Convergence, asymptotic periodicity, and finite-point blow-up in one- dimensional semilinear heat equations. (English) Zbl 0692.35013

This paper studies the semilinear heat equation for u(x,t) with a force f(u,t), with Dirichlet, Neumann or periodic boundary conditions. The problem consists principally of two cases: those solutions which exist for all time, and those which blow-up in a finite time. For either case, the paper studies the asymptotic behaviour of u(x,t). For a T-periodic force it is shown that any bounded global solution converges asymptotically to a T-periodic solution with a specific spatial structure. For the blow-up case it is shown that the blow-up set is finite and \(\lim_{t\to t_ 0}u(x,t)=\phi (x)\) with \(\phi\) (x) being a smooth function having at most a finite number of singular points.
Reviewer: B.Straughan


35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
Full Text: DOI


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