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On a heat problem involving the perturbed Hardy-Sobolev operator. (English) Zbl 1052.35067
Summary: Let $$\Omega$$ be a bounded domain in $$\mathbb{R}^n$$, $$n\geq 3$$ and $$0\in\Omega$$. It is known that the heat problem $$\partial u/ \partial t+L_{\lambda^*} u=0$$ in $$\Omega\times (0,\infty)$$, $$u(x,0) =u_0\geq 0$$, $$u_0\not\equiv 0$$, where $$L_{\lambda^*}: =-\Delta-\lambda^*/ | x|^2$$, $$\lambda^*:= \tfrac 14(n-2)^2$$, does not admit any $$H_0^1$$ solutions for any $$t>0$$. In this paper we consider the perturbation operator $$L_{\lambda^*q}: =-\Delta- \lambda^*q(x)/ | x |^2$$ for some suitable bounded positive weight function $$q$$ and determine the border line between the existence and non-existence of positive $$H_0^1$$ solutions for the above heat problem with the operator $$L_{\lambda^*q}$$. In dimension $$n=2$$, we have similar phenomena for the critical Hardy-Sobolev operator $$L^*:=-\Delta -(1/4| x|^2) (\log R/ | x|)^{-2}$$ for sufficiently large $$R$$.

##### MSC:
 35K20 Initial-boundary value problems for second-order parabolic equations 35R05 PDEs with low regular coefficients and/or low regular data 35A20 Analyticity in context of PDEs
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