Blow-Up of Solutions with Sign Changes for a Semilinear Diffusion Equation

This paper is concerned with the initial-boundary value problem utsuxxqlf u. in 0, 1.= 0, `., u x, 0.su0 x. in 0, 1., with the Dirichlet, Neumann, or periodic boundary condition. Here l )0 is a parameter, and f is an odd function of u satisfying f 9 0.)0 and some convexity condition. Let z U. be the number of times of sign changes for UgCw0, 1x. It is shown that there exists an increasing sequence of positive numbers lk4ks0, 1, 2, . . . such that any solution with z u0.sk blows up in finite time if l Glk , and there exists a global solution with z u0.sk if l -lk.