2020/07/14 by Rahul Raju Pattar, Pattar, Rahul Raju, N. Uday Kiran +1
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2007.07153
We investigate the behavior of the solutions of a class of certain strictly\nhyperbolic equations defined on [0,T]\× Rn in relation to a class of\nmetrics on the phase space. In particular, we study the global regularity and\ndecay issues of the solution to an equation with coefficients polynomially\nbound in x and with their t-derivative of order textnormalO(t-q),\nwhere q \∈ \[1,\(3)/(2)\). For this purpose, an appropriate\ngeneralized symbol class based on the metric is defined and the associated\nPlanck function is used to define an infinite order operator to perform\nconjugation. We demonstrate that the solution not only experiences a loss of\nregularity (usually observed for the case of coefficients bounded in x) but\nalso a decay in relation to the initial datum defined in a Sobolev space\ntailored to the generalized symbol class. Further, we observe that a precise\nbehavior of the solution could be obtained by making an optimal choice of the\nmetric in relation to the coefficients of the given equation. We also derive\nthe cone conditions in the global setting.\n