Abstract
We consider the nonlinear hyperbolic-type inequality
partial derivative(tt)u - Delta u >= vertical bar u vertical bar(p) + vertical bar del u vertical bar(q) + w(x), (t, x) is an element of (0, infinity) x R-+(N)
under the Dirichlet-type boundary condition
u(t, x', 0) >= 0, (t, x') is an element of (0, infinity) x RN-1,
where N >= 2, p, q > 1, and w >= 0. An optimal Fujita-type result is obtained for this problem. Namely, we prove that when w belongs to a certain functional space W, and p < 1+ 2/N-1 or q < 1 + 1/N, then there exists no global weak solution, while global solutions exist for some w is an element of W when p > 1 + 2/N-1 and q > 1 + 1/N. The obtained result shows an interesting phenomenon of discontinuity of the Fujita critical exponent, jumping from p = 1 + 2/N-1 to p = infinity as q reaches the value 1 + 1/N from above.