A New Critical Phenomenon for Semilinear Parabolic Problems

We study the Cauchy problem of the inhomogeneous semilinear parabolic equations 1u C up − ut C w D 0 on Mn 0;1‘ with initial value u0 0, where Mn is a Riemannian manifold with possibly nonnegative Ricci curvature. There is an exponent p which is critical in the following sense. When 1 < p p , the above problem has no global positive solution for any nonnegative w D wx‘ not identically zero and for any u0 0; when p > p the problem has a global positive solution for some w D wx‘ > 0 and some u0 0.