Very weak solutions to hypoelliptic wave equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Ruzhansky, Michael^1,2  Yessirkegenov, Nurgissa^1,3,4 

[1] Univ Ghent, Dept Math Anal Log & Discrete Math, Ghent, Belgium

[2] Queen Mary Univ London, Sch Math Sci, London, England

[3] Inst Math & Math Modelling, 125 Pushkin Str, Alma Ata 050010, Kazakhstan

[4] Imperial Coll London, Dept Math, 180 Queens Gate, London SW7 2AZ, England

关键词: Wave equation;    Rockland operator;    Graded Lie group;    Stratified group;    Heisenberg group;    Gevrey spaces;   

DOI  :  10.1016/j.jde.2019.09.020

来源: Elsevier

【 摘 要 】

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. (C) 2019 The Authors. Published by Elsevier Inc.

【 授权许可】

Free