The author studies the Cauchy problem for the inhomogeneous semilinear parabolic equation \(\Delta u + u^p - u_t + w = 0\) on \({\mathbb M}^n\times (0,\infty)\) with initial condition \(u_0\geq 0\). Here \({\mathbb M}^n\) is a Riemannian manifold and \(\Delta\) is the Laplace-Beltrami operator on \({\mathbb M}^n\). The main result is the existence of an exponent \(p^*\) which is critical in the following sense. If \(1<p<p^*\), the problem has no global positive solution for any nonnegative \(w\not\equiv 0\) and any \(u_0\geq 0\), while if \(p>p^*\), then there exist \(w>0\) and \(u_0\geq 0\) such that the problem has a global positive solution. In the case \(p=p^*\) the problem has no global positive solution if \(w\not\equiv 0\), provided \({\mathbb M}^n\) has nonnegative Ricci curvature.
In the special case that \(w\equiv 0\) and \(M^*={\mathbb R}^n\) a similar phenomenon was discovered by H. Fujita [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.34002)]. The critical exponent found by Fujita is \((n+2)/n\). It is interesting that in the nonhomogeneous case \(w\not\equiv 0\) with \({\mathbb M}^n={\mathbb R}^n\), the critical exponent is \(n/(n-2)\), which is larger than Fujita’s exponent.