Asymptotic energy equipartition for the wave equation on homogeneous trees
The paper deals with the wave propagation on a nonhomogeneous tree \({\mathcal X}\) of degree \(q+1\). Such a tree is a connected graph with no loops in which every vertex is adjacent to \(q+1\) other vertices. It is endowed with a natural distance function \(d\), where \(d(x,y)\) is the number of edges in the unique path from \(x\) to \(y\), and with a natural Laplace operator \({\mathcal L}\) defined by \[ {\mathcal L}f=f(x)-{\frac{1}{q+1}}\sum_{y:d(x,y)=1}f(y). \] If \(q=1\), then \({\mathcal X}\) may be identified with the set \({\mathbb Z}\) of all integers. The Cauchy problem for the modified wave equation \[ \partial^{2}_{t}u(x,t)=-({\mathcal L}-b)u(x,t); \quad u(x,0)=u_{0}(x),\quad \partial_{t}u(x,0)=u_{1}(x) \tag{1} \] on \({\mathcal X}\times {\mathbb R}^{+}\) is considered. As the main result it is proved in Theorem 2 that if the initial data \(u_{0}\) and \(u_{1}\) are in \(L^{2}({\mathcal X})\) and if \(u\) is the solution of the initial problem (1), then \[ \lim_{t\to +\infty}{\mathcal K}_{u}(t)=\lim_{t\to +\infty}{\mathcal P}_{u}(t) ={\frac{1}{2}}{\mathcal E}_{u}.\tag{2} \] Here \({\mathcal E}_{u}\) is the energy functional: \[ {\mathcal E}_{u}\equiv {\mathcal E}_{u}(t)={\tfrac{1}{2}} \left(\| d_{t}u(.,t)\|^{2}_{2}+<{\mathcal L}u(.,t),u(.,t)> -b\| u(.,t)\|^{2}_{2}\right), \] which does not depend on \(t\), and \({\mathcal K}_{u}(t)\) and \({\mathcal P}_{u}(t)\) are its kinetic and parabolic components: \[ {\mathcal K}_{u}(t)={\tfrac{1}{2}}\| d_{t}u(.,t)\|^{2}_{2},\;{\mathcal P}_{u}(t)={\tfrac{1}{2}}<({\mathcal L}-b)u(.,t),u(.,t)>. \] The corresponding result in the case \(q=1\) and \(b=0\) is also proved in Theorem 3 provided that \(u_{0}\) and \(u_{1}\) are in \(L^{2}({\mathbb Z})\). The proof of the result in (2) is based on the Fourier transform on \({\mathcal X}\), while the one in the case \(q=1\) and \(b=0\) is based on the classical Fourier transform on \(L_{1}({\mathbb Z})\).
scientific article; zbMATH DE number 1271756