Asymptotic behavior of solutions of semilinear heat equations on S1
We study the dynamical behavior of the initial value problem for the equation u t = u xx + f(u, u x ), x ∈ S 1 =R/Z, t > 0. One of our main results states that any C 1-bounded solution approaches either a single periodic solution or a set of equilibria as t → ∞. We also consider the case where the solution blows up in a finite time and prove that under certain conditions on f the blow-up set of any solution with nonconstant initial data is a finite set.
KeywordsParabolic Equation Equilibrium Solution Neumann Boundary Condition Linear Parabolic Equation Liapunov Function
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