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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 \(R\).

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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