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Critical states and minima for an energy with second-order gradients. (English) Zbl 0961.34035

The authors deal with a family of ordinary differential equations \[ 2cu_{5x} + 2bu_{3x} + \psi''_0 (u) u_x = 0 \tag{1} \] where the subscripts denote differentiation with respect to \(x\) and the prime symbol denotes differentiation with respect to \(u.\) The real numbers \(b > 0\) and \(c > 0\) are given constants, \(u\) is a smooth function \(\mathbb{R} \to \mathbb{R}\) and \(\psi_0\) is a given potential whose properties depend on the physical problem under investigation. Equation (1) arises in the theory of thermodynamical equilibrium of second-order materials, i.e. material for which the free-energy density depends not only on the concentration \(u(x)\) but also on its first gradient \(u'(x)\) and its second gradient \(u''(x).\) The authors find all bounded solutions to (1) which are perturbation of homogeneous solutions \(u(x) = a\) for all \(x \in \mathbb{R}\) and such that for every \(x_0 \in \mathbb{R}\) the limit \(\lim_{L \to \infty}\int_{x_0 - L}^{x_0 + L} u(x)dx\) exists and is independent of \(x_0.\) There are four types of bounded solutions to (1): periodic solutions, quasi-periodic solutions, homoclinic to zero solutions, and solutions which start (at \(x = -\infty)\) from a periodic solution and end (at \(x = +\infty)\) at the same periodic solution with a phase shift. A selected principle for second-order material stemming from energy considerations is introduced.

MSC:

34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
82B30 Statistical thermodynamics
34C25 Periodic solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
82B35 Irreversible thermodynamics, including Onsager-Machlup theory
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