Zur Abhängigkeit der Lösung von ut + (h(u))x = 0 von ihren Anfangswerten

Abstract

AbstractThe above equation has some remarkable properties. In general a global solution exists in a weak sense only, and this solution is not reversible in time. Furthermore it is known, that the solutions for different initial values can coincide for all t ⩾ t0 > 0, and the set of the initial values with this property is convex. Conditions assuring that this set contains only one element are given. This means a weak form of time‐reversibility.As a global solution exists only in the weak sense, the classical question concerning dependence of the solution on the initial values needs some modification. This problem is dealt with in suitable L1‐norms. It is shown, that the L1‐norm of the difference of two weak solutions with respect to the space variable does not increase in time.