Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations

AbstractWe consider the initial value problem for the semilinear heat equation ut = uxx + f(u,t) (0 < x < L, t > 0) under the Dirichlet, the Neumann, or the periodic boundary conditions. We show that each solution—whether it exists globally for t > 0 or blows up in a finite time—possesses an “asymptotic profile” in a certain sense and tends to this profile as time increases. In the special case where f(u, t + T) ≡ f(u, t) for some T > 0, among other things, the above statement is interpreted as saying that any bounded global solution converges as t → ∞ to a time T-periodic solution having some specific spatial structure. In the case where the solution blows up in a finite time (say at t = t0), assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function, we prove that the blow-up set is a finite set and that limt↑t0 u(x, t) = ϑ(x) exists, with ϑ being a smooth function having at most finitely many singular points