Fast spatial behavior in higher order in time equations and systems

In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before.


Introduction
Parabolic high-order (in time) equations arise in the study of viscoelasticity, fluid mechanics or heat conduction. We can cite the work of Lebedev and Gladwell [16], where the authors propose high order in time viscoelastic solids. We can also consider the generalized Burgers fluids [33], which correspond to a parabolic third order in time equation (anti-plane shear). Moreover, we can recall the recent theories concerning dual-phase-lag [34] and three-phase lag [4] for the heat conduction. In short, we can say that parabolic high-order equations model a big quantity of thermomechanical problems.
The knowledge of the spatial behavior of the solutions for equations and systems is an important topic in mechanics and mathematics. From a mechanical point of view, it is related to the Saint-Venant principle and, from a mathematical point of view, with the Phragmén-Lindelöf principle. Mathematical studies about the spatial behavior have been proposed for elliptic, hyperbolic and parabolic equations [2,[5][6][7][8][11][12][13][14]19,20,[23][24][25]. The list of contributions in this theory is huge, but we want to focus our attention to the parabolic case. Perhaps, the first contribution in this line was done by Knowles [15], where the exponential decay for the solutions was obtained. However, it is worth recalling the work of Horgan et al [9] and extended by Horgan and Quintanilla [10] for functionally graded materials. These contributions provide spatial decay estimates of the kind of the exponential of the "square" of the distance to the boundary where the perturbations hold. They represent an improvement in the sense that the spatial decay for the transient classical heat equation is faster than the spatial decay for the static heat equation. Later, some extensions to these contributions were proposed [28,29]. Furthermore, the combination with the elastic equation has been also considered [17,31]. However, the first contribution concerning the spatial behavior for high order (n-order) of a partial differential equation was given in [32]. In this last contribution, the parabolic (and hyperbolic) transient problem was studied with the help of a weighted Poincaré inequality. In the parabolic case, an exponential decay (linear in the distance to the bounded boundary) was obtained.
In this paper, we want to improve this last result. We are going to obtain a Phragmén-Lindelöf alternative for a function defined on the cross-section and we will prove that the decay is of the type obtained in [9,10]. We also study the thermoelastic problem when the heat conduction is determined by a high-order parabolic equation. It is worth recalling that in a recent paper [30] the author showed that the decay would be faster than any exponential of a linear expression of the distance. Here, we give a new precise decay improving the ones presented previously. Although the arguments proposed have been considered in many other contributions by different authors, in this work we introduce new functionals which allow us to obtain the improvement in the knowledge of the decay.
In the next section, we propose the parabolic high-order problem that we will study later. To this end, we need to recall several inequalities which are summarized in the third section. In the fourth section, we obtain a Phragmén-Lindelöf alternative for a cross-sectional measure. In the fifth section, we prove a faster decay estimate. In the sixth section, we consider the thermoelastic problem and we prove a decay estimate of the type of the exponential of a second order polynomial. Finally, we give some examples where the results obtained can be applied.

The problem
In this paper, we study the spatial behavior of the problem determined by the equation: in a semi-infinite cylinder (or strip) B = [0, ∞) × D, with the boundary conditions: and the initial conditions: In Eq. (2.1) we have used the notations: We refer the reader to Sect. 7 for some specific examples of these higher-order equations. As usual, we need to impose that to guarantee the compatibility of the conditions. In this paper, we assume that a n+1 > 0 and b n > 0. Of course, the case a n+1 < 0 and b n < 0 can be considered in a similar way. We note that the existence of the solutions to problem (2.1)-(2.3) as well as their regularity can be obtained in view of the results in [21], once we combine these ideas with the ones presented in the appendix of [22].

Some useful inequalities
To obtain our results it will be useful to recall (and to deduce) several inequalities. First, we recall the weighted Poincaré inequality which states that where ω > 0 and f (0) = 0. From the above inequality we can deduce several inequalities which will be useful in our approach. In view of the inequality (3.1), the systematic use of the Hölder and A-G inequalities allows us to obtain where C 1 is a positive calculable constant. The next inequality we consider is the following one: where ω is large enough, which can be obtained in a similar way. We will also need the following inequalities: where ω is again large enough.

Phragmén-Lindelöf alternative
In this section, we obtain a Phragmén-Lindelöf alternative for the solutions to our problem (2.1)-(2.3) for a measure defined in the cross section of the cylinder. We first define the function But we find that +b n u (n) (a 1u + · · · + a n u (n) ),
Therefore, in view of the inequalities provided in the previous section, it follows that whenever ω is large enough. Thus, it leads where λ 2 is the known Poincaré constant for the cross section D. Inequality (4.3) has been previously studied in the context of spatial estimates (see [18]). From here we can obtain that either for every z ≥ z 0 and where G(z 0 , t) > 0, or the exponential decay is satisfied. So we can deduce the following.

Fast decay
In this section, we prove a decay estimate for the solutions to problem (2.1)-(2.3) of the type of the exponential of the distance to the part of the boundary where the perturbations are imposed. First, we note that From (3.2) and (4.2) we also find that where C is a computable positive constant depending on the constitutive coefficients and ω of the form: Inequality (5.1) is well-known (see Equation (3.16) in [10]). If we denote by P (z, t) = G(z, t) 1/2 we can write We also note that P (z, 0) = 0 for z ≥ 0 and where g(0) = 0. Let P (z, t) = exp −λ 2 tC Φ(z, t). It then follows that An upper bound for Φ(z, t) follows from the maximum principle by using the solution to the problem η t = Cη zz with the initial condition η(z, 0) = 0, when z ≥ 0, and the boundary condition: We know that P (z, t) ≤ exp −λ 2 tC η(z, t).
The function η(z, t) is well known (see Carslaw and Jaeger [3, p. 64]) and so, we have Therefore, Since we know that holds. We remark that we can choose ω large enough to guarantee that the decay is "almost" of the type exp − a n+1 z 2 8b n t .

Thermoelastic system
In this section, we extend the estimates obtained in the previous section to the thermoelastic case, that is, a fast decay of the decaying solutions. Thus, we consider the system: with the boundary conditions: It is worth noting that here u i is the displacement vector, θ is the temperature, λ and μ are the Lamé constants, ρ is the mass density, c is the heat capacity and β is the coupling coefficient.
To make the calculations easier we assume that We also assume that a n+1 > 0 and b n > 0. In order to study the problem it is worth writing the displacement equation as We can define the function In this section, we want to obtain a new spatial decay estimate. Therefore, we assume that We obtain that i,j (a 1ui,j + a 2üi,j + · · · + a n u In a similar way, we also havẽ i,i (a 1uj,j + a 2üj,j + · · · + a n u We then obtain i,i )û j,j + cb n θ (n) (a 1θ + · · · + a n θ (n) ) −c(b 0θ + · · · + b n−1 θ (n) )θ.

ZAMP
Fast spatial behavior in higher order Page 9 of 13 102 By choosing ω large enough we can obtain We consider now and so, we have From here, the argument is again standard (see, for instance, [26,27]). We can obtain the existence of two positive constants β 1 and R such that An argument similar to the one proposed in the previous section shows that where N (z, t) = z (4πR) 1/2 , and a change of variable implies that N (z, t) = erfc z (4Ct) 1/2 . Therefore, we conclude that where Rs G 1 (0, s).
We remark that we can obtain upper bounds for this function A(t) in terms of the boundary conditions following the arguments already used in [17,27].
It is clear that these estimates imply that the decay at the infinite is of the type of exp −z 2 4Rt which we summarize as follows.
We note that, for ω large enough, we can choose R as near as we want to the value 2 b n ca n+1 .
Therefore, asymptotically the rate of decay that we have obtained for the function G 1 approaches to exp − ca n+1 z 2 8b n t .

A few examples
In this section we give several elementary examples where the results obtained in this paper can be applied.

Parabolic equation
We give here several examples of parabolic equations of higher order. The first example corresponds to the linearized form of generalized Burgers' fluid. From [33] we know that the system determining the evolution of this fluid is given by where ρ, λ 1 , λ 2 , η 1 , η 2 and η 3 are positive constants.