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Very weak solutions to hypoelliptic wave equations

Authors
  • Ruzhansky, Michael
  • Yessirkegenov, Nurgissa
Publication Date
Jan 01, 2020
Source
Ghent University Institutional Archive
Keywords
Language
English
License
Green
External links

Abstract

In this paper we study the Cauchy problem for the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups when the time-dependent non-negative propagation speed is regular, Holder, and distributional. For Holder coefficients we derive the well-posedness in the spaces of ultradistributions associated to Rockland operators on graded groups. In the case when the propagation speed is a distribution, we employ the notion of "very weak solutions" to the Cauchy problem, that was already successfully used in similar contexts in [12] and [20]. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique "very weak solution" in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples include the time dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or p-evolution equations for higher order operators on R-n or on groups, the results already being new in all these cases.

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