## A very singular solution of the heat equation with absorption.(English)Zbl 0627.35046

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).
Reviewer: L.Kaljakin

### MSC:

 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations
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### References:

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