Stabilization of the solutions of boundary-value problems for the multidimensional heat-conduction equation (Q581733)

The authors consider the equation \[ \partial_ tu=\Delta u,\tag{1} \] , in the domain \(R_{xx}=\{(t,x): t>0, x=(x',x_ n), x'\in \mathbb R^{n-1},x_ n>0\}\), with the conditions \[ u(0,x)=0, x\in \mathbb R^ n_+=\{x: x'\in \mathbb R^{n-1},x_ n>0\},\tag{2} \] and \[ \sum_{| k'| +k_ n\leq r}b_{k',k_ n}\partial^{k'}_{x'}\partial_{x'}^{k_ n}u|_{x_ n=0}=g(t,x'),\tag{3} \] in \(\mathbb R_+=\{(t,x'), t>0, x'\in \mathbb R^{n-1}\}\), \((k'=(k_ 1,...,k_{n-1})\), \(| k'| =k_ 1+...+k_ n).\) The function \(g(t,x')\) has the following properties: (a) \(g(t,x')\) is continuous and bounded in \(\mathbb R_+.\) (b) \(g(t,x')\to g_ 0(x')\) for \(t\to \infty\), uniformly with respect to \(x'\in \mathbb R^{n-1}\); this property is named the stabilization property of \(g(t,x').\) It is proved that a bounded solution \(u(t,x)\) for the problem (1), (2), (3) uniformly converges in \(\mathbb R^ n_+\) to a bounded harmonic function \(U_ 0(x)\).