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Asymptotic behavior of solutions of semilinear heat equations on \(S^ 1\). (English) Zbl 0671.35039

Nonlinear diffusion equations and their equilibrium states II, Proc. Microprogram, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 13, 139-162 (1988).
[For the entire collection see Zbl 0643.00016.]
The author studies the initial value problem for the equation \[ u_ t=u_{xx}+f(u,u_ x),\quad x\in C^ 1={\mathbb{R}}/{\mathbb{Z}},\quad t>0. \] He investigates the asymptotic behaviour of the solutions. First he shows that any \(C^ 1\)-bounded solution tends either to a time-periodic solution or to a set of equilibria as \(t\to \infty\). Then he considers the case where the solution blows up in a finite time and proves that the blow-up set of any solution with nonconstant initial data is a finite set under certain conditions on f.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35B35 Stability in context of PDEs

Citations:

Zbl 0643.00016