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Energy flow in formally gradient partial differential equations on unbounded domains. (English) Zbl 1003.35085
The authors study the longtime behavior of solutions of the following damped hyperbolic equation in an unbounded domain \(\Omega=\mathbb R^N\): \[ \alpha u_{tt}+u_t=\Delta_xu+F(x,u),\quad \alpha\geq 0. \tag{1} \] In contrast to the case of bounded domains \(\Omega\) where equation (1) has a gradient structure and, thus, possesses a global Lyapunov function, the energy of a typical solution of (1) is infinite now and, therefore, we do not have finite Lyapunov function in this case. Nevertheless, as it is shown in the paper, the formal gradient structure of equation (1) imposes essential restrictions on the dynamics generated by this equation at least in the case \(N\leq 2\). In particular, the nonexistence of nontrivial time-periodic orbits is proven, the lower bounds for the time needed for a bounded trajectory to return in a small neighborhood of the initial point are given and the convergence on average of any bounded trajectory to the set of equilibria is established. Moreover, some counterexamples to these results in three-dimensional case (\(\Omega=\mathbb R^3\)) are also given.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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