New critical behaviors for semilinear wave equations and systems with linear damping terms

Abstract

We consider the inhomogeneous semilinear wave equation with linear damping ∂ttu−Δu+∂tu=|u|p+w(t,x) in , where p>1 and w≢0. A general criterium for the nonexistence of global weak solutions is established. In the particular case w=ω(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 (p∗(N)=∞ if N∈{1,2}) and ω ≥ 0, then the problem admits no global weak solution; if p>p∗(N), then a global solution exists, for some ω>0 and suitable initial values. Moreover, when N ≥ 3 and p>p∗(N), we prove that for σ<N, if and ω(x) ≥ C|x|−σ for |x| large, then there is no global weak solution; if σ∗ ≤ σ<N, then a global solution exists for some ω>0 with ω(x) ≤ C|x|−σ for |x| large, and suitable initial values. Next, we extend our study to the case of systems.