Electron. J. Diff. Eqns., Vol. 2002(2002), No. 97, pp. 1-19.

Nonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains

Abdallah El Hamidi & Gennady G. Laptev

Abstract:
In this paper, we obtain nonexistence results for global solutions to the system of higher-order semilinear partial differential inequalities
$$\displaylines{ \frac{\partial^k u_i}{\partial t^k}-\Delta (a_i (x,t) u_i (x,t)) \geq t^{\gamma_{i+1}}|x|^{\sigma_{i+1}} |u_{i+1} (x,t) |^{p_{i+1}}, \quad 1 \leq i \leq n, \cr u_{n+1}=u_1, }$$
in cones and cone-like domains in $\mathbb{R}^N$, $t greater than 0$. Our results apply to nonnegative solutions and to solutions which change sign. Moreover, we provide a general formula of the critical exponent corresponding to this system. Our proofs are based on the test function method, developed by Mitidieri and Pohozaev.

Submitted March 10, 2002. Published November 14, 2002.
Math Subject Classifications: 35G25, 35R45.
Key Words: nonexistence, blow-up, higher-order differential inequalities, critical exponent.

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Abdallah El hamidi
Laboratoire de Mathematiques
Universite de La Rochelle
Avenue Michel Crepeau
17000 La Rochelle, France
e-mail: aelhamid@univ-lr.fr
Gennady G. Laptev
Department of Function Theory
Steklov Mathematical Institute
Gubkina 8, 117966 Moscow, Russia
e-mail: laptev@home.tula.net

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