Viscous shocks and long-time behavior of scalar conservation laws

We are interested in the long-time dynamics of viscous scalar conservation laws,

with smooth and strictly convex flux function $f:\mathbb{R}\to\mathbb{R}$, that is, $f^{\prime\prime}(u)>0$ for all $u\in\mathbb{R}$. A typical example is Burgers’ equation where $f(u)=u^{2}/2$. The Cauchy problem for (1.1) is globally well-posed in the space $L^{\infty}(\mathbb{R})$, see e.g. [29]. More precisely, given initial data $u_{0}\in L^{\infty}(\mathbb{R})$, equation (1.1) has a unique global solution $u\in C^{0}((0,+\infty),L^{\infty}(\mathbb{R}))$ such that $u(t,\cdot)$ converges to $u_{0}$ in the weak-$*$ topology of $L^{\infty}(\mathbb{R})$ as $t\to 0+$. By parabolic regularity, the function $u(t,x)$ is smooth for all positive times. For any $t>0$, let $\mathcal{S}_{t}:L^{\infty}(\mathbb{R})\to L^{\infty}(\mathbb{R})$ be the nonlinear map defined by $u(t,\cdot)=\mathcal{S}_{t}(u_{0})$, where $u(t,x)$ is the solution of (1.1) with initial data $u_{0}\in L^{\infty}(\mathbb{R})$. We also write $\mathcal{S}_{0}=\mathbbold{1}$, the identity map.

To a first approximation, the long-time behavior of $\mathcal{S}_{t}(u_{0})$ as $t\to\infty$ is described by the collection of all limit points, usually referred to as the $\omega$-limit set. The unboundedness of the spatial domain $\mathbb{R}$ implies a typical lack of compactness of the trajectory $\{\mathcal{S}_{t}(u_{0})\,|\,t>0\}$, and the $\omega$-limit set may indeed be empty when convergence is measured in the uniform topology defined by the norm in $L^{\infty}(\mathbb{R})$. It is therefore preferable to rely on the local topology induced by $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$, which is the topology of uniform convergence on compact intervals $[-R,R]\subset\mathbb{R}$. The $\omega$-limit set is commonly defined as follows:

but this definition assigns a particular role to the laboratory frame and is not invariant under Galilean transformations. A somewhat richer description of asymptotic behavior is obtained by considering the set of limit points modulo translations,

where $(\mathcal{T}_{y}u)(x)=u(x-y)$. Note that we use the zero subscript in the definition of $\omega_{0}$ in (1.2) to emphasize the fixed origin in the definition of locally uniform convergence in (1.2).

Fairly standard results assert that both $\omega_{0}(u_{0})$ and $\omega(u_{0})$ are non-empty, compact, connected, fully invariant, attractive, and chain recurrent (up to translations) in the topology of $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$; see Propositions 3.2 and 3.3. Full invariance implies in particular that for any $v_{0}$ in the $\omega$-limit set, there exists a solution $v(t,x)$ of (1.1) that is defined for all $t\in\mathbb{R}$ and satisfies $v(0,\cdot)=v_{0}$. We refer to such solutions, defined for all positive and negative times, as entire solutions. Describing all possible long-term dynamics can then be rephrased as describing all subsets of the family of entire solutions that can occur as $\omega$-limit sets for bounded initial data.

The results we present can be seen as small steps in this direction. Somewhat trivial candidates for the $\omega$-limit sets are first spatially constant states $v(x)\equiv m$ and then viscous shocks, found as traveling-wave solutions $v(t,x)=\phi_{\beta,\alpha}(x-ct)$ with $\phi(-\infty)=\beta>\phi(+\infty)=\alpha$, $\phi^{\prime}(\xi)<0$ for all $\xi$, and $c$ given by the Rankine-Hugoniot formula $c_{\beta,\alpha}=(f(\beta)-f(\alpha))/(\beta-\alpha)$. It is known since the classical work of Il’in and Oleinik [15] that large sets of initial data give rise to solutions of (1.1) that converge uniformly to shocks as $t\to+\infty$, see also [8]. In Proposition 4.1 below, we show that this is the case for all initial data that are monotonically decreasing, without any assumption on the rate at which the limits are approached as $x\to\pm\infty$. At this level of generality, we cannot prove convergence to a fixed translate of the shock as $t\to+\infty$. In fact, as discussed in Remark 4.2, there exist monotone initial data $u_{0}\in L^{\infty}(\mathbb{R})$ with $u_{0}(-\infty)=\beta>u_{0}(+\infty)=\alpha$ and $c_{\beta,\alpha}=0$ such that, for instance, $\omega_{0}(u_{0})=\beta$; in particular one observes that $\mathcal{T}_{y}\phi_{\beta,\alpha}\notin\omega_{0}(u_{0})$ for all $y\in\mathbb{R}$.

Our first general result establishes a property reminiscent of the Poincaré-Bendixson theorem, in the sense that it describes the long-time behavior of solutions with initial data in $\omega$-limit sets.

In other words, if $v\in\omega(u_{0})$ is nonconstant, the $\omega$-limit set $\omega(v)$ consists of all translates of a viscous shock $\phi_{\beta,\alpha}$, together with the constant states $\alpha$ and $\beta$ that arise as limits of the shock profile at $\pm\infty$. The proof relies on the simple observation that any such $v$ is necessarily monotonically decreasing, as a consequence of Oleinik’s inequality (2.2). We can thus invoke Proposition 4.1 to determine the $\omega$-limit $\omega(v)$, which is included in $\omega(u_{0})$ since the latter set is invariant under the dynamics of (1.1).

Our remaining results focus on the specific case of Burgers’ equation, which through the Cole-Hopf transformation allows for a somewhat explicit representation of any solution in terms of its initial data. Interestingly, as pointed out in [16], bounded entire solutions of Burgers’ equation can be represented in terms of probability measures $\mu$ on the real line,

This remarkable formula gives, in particular, an explicit characterization of candidates for elements in $\omega$-limit sets. In Section 5 we give a short proof of the representation (1.5), showing that the measure $\mu$ is unique and supported in the closure of the range of the entire solution $u$. We also relate the measure $\mu$ to backward-in-time asymptotics of the entire solution $u$. A striking result in this direction is:

It is also possible to determine the asymptotic behavior of $u(t,x)$ as $t\to-\infty$ in Galilean frames with speeds $c\notin\mathop{\mathrm{supp}}(\mu)$. In that case we define

We refer to Propositions 5.6 and 5.8 below for more general statements, which also cover the somewhat delicate situation where $c=(m_{+}(c)+m_{-}(c))/2$. Other properties of the measure $\mu$, such as the presence of atoms, can also be detected in the ancient behavior of the corresponding entire solution $u$.

Lastly, we show that out of this plethora of entire solutions, the $\omega$-limit set may contain elements that are not simply shocks or constants.

The construction is carried out in a somewhat explicit fashion in Section 6. In the terminology of dynamical systems, the $\omega$-limit set contains a heteroclinic trajectory connecting the zero solution to a steady shock $\phi$, as well as a continuous family of steady shocks interpolating between $\phi$ and $0$. This can be compared to a famous example of coarsening dynamics due to Eckmann and Rougemont [5], also rigorously studied by Poláčik [21, 22], where the $\omega$-limit set is a heteroclinic loop.

We first recall some basic properties of scalar conservation laws of the form (1.1).

In this section we establish the properties of the $\omega$-limit sets (1.2), (1.3) that were announced in the introduction. Here and in what follows, we denote by $d$ the distance on $L^{\infty}(\mathbb{R})$ defined by

As is easily verified, on any bounded set $\Sigma\subset L^{\infty}(\mathbb{R})$, the topology defined by the distance (3.1) coincides with the topology of $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$, namely the topology of uniform convergence on compact subsets of $\mathbb{R}$.

In view of the properties recalled in Section 2, for any initial data $u_{0}\in L^{\infty}(\mathbb{R})$ the solution $u(t,\cdot)=\mathcal{S}_{t}(u_{0})$ of (1.1) belongs for all times to the ball

The following standard result plays a fundamental role:

• Proof . It is easy to check that $\Sigma(u_{0})$ is closed in $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$, and the properties recalled in Section 2 imply that the semiflow $\mathcal{S}_{t}$ maps the ball $\Sigma(u_{0})$ into itself. The key point is the continuous dependence of the solution $\mathcal{S}_{t}(u)$ upon the initial data $u\in\Sigma(u_{0})$, in the topology of $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$. This a rather standard result for parabolic PDEs on unbounded domains, see e.g. [19]. For the reader’s convenience, the argument showing continuity is reproduced in Section A.2 below.  

We are now in position to establish the main properties of the $\omega$-limit set (1.2).

• Proof . Smoothing properties of the parabolic equation (1.1) and a priori bounds for the solutions and their derivatives guarantee that, for any $u_{0}\in L^{\infty}(\mathbb{R})$ and any $k\in\mathbb{N}$, the solution $\mathcal{S}_{t}(u_{0})$ is uniformly bounded in $C^{k}_{b}(\mathbb{R})$ for $t\geq 1$. This does not imply that the forward trajectory $\gamma_{+}(u_{0}):=\{\mathcal{S}_{t}(u_{0})\,;\,t\geq 0\}$ is compact in $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$, because in general the map $t\mapsto\mathcal{S}_{t}(u_{0})$ is not continuous at $t=0$ in that topology. However, for any $T>0$, the trajectory $\gamma_{+}(\mathcal{S}_{T}(u_{0}))=\{\mathcal{S}_{t}(u_{0})\,;\,t\geq T\}$ is relatively compact and connected in $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$. Topological properties (a) and dynamic properties (b) of the $\omega$-limit set $\omega_{0}(u_{0})$ follow in a standard fashion. We include some details here for later reference.

  1) Compactness and attractivity. It is easy to verify that the relation (3.3) is equivalent to the definition (1.2). Now (3.3) shows that $\omega_{0}(u_{0})$ is the intersection of a decreasing family of non-empty compact sets, so that $\omega_{0}(u_{0})$ is itself compact and non-empty. By the same argument, if $\mathcal{N}$ is any neighborhood of $\omega_{0}(u_{0})$ in $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$, there exists $T>0$ such that $\gamma_{+}(\mathcal{S}_{T}(u_{0}))\subset\mathcal{N}$, which proves attractivity.

  2) Connectedness. We argue by contradiction: if $\omega_{0}(u_{0})=A_{1}\cup A_{2}$ where $A_{1},A_{2}$ are non-empty disjoint closed sets, then $A_{1},A_{2}$ are in fact compact and are therefore separated by a distance $\varepsilon>0$. If $\mathcal{N}_{1},\mathcal{N}_{2}$ are $\varepsilon/3$-neighborhoods of $A_{1},A_{2}$, respectively, then $\mathcal{N}_{1},\mathcal{N}_{2}$ are non-empty disjoint open sets, and the attractivity property shows that, for $T>0$ sufficiently large, the connected forward orbit $\gamma_{+}(\mathcal{S}_{T}(u_{0}))$ is contained in the neighborhood $\mathcal{N}:=\mathcal{N}_{1}\cup\mathcal{N}_{2}$, without being included in either $\mathcal{N}_{1}$ or $\mathcal{N}_{2}$, which is clearly impossible.

  3) Full invariance. If $v_{0}\in\omega_{0}(u_{0})$ there exists a sequence $t_{k}\to+\infty$ such that $\mathcal{S}_{t_{k}}(u_{0})\to v_{0}$ in $L^{\infty}_{\mathrm{loc}}$. By Lemma 3.1, for any $t>0$, we thus have

  $$\mathcal{S}_{t}(v_{0})\,=\,\mathcal{S}_{t}\bigl{(}\lim_{k\to\infty}\mathcal{S}_{t_{k}}(u_{0})\bigr{)}\,=\,\lim_{k\to\infty}\mathcal{S}_{t_{k}+t}(u_{0})\,\in\,\omega_{0}(u_{0})\,,$$

  which proves that $\mathcal{S}_{t}(\omega_{0}(u_{0}))\subset\omega_{0}(u_{0})$. Similarly, we can extract a subsequence (still denoted by $t_{k}$) such that $\mathcal{S}_{t_{k}-t}(u_{0})\to v_{-t}\in\omega(u_{0})$, where $v_{-t}$ satisfies $\mathcal{S}_{t}(v_{-t})=v_{0}$. Altogether, this shows that $\mathcal{S}_{t}(\omega_{0}(u_{0}))=\omega_{0}(u_{0})$ for all $t\geq 0$. It is easy to deduce that, given any $v_{0}\in\omega_{0}(u_{0})$, there exists an entire solution $v\in C^{0}(\mathbb{R},L^{\infty}_{\mathrm{loc}}(\mathbb{R}))$ of (1.1) such that $v(0)=v_{0}$.

  4) Chain recurrence. This is a consequence of continuity and attractivity, which can be established as follows. Fix $\varepsilon,T>0$ and take $v_{0}\in\omega_{0}(u_{0})$. By continuity, there exists $\delta\in(0,\varepsilon/2)$ such that, for all $u_{1},u_{2}\in\Sigma(u_{0})$ such that $d(u_{1},u_{2})<\delta$, one has $d(\mathcal{S}_{t}(u_{1}),\mathcal{S}_{t}(u_{2}))<\varepsilon/2$ for all $t\in[T,2T]$. By attractivity, we can then choose $t_{*}>0$ such that $\mathop{\mathrm{dist}}(\mathcal{S}_{t}(u_{0}),\omega_{0}(u_{0}))<\delta$ for all $t\geq t_{*}$. Now, we take $T_{0}\geq t_{*}$ such that $d(\mathcal{S}_{T_{0}}(u_{0}),v_{0})<\delta$, and also $T_{*}\geq t_{*}+T$ such that $d(\mathcal{S}_{T_{*}}(u_{0}),v_{0})<\delta$. For some $N\in\mathbb{N}^{*}$, we define intermediate times $T_{1},\dots,T_{N}$ such that $T_{N}=T_{*}$ and

  $$t_{j}\,:=\,T_{j+1}-T_{j}\,\in\,[T,2T]\,,\quad\hbox{for all }\,j\in\{0,\dots,N-1\}\,.$$

  Finally we denote $\tilde{u}_{j}=\mathcal{S}_{T_{j}}(u_{0})$ for $j=0,\dots,N$, and we take $v_{j}\in\omega_{0}(u_{0})$ such that $d(v_{j},\tilde{u}_{j})<\delta$. Note that $v_{0}$ is given from the beginning, and we can take $v_{N}=v_{0}$. We claim that the sequence $v_{j}$ for $j=0,\dots,N$ is the desired pseudo-orbit. Indeed for $j=0,\dots,N-1$ we have $\tilde{u}_{j+1}=\mathcal{S}_{t_{j}}(\tilde{u}_{j})$, hence

  $$d\bigl{(}v_{j+1},\mathcal{S}_{t_{j}}(v_{j})\bigr{)}\,\leq\,d\bigl{(}v_{j+1},\tilde{u}_{j+1}\bigr{)}+d\bigl{(}\mathcal{S}_{t_{j}}(\tilde{u}_{j}),\mathcal{S}_{t_{j}}(v_{j})\bigr{)}\,<\,\delta+\varepsilon/2\,<\,\varepsilon\,,$$

  where we used the uniform continuity of $\mathcal{S}_{t_{j}}$ and the fact that $t_{j}\in[T,2T]$.

  5) Assertion (c) is an easy consequence of parabolic smoothing and Oleinik’s inequality (2.2).  

We next consider the larger $\omega$-limit set (1.3), where limit points are considered up to translations in space. The analogue of Proposition 3.2 is:

• Proof . The proof is completely parallel to that of Proposition 3.2, and we just indicate here the main differences. The starting point is the formula (3.4), which is easily derived from the definition (1.3). Since the space-time trajectory $\bigl{\{}\mathcal{T}_{y}\mathcal{S}_{t}(u_{0})\,;\,t\geq T,\,y\in\mathbb{R}\bigr{\}}$ is relatively compact in $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$ for any $T>0$, we see that $\omega(u_{0})$ is non-empty and compact as the decreasing intersection of non-empty compact sets. Moreover, if $\mathcal{N}$ is any neighborhood of $\omega(u_{0})$ in $L^{\infty}_{\mathrm{loc}}(\mathbb{R})$, we have $\bigl{\{}\mathcal{T}_{y}\mathcal{S}_{t}(u_{0})\,;\,t\geq T,\,y\in\mathbb{R}\bigr{\}}\subset\mathcal{N}$ for any sufficiently large $T>0$, which means that $\omega(u_{0})$ attracts the trajectory $\mathcal{T}_{y}\mathcal{S}_{t}(u_{0})$ uniformly in $y\in\mathbb{R}$ as $t\to+\infty$. As a consequence, since the space-time trajectory is connected for all $T>0$, the same argument as in Proposition 3.2 shows that $\omega(u_{0})$ is a connected set. There is no difference either in the reasoning showing that $\mathcal{S}_{t}(\omega(u_{0}))=\omega(u_{0})$ for all $t\geq 0$. Finally, the definition (1.3) immediately implies that $\mathcal{T}_{y}(\omega(u_{0}))=\omega(u_{0})$ for all $y\in\mathbb{R}$, and the boundedness properties (c) are established exactly as before.

  The main difference we would like to point out is that $\omega(u_{0})$ is not chain recurrent in the sense of Proposition 3.2, but only in a weaker sense that can be called “chain recurrence up to translations”. The precise definition is as follows: for each $v_{0}\in\omega(u_{0})$ and any $T,\varepsilon>0$, there exist finite sequences $v_{j}\in\omega(u_{0})$, $t_{j}\geq T$, and $y_{j}\in\mathbb{R}$ for $0\leq j\leq N-1$, such that $v_{N}=v_{0}$ and $d(v_{j+1},\mathcal{T}_{y_{j}}\mathcal{S}_{t_{j}}(v_{j}))<\varepsilon$ for all $j\in\{0,\dots,N{-}1\}$. In other words, the definition of the $(\varepsilon,T)$-pseudo-orbit involves spatial shifts $y_{j}$ in addition to the time shifts $t_{j}$, which is natural in view of (1.3). The existence of such a pseudo-orbit for all $v_{0}\in\omega(u_{0})$, all $\varepsilon>0$, and all $T>0$ is established by the same argument as in Proposition 3.2.  

Although the $\omega$-limit sets (1.2), (1.3) are relatively easy to define and enjoy the nice properties listed in Propositions 3.2 and 3.3, it is notoriously difficult to compute them for arbitrary initial data. In the case of equation (1.1), general results in this direction are only available under monotonicity assumptions. If $u_{0}$ is increasing, the solution $\mathcal{S}_{t}(u_{0})$ remains increasing for all $t>0$ by the maximum principle, and $\|\partial_{x}\mathcal{S}_{t}(u_{0})\|_{L^{\infty}}\to 0$ as $t\to+\infty$ by Oleinik’s inequality (2.2). This implies that $\omega(u_{0})$ consists of the constant states $u\equiv\gamma$ for all $\gamma\in[\alpha,\beta]$, where $\alpha,\beta$ are as in (2.1). On the other hand, if $u_{0}$ is decreasing, Proposition 4.1 below implies that $\omega(u_{0})$ is the set of all translates of the viscous shock $\phi_{\beta,\alpha}$, supplemented with the constant states $u\equiv\alpha$ and $u\equiv\beta$. A similar conclusion is reached if $u_{0}$ satisfies the assumptions of Il’in and Oleinik’s result [15]. Incidentally, we observe in this example that $\omega(u_{0})$ is a heteroclinic orbit in the terminology of dynamical systems, so that $\omega(u_{0})$ is not chain recurrent.

More generally, if $u_{0}\in L^{\infty}(\mathbb{R})$ and if there is a $v\in\omega(u_{0})$ that is not a constant, then $v$ is decreasing by Proposition 3.3, and since $\omega(v)\subset\omega(u_{0})$ we deduce that $\omega(u_{0})$ contains the translates of a viscous shock, as asserted in (1.4). So we see that Proposition 1.1 is a direct consequence of Propositions 3.3 and 4.1. In addition we have

This statement can be compared with a result by S. Slijepčević and the first author [9] which shows that, for a general class of dissipative systems including reaction-diffusion equations on the real line, the $\omega$-limit set of a bounded trajectory always contains an equilibrium. For the viscous conservation law (1.1), where all Galilean frames are equivalent, the role of equilibria is played by the constant states and the viscous shocks.

From a different perspective, one may wonder which collections of entire solutions $v\in C^{0}(\mathbb{R},L^{\infty}_{\mathrm{loc}}(\mathbb{R}))$ of (1.1) may occur as $\omega$-limit sets of bounded initial data. From the results presented thus far, only non-empty, compact, connected, invariant, and chain-recurrent sets are candidates. In general, compact, connected sets with a chain-recurrent flow are precisely the possible $\omega$-limit sets of flows, in the sense that any such set is topologically conjugated to an $\omega$-limit set of some flow [7]. It is however not clear at all if any compact, connected, invariant, and chain-recurrent set within the family of entire solutions is realized as the $\omega$-limit set of the particular flow generated by the conservation law (1.1).

The main result of this section is:

The remainder of this section is devoted to the proof of Proposition 4.1. Assume that the initial data $u_{0}\in L^{\infty}(\mathbb{R})$ are nonincreasing and satisfy (4.1) for some $\alpha<\beta$. The solution $u(t)=\mathcal{S}_{t}u_{0}$ of (1.1) is smooth for positive times, and the strong maximum principle implies that $\partial_{x}u(t,x)<0$ for all $t>0$ and all $x\in\mathbb{R}$. On the other hand, using for instance Lemma 3.1, it is not difficult to verify that the limits of $u(t,x)$ as $x\to\pm\infty$ are independent of time. As a consequence, for each $t>0$, there exists a unique point $s(t)\in\mathbb{R}$ such that

Moreover $s(t)$ is a smooth function of time thanks to the implicit function theorem.

• Proof . We use the monotonicity of the evolution map $\mathcal{S}_{t}$ to compare the solution $u(t)=\mathcal{S}_{t}u_{0}$ with suitably translated viscous shocks. Take $\varepsilon>0$ small enough so that

  $$0\,<\,\varepsilon\,<\,\frac{\beta-\alpha}{2}\,,\qquad\hbox{hence}\quad\alpha+\varepsilon\,<\,\frac{\alpha+\beta}{2}\,<\,\beta-\varepsilon\,.$$

  (4.5)

  Since $u_{0}$ is nonincreasing and satisfies (4.1), there exist $x_{+}(\varepsilon)\in\mathbb{R}$ and $x_{-}(\varepsilon)\in\mathbb{R}$ such that

  $$\phi_{\beta-\varepsilon,\alpha-\varepsilon}(x-x_{-}(\varepsilon))\,\leq\,u_{0}(x)\,\leq\,\phi_{\beta+\varepsilon,\alpha+\varepsilon}(x-x_{+}(\varepsilon))\,,\qquad\forall\,x\in\mathbb{R}\,,$$

  (4.6)

  where $\phi_{\beta\pm\varepsilon,\alpha\pm\varepsilon}$ denotes the viscous shock connecting $\beta\pm\varepsilon$ with $\alpha\pm\varepsilon$. In fact, it is straightforward to verify that (4.6) holds as soon as $x_{+}(\varepsilon)\gg 1$ and $x_{-}(\varepsilon)\ll-1$ are sufficiently large, depending on $\varepsilon$. By monotonicity, we deduce from (4.6) that

  $$\phi_{\beta-\varepsilon,\alpha-\varepsilon}\bigl{(}x-c_{-}(\varepsilon)t-x_{-}(\varepsilon)\bigr{)}\,\leq\,u(t,x)\,\leq\,\phi_{\beta+\varepsilon,\alpha+\varepsilon}\bigl{(}x-c_{+}(\varepsilon)t-x_{+}(\varepsilon)\bigr{)}\,,$$

  (4.7)

  for all $t\geq 0$ and all $x\in\mathbb{R}$, where the speeds $c_{\pm}(\varepsilon)$ are given by the Rankine-Hugoniot formulas

  $$c_{+}(\varepsilon)\,:=\,\frac{f(\beta{+}\varepsilon)-f(\alpha{+}\varepsilon)}{\beta-\alpha}\,,\qquad c_{-}(\varepsilon)\,:=\,\frac{f(\beta{-}\varepsilon)-f(\alpha{-}\varepsilon)}{\beta-\alpha}\,.$$

  (4.8)

  On the other hand, due to the second inequality in (4.5), there exist $s_{+}(\varepsilon)\in\mathbb{R}$ and $s_{-}(\varepsilon)\in\mathbb{R}$ such that

  $$\phi_{\beta+\varepsilon,\alpha+\varepsilon}\bigl{(}s_{+}(\varepsilon)\bigr{)}\,=\,\phi_{\beta-\varepsilon,\alpha-\varepsilon}\bigl{(}s_{-}(\varepsilon)\bigr{)}\,=\,\frac{\alpha+\beta}{2}\,.$$

  (4.9)

  In view of (4.9), we deduce from (4.7) that the shift function defined by (4.3) satisfies

  $$c_{-}(\varepsilon)t+x_{-}(\varepsilon)+s_{-}(\varepsilon)\,\leq\,s(t)\,\leq\,c_{+}(\varepsilon)t+x_{+}(\varepsilon)+s_{+}(\varepsilon)\,,\qquad\forall\,t>0\,.$$

  (4.10)

  In particular we infer from (4.10) that

  $$c_{-}(\varepsilon)\,\leq\,\liminf_{t\to+\infty}\frac{s(t)}{t}\,\leq\,\limsup_{t\to+\infty}\frac{s(t)}{t}\,\leq\,c_{+}(\varepsilon)\,.$$

  (4.11)

  Finally it is clear from (4.8) that $c_{\pm}(\varepsilon)\to c$ as $\varepsilon\to 0$, which concludes the proof of (4.4).