Blow-up of sign-changing solutions for a one-dimensional nonlinear diffusion equation

Abstract

This paper is concerned with a spatially inhomogeneous semilinear diffusion equation on a bounded interval under the Neumann or Dirichlet boundary conditions. Assuming that the nonlinearity satisfies a rather general growth condition, we consider the blow-up and global existence of sign-changing solutions. It is shown that for some nonnegative integer k depending on the linearized operator at a trivial solution, the solution blows up in finite time if an initial value changes its sign at most k times, whereas there exist stationary solutions with more than k zeros. The proof is based on the intersection number principle combined with a topological method.