On the necessary conditions of global existence to a quasilinear inequality in the half-space

We consider functions u(t, x) satisfying the inequality: partial derivative u/partial derivative t greater than or equal to L[u(p)] + \u\(q), p > 0, q > 1, q > p, for all t greater than or equal to 0, x epsilon R-n, where L[v]:= Sigma(\alpha\=m) D-alpha(a(alpha)(t,x,v)v) is a differential operator of order m, a(alpha)(t,x,v) epsilon L-infinity. We prove that u(t,x) drop 0 if 1 < q, 0 < p < q less than or equal to p + m/n, u epsilon L-loc(q), integral(Rn) u(0, x) dx less than or equal to 0. This result generalizes some results of H. Fujita, K. Hayakawa, V.A. Galaktionov, S.P. Kurdyumov, A.P. Mikhailov, A.A. Samarskii, V.A. Kondratiev and S.D. Eidelman. We prove also a similar result for systems of inequalities. Another our theorem generalizes a theorem of P. Meier. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.