BLOW-UP OF SOLUTIONS WITH SIGN CHANGES FOR A SEMILINEAR DIFFUSION EQUATION

This paper is concerned with the initial-boundary value problem [GRAPH ICS] with the Dirichlet, Neumann, or periodic boundary condition. Here lambda > 0 is a parameter, and f is an odd function of u satisfying f '(0) > 0 and some convexity condition. Let z(U) be the number of times of sign changes for U is an element of C[0, 1]. It is shown that ther e exists an increasing sequence of positive numbers {lambda(k)}(k) = 0 ,1,2,... such that any solution with z(u(0)) = k blows up in finite ti me if lambda greater than or equal to lambda(k) and there exists a glo bal solution with z(u(0)) = k if lambda < lambda(k). (C) 1996 Academic Press, Inc.