Abstract
We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation ∂tu=Δu+|u|p+tσw(x) in (0,T)×RN, where N≥2, p>1, σ>−1, σ≠0 and w≢0. The novelty of this paper lies in considering a forcing term (tσw(x)) which depends on both of time and space. We show that there is an exponent p⁎(σ) which is critical in the following sense: the solution of the above problem blows up in finite time when 1<p<p⁎(σ) and ∫RNw(x)dx>0, while global solution exists for suitably small initial data and w belonging to certain Lebesgue spaces when p≥p⁎(σ). Our obtained results show that the forcing term induces an interesting phenomenon of discontinuity of the critical exponent p⁎(σ). Namely, we found that limσ→0−p⁎(σ)≠limσ→0+p⁎(σ). Furthermore, limσ→0−p⁎(σ) coincides with the critical exponent of the above problem with σ=0.