Seminár z kvalitatívnej teórie diferenciálnych rovníc

Seminar on Qualitative Theory of Differential Equations

Thursday 22.2.2018 at 14:00 Lecture room M-223

Eiji Yanagida (Tokyo Institute of Technology):

Blow-up of sign-changing solutions for a one-dimensional semilinear parabolic equation

Abstract:

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.