Convergence of solutions of one-dimensional semilinear parabolic equations
Hiroshi Matano · Kyoto journal of mathematics · 1978
The idea of co-limiting sets borrowed from the theory of dynamical systems has been employed in many authors' works including [1] and [5 ] to investigate the asymptotic properties of solutions of autonomous parabolic initial-boundary value problem s. In the case of parabolic systems, it is well-known that the w-limiting set of a solution often contains plural elements (in fact infinitely many elements). But, to the best of our knowledge, it has not yet been made clear in the case of single equations whether there exists such a solution as has plural co-limiting points. The present paper forms part of the answer to this question. That is, we show that the co-limiting set of any solution contains at most one element providing that the space dimension is one. This result leads to the conclusion that in the case of single onedimensional equations any solution that neither blows up in a finite time nor grows up as t—*oo should converge to some equilibrium solution as t tends to infinity.