On asymptotics of the Green function and Poisson kernels of a mixed parabolic problem in a cone. I
The author considers the initial-boundary value problem: \(\sum_{2bk+| \alpha | \leq 2bm}a_{\alpha k}D_ x^{\alpha}\partial_ t^ ku=0\) on \(K\times (0,\infty)\), \(\sum_{2bk+| \alpha | \leq m_ j}b_{j\alpha k}D_ x^{\alpha}\partial_ t^ ku=g_ j\) \((j=1,...,mb)\) on (\(\partial K\setminus 0)\times (0,\infty)\), where K is a cone with a vertex at the origin of \({\mathbb{R}}^ n\), b is a fixed positive integer and \(a_{\alpha k}\), \(b_{j\alpha k}\) are positiv-homogeneous functions. Under some conditions imposed on differential operators he proves existence of the unique solution to the above problem in an appropriate class of functions. Some special solutions of the homogeneous problem are also given. Similar results are obtained for an adjoint parabolic problem. These results are derived by considering an associated parametrized elliptic problem. In the latter case formulas for an asymptotic form of the solutions, and for the coefficients of the expansions are established as well.
scientific article; zbMATH DE number 4202852