Abstract
We study the hyperbolic type differential inequality
u(tt)(t, x, y) - E(l)u(t, x, y) >= vertical bar u(t, x, y)vertical bar(p), (t, x, y) is an element of (0, infinity) x D-1 x D-2
under the boundary conditions
(t, x, y) >= f (x), (t, x, y) is an element of (0,infinity) x partial derivative D-1 x D-2,
u(t, x, y) >= g(y), (t, x, y) is an element of (0,1) x partial derivative D1 x partial derivative D-2,
where p > 1, D-k = {z is an element of R-k(N) : vertical bar z vertical bar >= 1}, k = 1, 2, N-k >= 2, f is an element of L-1(partial derivative D-1), g is an element of L-1(partial derivative D-2), and L-l, l is an element of R, is the Grushin operator
L(l)u - Delta xu + vertical bar x vertical bar(2l)Delta(y)u.
We obtain sufficient conditions depending on p, l, N-1, N-2, f, and g, for which the considered problem admits no global weak solution. We discuss separately the four cases: N-1 = N-2 = 2; N-1 = 2, N-2 >= 3; N-1 >= 3, N-2 = 2; N-1,N-2 >= 3.