Asymptotically homogeneous solutions to differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter group
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We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive. The infinitesimal matrix may contain a nilpotent part. In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have asymptotically homogeneous solutions in the critical case.
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- 6.I. M. Gel’fand and G. E. Shilov, Generalized Functions and Operations on Them (Fizmatgiz, Moscow, 1959); Engl. transl: Generalized Functions, Vol. 1: Properties and Operations (Academic, New York, 1964).Google Scholar