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

• Published in: Journal of Differential Equations Vol. 78; no. 1; pp. 160 - 190
• Main Authors: Chen, Xu-Yan, Matano, Hiroshi
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.03.1989; Academic Press
• Subjects: Exact sciences and technology; Function theory, analysis; Mathematical methods in physics; Physics; Parabolic equation; Semi linear equation; Boundary value problem; Heat equation; Initial value problem; Partial differential equation; Solution
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/0022-0396(89)90081-8

We consider the initial value problem for the semilinear heat equation u t = u xx + 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 = t 0), 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 lim t↑ t 0 u( x, t) = ϑ( x) exists, with ϑ being a smooth function having at most finitely many singular points.