On the Poisson kernel of some pseudo-differential boundary value problem (Q2761550)

Let \(x'=(x_1,\ldots,x_{n-1})\in \mathbb{R}^{n-1}\); \(\mathbb{R}_{+}^{n}=\{x=(x_1,\ldots,x_{n})\in \mathbb{R}^{n}|x_{n}>0\}\), \(n>1\); \(\Pi^{+}=\{(t,x)|t\in(0,\infty), x\in \mathbb{R}^{n}_{+}\}\); \(\Pi'=\{(t,x')|t\in(0,\infty), x'\in \mathbb{R}^{n-1}\}\); \(\gamma\geq 1\); \(0<\beta<n-1\); the function \(u:\Pi'\to \mathbb{R}\) has one continuous derivative on \(t\in (0,\infty)\) and \([\gamma]+1\) continuous derivatives on \(x_{i}, i=1,\ldots,n\). The author considers the boundary value problem NEWLINE\[NEWLINE\partial_{t}u(t,x)=a^2\partial^2_{x_{n}}u(t,x)+b(A_{\gamma}u)(t,x),\quad (t,x)\in \Pi^{+},NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(t,x)|_{t=0}=0,\;x\in \mathbb{R}_{+}^{n}, \quad A_{\beta}\partial_{x_{n}}u(t,x)|_{x_{n}=0}=f(t,x'),\;(t,x')\in\Pi',NEWLINE\]NEWLINE where \(a\neq 0\), \(b>0\) are constants, \(A_{\gamma}\), \(A_{\beta}\) are pseudodifferential operators. The explicit form of the solution and Poisson kernel of the considered boundary value problem is obtained.