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MATHEMATICA BOHEMICA, Vol. 127, No. 2, pp. 301-310 (2002)
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# Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

## P. Polacik

* P. Polacik*, Institute of Applied Mathematics, Comenius University, Mlynska dolina, 842 48 Bratislava, Slovakia; e-mail: ` polacik@fmph.uniba.sk`

**Abstract:**
We consider three types of semilinear second order PDEs on a cylindrical domain $\Omega \times (0,\infty )$, where $\Omega $ is a bounded domain in ${\R }^N$, $N\ge 2$. Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of $\Omega \times (0,\infty )$ is reserved for time $t$, the third type is an elliptic equation with a singled out unbounded variable $t$. We discuss the asymptotic behavior, as $t\to \infty $, of solutions which are defined and bounded on $\Omega \times (0,\infty )$.

**Keywords:** parabolic equations, elliptic equations, hyperbolic equations, asymptotic behavior, center manifold

**Classification (MSC2000):** 35B40, 35K55, 35L70, 35J25

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