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Some nonexistence results for higher-order evolution inequalities in cone-like domains
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## Some nonexistence results for higher-order evolution inequalities in cone-like domains

### Gennady G. Laptev

**Abstract.**
Nonexistence of global (positive) solutions of semilinear
higher-order evolution inequalities
\begin{equation*}
\frac{\partial^k u}{\partial t^k}-\Delta u^m\ge |u|^q,\quad
\frac{\partial^k u}{\partial t^k}-\Delta u\ge |x|^\sigma u^q,\quad
\frac{\partial^ku}{\partial t^k}-\text{\rm div}\, (|x|^\alpha Du)\ge u^q
\end{equation*}
with $k=1,2,\dots$, in cone-like domains is studied. The critical exponents
$q^*$ are found and the nonexistence results are proved for $1 < q\le q^*$.
Remark that the corresponding result for $k=1$ (parabolic problem) is sharp.

*Copyright 2001 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**07** (2001), pp. 87-93
- Publisher Identifier: S 1079-6762(01)00098-1
- 2000
*Mathematics Subject Classification*. Primary 35G25; Secondary 35R45, 35K55, 35L70
- Received by the editors April 7, 2001
- Posted on October 15, 2001
- Communicated by Guido Weiss
- Comments (When Available)

**Gennady G. Laptev**

Department of Function Theory, Steklov Mathematical Institute, Gubkina Street 8, Moscow, Russia

*E-mail address:* `laptev@home.tula.net`

The author was supported in part by RFBR Grant #01-01-00884.

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