueness in the analytic case using an abstract Cauchy–Kovalevska theorem.