Asymptotics of the Cauchy problem for thermal conductivity equation for large solution time (Q1363995)

Let \(u(t,x)\) be a solution to the heat equation \(\partial_tu=\Delta u\), \(x\in\mathbb{R}^n\), \(t>0\) with continuous and bounded initial function: \(|u(0,x)|< C\) for all \(x\in\mathbb{R}^n\). Suppose there exist constants \(\nu>0\), \(A_0\) and \(A_1\) such that \[ u(t,x)= A_0+{A_1\over t^\nu}+ {Q(t,x)\over t^\nu} \] with \(Q(t,x)\to 0\) as \(t\to\infty\) uniformly with respect to \(x\in\mathbb{R}^n\). Then the following is proved: For \(\alpha\geq 2[\nu]+1\) the Riesz spherical means \(S^\alpha_Ru(0,x)\) of order \(\alpha\) of the initial function have the following asymptotics for \(R\to\infty\): \[ S^\alpha_Ru(0,x)= A_0+{A_1C_1\over R^{2\nu}}+ {Q_1(R,x)\over R^{2\nu}} \] with \(Q_1(R,x)\to 0\) as \(R\to\infty\) uniformly with respect to \(x\in\mathbb{R}^n\), and \[ C_1=2^\nu\Gamma\Biggl({N+2\alpha+2\over 2}\Biggr)\Biggl/\Gamma\Biggl({N+2\alpha+ 2-2\nu\over 2}\Biggr). \]