PDE and extrafunctions
Based on the first author's construction of hyperreal numbers and hyperfunctionals [Lect. Notes Pure Appl. Math. 238, 81--117, Dekker, New York (2004; Zbl 1094.46508)]. The present paper investigates solvability of linear first order Cauchy problems in a similar setting of generalized functions, the space of extrafunctions. It is shown that solutions exist under mild assumptions, although uniqueness is not attainable. Finally, the relationship between extrafunctions and Schwartz-, Egorov-, and Colombeau generalized functions is explored. It should be noted, though, that several statements on Colombeau generalized functions included in the article are incorrect (e.g., it is claimed that the product of smooth functions is not preserved when embedding into the Colombeau algebra although in fact this is one of the distinguishing features of the construction). A similar criticism has to be made on the handling of the interrelationship between the various solution concepts.
scientific article; zbMATH DE number 2144257
scientific article