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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: V. A. Galaktionov and I. V. Kamotski
Journal: Trans. Amer. Math. Soc. 362 (2010), 4117-4136
MSC (2000): Primary 35K55, 35K40
Published electronically: March 17, 2010
MathSciNet review: 2608398
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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$,

   $\displaystyle \mbox{$ u_t= \Delta u + \frac c{\vert x\vert^2} u \quad \mbox{in} \quad \Omega \times (0,T), \quad u \big\vert _{\partial \Omega}=0, $}$

in the supercritical range

   $\displaystyle \mbox{$ c> c_{\rm 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

   $\displaystyle \mbox{$ V(x) = \frac c{\vert x\vert^2} \mapsto V_n(x)=\min \big\{ \frac c{\vert x\vert^2}, n \big\} (n \ge 1) $}$

diverges, and, as $ n \to \infty$,

$\displaystyle u_n(x,t) \to +\infty$   in$\displaystyle \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

   $\displaystyle \mbox{$ u_t =-(-\Delta)^m u + \frac c{\vert x\vert^{2m}} u \quad ... ...quad \Omega \times (0,T), \quad \mbox{where} \quad c>c_{\rm 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

$\displaystyle u_0(x)$   is continuous at $ x=0$ and$\displaystyle \par \quad 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{\vert u\vert^p}{\varepsilon^{2m}+\vert x\vert^{2m}}$ with $ p>1$. Applications to other, linear and semilinear, Schrödinger and wave PDEs are discussed.


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Additional Information

V. A. Galaktionov
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: vag@maths.bath.ac.uk

I. V. Kamotski
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
Email: ivk20@maths.bath.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04855-5
PII: S 0002-9947(10)04855-5
Keywords: Parabolic equations with singular potentials, Hardy inequality, nonexistence, regular approximations, oscillatory solutions
Received by editor(s): February 4, 2008
Published electronically: March 17, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.