Seminar on Qualitative Theory of Differential Equations - Eiji Yanagida (22.2.2018)
Thursday 22.2.2018 at 14:00 hod., Lecture room M/223
By: Pavol Quittner
Eiji Yanagida (Tokyo Institute of Technology):
Blow-up of sign-changing solutions for a one-dimensional semilinear parabolic equation
This talk is concerned with a nonlinear parabolic equation on a bounded interval with the Dirichlet or Neumann boundary condition, where the nonlinearity is superlinear and spatially inhomogeneous. Under rather general conditions on the nonlinearity, we consider the blow-up and global existence of sign-changing solutions. It is shown that there exists a nonnegative integer $k$ such that the solution blows up in finite time if the initial value changes its sign at most $k$ times, whereas there exists a stationary solution with more than $k$ zeros. This result is an extension of Mizoguchi-Yanagida (1996) which dealt with an odd and spatially homogenous nonlinearity. The proof is based on an intersection number argument combined with a topological method.