Time decay estimates of solutions to the mixed problem for heat equations in a half space (Q2434901)

In this paper the authors consider the following initial-boundary problem for the heat equation of the form \[ \partial_tu-\Delta u=0,\quad t>0,\quad x\in{\mathbb R}_+^n \tag{1.1} \] \[ Bu:=(c\partial x_n+\sum_{j=1}^{n=1}b_j\partial_{x_j}+d)u|_{x_n=0}=0,\quad t>0,\;\;x\in{\mathbb R}^n \tag{1.2} \] \[ u|_{t=0}=u_0(x);\quad x\in{\mathbb R}_\tau^n \tag{1.3} \] where \({\mathbb R}_+^n=\{x'=(x,x_n): x'\in{\mathbb R}^{n-1},x_n>0\}\) is the half space, \(u=u(t,x),d\), \(c\), and \(b_j\) are complex constants. \noindent The main aim of this paper is to give the integral of the solutions to the mixed problem (1.1)--(1.3) and to obtain by use of these expressions the time decay estimate of the solutions of the problem (1.1)--(1.3) under the strong Lopatinski condition. \noindent Taking the fact that the authors consider the equation (1.1) in the space \({\mathbb R}^{n-1}\), they use the Fourier transform with respect to the tangential variable \(x\). In the proof they use the \(L^p\) boundedness of the singular integral operators of Calderon-Zygmund, the Hölder inequality, the Riesz-Thorin interpolation theorem and some strong estimates. \noindent The authors notice that their method can be extended to obtain the time decay estimates of solutions to the mixed problem for the Stokes equation in a half space.