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Critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations. (English) Zbl 1422.35112

Let \(n\geq 3, \lambda_{*}=\frac{(n-2)^2}{4}\). In \(L^2(\mathbb R^n)\), the heat semigroup \(e^{-tH}\) generated by \(H=-\Delta-\frac{\lambda_{*}}{|x|^2},\) with domain \[ D(H)=\{u\in L^2\cap W^{2,2}(\mathbb R^n\backslash \{0\}): Hu\in L^2(\mathbb R^n),\ |x|^{-n(\frac{1}{2}-\frac{n-2}{2n})} \big|\log |x|\big|^{-1}u\in L^2(B_{\frac{1}{2}})\}, \] where \(B_{\frac{1}{2}}\) denotes the ball centered at the origin with radius \(\frac{1}{2}\), is investigated. The extension of the semi-group \(e^{-tH}\) to the Lorentz spaces \(L^{p,\sigma}(\mathbb R^n)\) with \(\frac{2n}{n+2}<p<\frac{2n}{n-2}\), and \(1\leq\sigma\leq\infty,\) \(p=\frac{2n}{n+2}\) and \(\sigma =1\), and \(p=\frac{2n}{n-2},\ \sigma=\infty,\) is stated.
Dissipative estimates for \(\|e^{-tH} u_0\|_{L^{q,1}(\mathbb R^n)},\ \frac{2n}{n+2}<q<\frac{2n}{n-2}\) or \(\|e^{-tH}u_0\|_{L^{\frac{2n}{n-2},\infty}(\mathbb R^n)},\) in terms of \(\|u_0\|\) in suitable spaces, are stated (Theorem 1.3).
As an application, an existence theorem for a time global weak solution of problem \[ \partial_t u-\Delta u -\frac{\lambda_{*}}{|x|^2} u=u^p,\quad t>0,\ x\in\mathbb R^n\tag{1} \] with the initial condition \[ u(0)=u_0,\quad x\in\mathbb R^n \] is obtained, namely: Let \(1+\frac{4}{n+2}<p<1+\frac{4}{n-2}.\) Then there exists a constant \(\epsilon_0>0\) such that for any \(u_0\in L^{\frac{n}{2}(p-1)}(\mathbb R^n)\) with \(\|u_0\|_{L^{\frac{n}{2}(p-1)}}\leq \epsilon_0\), there exists a time global weak solution \(u\in C([0,\infty);L^{\frac{n}{2}(p-1)}(\mathbb R^n))\) for problem (1), (Theorem 1.4).

MSC:

35K55 Nonlinear parabolic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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