On nonexistence of Baras–Goldstein type for higher-order parabolic equations with singular potentials

Abstract:

The celebrated result by Baras and Goldstein (1984) established that the heat equation with singular inverse square potential in a smooth bounded domain $\Omega \subset \mathbb {R}^N$, $N \ge 3$, such that $0 \in \Omega$, \[ u_t= \Delta u + \frac c{|x|^2} u \;\; \text {in} \;\; \Omega \times (0,T), \;\; u \big |_{\partial \Omega }=0, \] in the supercritical range \[ c > c_{\mathrm {Hardy}}(1) = \big (\frac {N-2}2\big )^2, \] does not have a solution for any nontrivial $L^1$ initial data $u_0(x) \ge 0$ in $\Omega$ or for a positive measure. Namely, it was proved that a regular approximation of a possible solution by a sequence $\{u_n(x,t)\}$ of classical solutions of uniformly parabolic equations with bounded truncated potentials given by \[ V(x) = \frac c{|x|^2} \mapsto V_n(x)=\min \big \{ \frac c{|x|^2}, \, n \big \} \,\,\, (n \ge 1) \] diverges, and, as $n \to \infty$, \[ u_n(x,t) \to +\infty \quad \mbox {in} \quad \Omega \times (0,T). \] In the present paper, we reveal the connection of this “very singular” evolution with a spectrum of some “limiting” operator. The proposed approach allows us to consider more general higher-order operators (for which Hardy’s inequalities were known since Rellich, 1954) and initial data that are not necessarily positive. In particular it is established that, under some natural hypothesis, the divergence result is valid for any $2m$th-order parabolic equation with singular potential \[ u_t = -(-\Delta )^m u + \frac c{|x|^{2m}}\, u \;\; \mathrm {in} \;\; \Omega \times (0,T), \;\; \mathrm {where} \;\; c>c_{\mathrm {H}}(m), \,; m \ge 1, \] with zero Dirichlet conditions on $\partial \Omega$ and for a wide class of initial data. In particular, typically, the divergence holds for any data satisfying \[ u_0(x) \;\; \text {is continuous at $x=0$ and $u_0(0)>0$}. \] Similar nonexistence (i.e., divergence as $\varepsilon \to 0$) results are also derived for time-dependent potentials $\varepsilon ^{-2m}q(\frac {x}{\varepsilon }, \frac {t}{\varepsilon ^{2m}})$ and nonlinear reaction terms $\frac {|u|^p}{\varepsilon ^{2m}+|x|^{2m}}$ with $p>1$. Applications to other, linear and semilinear, Schrödinger and wave PDEs are discussed.