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On the possible decay of solutions of second order partial differential equations. (Russian) Zbl 0736.35009

The author is interested in the asymptotic behavior of second order partial differential equations on unbounded domains. More precisely the basic three types of p.d.e.’s (elliptic, parabolic and hyperbolic) are considered separately and the question of decaying at infinity faster than exponentially is asked. For elliptic problems the author is interested in the Poisson problem in \(\mathbb{R}^ 2\), \(\Delta u=q(x)\) where \(q\) is assumed to be bounded. The result proved is amazingly sharp namely it is shown that solutions behaving as \(O(\exp(-c| x|^{4/3}))\) (for some positive \(c\)) at infinity exist while a solution behaving as \(O(\exp(-c| x|^{4/3}))\) (for all positive \(c\)’s) vanishes identically. For the evolution equations it is considered the time asymptotic behavior for equations defined on compact manifolds without boundary. Superexponential decay at infinity is shown not to be possible for hyperbolic equations while for the parabolic ones the answer depends on the spectral properties of the associated elliptic problem.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L10 Second-order hyperbolic equations
35K10 Second-order parabolic equations
35J15 Second-order elliptic equations
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