Abstract
We consider the inhomogeneous semilinear wave equation with linear damping partial differential (tt)u-Delta u+ partial differential (t)u=|u|(p)+w(t,x) in (0,infinity)xRN, where p>1 and w not equivalent to 0. A general criterium for the nonexistence of global weak solutions is established. In the particular case w=omega(x), we obtain the critical exponent in the sense of Fujita (first critical exponent) and the critical exponent in the sense of Lee and Ni (second critical exponent) for the considered problem. Namely, we show that, if 1*(N)=infinity if N is an element of{1,2}) and omega >= 0, then the problem admits no global weak solution; if p>p*(N), then a global solution exists, for some omega>0 and suitable initial values. Moreover, when N >= 3 and p>p*(N), we prove that for sigma<N, if sigma<sigma*:=2pp-1 and omega(x) >= C|x|(-sigma) for |x| large, then there is no global weak solution; if sigma* <= sigma<N, then a global solution exists for some omega>0 with omega(x) <= C|x|(-sigma) for |x| large, and suitable initial values. Next, we extend our study to the case of systems.