Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 97, pp. 1-19.
Title: Nonexistence of solutions to systems of higher-order semilinear
inequalities in cone-like domains
Authors: Abdallah El Hamidi (Univ. de La Rochelle, France)
Gennady G. Laptev (Steklov Mathematical Institute, Russia)
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 $\mathbb{R}^N$, $t>0$.
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.