Asymptotic behaviour of solutions of semilinear parabolic equations

The asymptotic behaviour of solutions of a second-order semilinear parabolic equation is analyzed in a cylindrical domain that is bounded in the space variables. The dominant term of the asymptotic expansion of the solution as $t\to+\infty$ is found. It is shown that the solution of this problem is asymptotically equivalent to the solution of a certain non-linear ordinary differential equation.
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