A very singular solution of the heat equation with absorption
The authors consider the nonlinear parabolic equation \((1)\quad u_ t- \Delta u+u^ p=0,\) \(t>0\), \(x\in {\mathbb{R}}^ n\) with condition \(u(x,t)>0\) on \({\mathbb{R}}^ n\times (0,\infty)\) and initial condition \(u(x,t)|_{t=0}=0\) for all \(x\in {\mathbb{R}}^ n\setminus \{0\}\). It is shown that the problem has a unique solution of the form \(u(x,t)=(1/t^{1/(p-1)})f(| x| /\sqrt{t})\) with a smooth function f(\(\eta)\) on [0,\(\infty)\) satisfying \(f'(0)=0\) and \(\eta^{2/(p-1)}f(\eta)\to 0\) as \(\eta\to \infty\). This solution u(x,t) is more singular than the fundamental solution of the heat equation \((1/(4\pi t)^{n/2})e^{-| x|^ 2/t}\) or the solution of nonlinear equation (1) with initial condition \(u(x,t)|_{t=0}=\delta (x)\) at the point (0,0).
scientific article; zbMATH DE number 4019592
scientific article