Asymptotics of solutions of the heat equation in cones and dihedra (Q2909658)

Let \(K=\{ x'=(x_1,...,x_m):\, |x'|/|x'|\in \Omega \}\) be a cone in \(\mathbb R^m\), \(2\leq m\leq n\), and \(D=\{ x=(x',x''):\, x'\in K,\, x''\in \mathbb R^{n-m}\} \). The authors derive the asymptotics of the first boundary problem for the heat equation near vertices of cones and edges. They estimate the reminder term in weighted Sobolev spaces \(L_{p,q;\beta}(D\times R)\) with finite norm NEWLINE\[NEWLINE \| u \|_{L_{p,q;\beta}(D\times R)}=\left(\int_R\left( \int_D|x'|^{p\beta}|u(x,t)|^pdx\right)^{q/p}dt\right)^{1/q}. NEWLINE\]NEWLINE The paper generalized the previous results of the first author and \textit{V. G. Maz'ya} [Sov. Math. 31, No. 3, 49--57 (1987); translation from Izv. Vyssh. Uchebn. Zaved, Mat. 1987, No. 3 (298), 37--44 (1987; Zbl 0675.35048)] to the case \(p,q\neq 2\) and to the edges. The approach is based on asymptotic representations of Green's function from the author [Z. Anal. Anwend. 10, No. 1, 27--42 (1991; Zbl 0784.35040)].