On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations

TL;DR

This paper investigates the long-term behavior of solutions to subquadratic viscous Hamilton-Jacobi equations with Dirichlet boundary conditions, revealing different asymptotic behaviors depending on the growth rate of the gradient and the ergodic constant.

Contribution

It extends the understanding of large time behavior for subquadratic cases, showing new phenomena when the gradient growth is at or below a critical threshold.

Findings

For m > 3/2 and c > 0, solutions behave similarly to the superquadratic case.

When m ≤ 3/2 or c = 0, solutions can become unbounded from below after adjusting by ct.

Different asymptotic behaviors are characterized based on the gradient growth and ergodic constant.

Abstract

In this article, we are interested in the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi Equations. In the superquadratic case, the third author has proved that these solutions can have only two different behaviors: either the solution of the evolution equation converges to the solution of the associated stationary generalized Dirichlet problem (provided that it exists) or it behaves like $-ct+\varphi (x)$ where $c\geq0$ is a constant, often called the "ergodic constant" and $\varphi$ is a solution of the so-called "ergodic problem". In the present subquadratic case, we show that the situation is slightly more complicated: if the gradient-growth in the equation is like $|Du|^m$ with $m>3/2,$ then analogous results hold as in the superquadratic case, at least if $c>0.$ But, on the contrary, if $m\leq 3/2$ or $c=0,$ then another different behavior appears since $u(x,t) + ct$ can be unbounded from below where $u$ is the solution of the subquadratic viscous Hamilton-Jacobi Equations.

Equations

$$u_t-\Delta u+|Du{|}^m$$

$$u(x,0)$$

$$u(x,t)$$

$$u_0(x)=g(x)for\; allx\in \partial \Omega .$$

$$-\Delta v+|Dv{|}^m=f(x)in\; \Omega ,$$

$$v(x)=g(x)on\partial \Omega ,$$

$$\left\{\begin{array}[]{rcl}-\Delta\varphi+|D\varphi|^{m}&=&f(x)+c\quad\hbox{in }\Omega\;,\\ -\Delta\varphi+|D\varphi|^{m}&\geq&f(x)+c\quad\hbox{on }{\partial\Omega}\;.\end{array}\right.$$

$$\left\{\begin{array}[]{rcl}-\Delta\varphi+|D\varphi|^{m}&=&f(x)+c\quad\hbox{ in }\Omega\;,\\ \varphi(x)&\to&+\infty\quad\hbox{ when }x\to{\partial\Omega}\;.\end{array}\right.$$

$$-\Delta \phi +|D\phi {|}^m=f(x)+\tilde{c}in\; \Omega ,$$

$$\phi (x)=C^{\ast }{d}^{-\frac{2-m}{m-1}}(x)(1+o(1))asd(x)\to 0,\; with{C}^{\ast }=\frac{(m-1{)}^{\frac{m-2}{m-1}}}{(2-m)}.$$

$$\phi (x)=|\log d(x)|(1+o(1))asd(x)\to 0.$$

$$|D\phi (x)|\le \Lambda d^{-\frac{1}{m-1}}(x)in\Omega .$$

$$\underset{d(x)\to 0}{\lim }{d}^{\frac{1}{m-1}}(x)D\phi (x)=c_m\nu (x)$$

$$|D^2\phi (x)|\le K_0{d}^{-\alpha -2}(x)with\alpha =\frac{2-m}{m-1}.$$

$$|D^2\phi (x)|\le K_1|D\phi (x){|}^m$$

$$\left\{\begin{array}[]{rl}&|\varphi(x)|\leq K_{1}\,|D\varphi(x)|^{2-m}\quad\quad\quad\quad\hbox{ if }1<m<2,\\ &\\ &|\varphi(x)|\leq|\log(|D\varphi(x)|)|+K_{1}\quad\quad\quad\hbox{ if }m=2.\end{array}\right.$$

$$D\psi (x)=r^{\alpha +1}D\phi (x_0+rx)\text{ and }{D}^2\psi (x)=r^{\alpha +2}{D}^2\phi (x_0+rx).$$

$$-\Delta \psi +|D\psi {|}^m=r^{\alpha +2}(f(x_0+rx)+c)for\; anyx\in B(0,1).$$

$$\Vert D\psi {\Vert }_{L^{\infty }(B(0,\frac{1}{2}))}\le K$$

$$\Vert D^2\psi {\Vert }_{L^{\infty }(B(0,\frac{1}{2}))}\le K^{\prime }$$

$$|D^2\phi (x_0+rx)|\le K^{\prime }{r}^{-\alpha -2}for\; allx\in B(0,1/2).$$

$$|D\phi (x)|\ge K{d}^{-\frac{1}{m-1}}(x)$$

$$|D^{2}\varphi(x)|\leq K_{0}\,d^{-\alpha-2}(x)=K_{0}d^{-\frac{m}{m-1}}(x)\leq K_{0}\biggl{(}\frac{1}{K}|D\varphi(x)|\biggr{)}^{m}$$

$$c_{\Omega }\le c_{{\Omega }^{\prime }}.$$

$$c_{\Omega }<c_{{\Omega }^{\prime }}.$$

$$c_{\eta }\downarrow c_{\Omega }as\eta \downarrow 0$$

$$\underset{x\to x_0}{\lim }(\mu {\phi }_{{\Omega }^{\prime }}-{\phi }_{\Omega })(x)=\mu {\phi }_{{\Omega }^{\prime }}(x_0)-\underset{x\to x_0}{\lim }{\phi }_{\Omega }(x)=-\infty .$$

$$(\mu {\phi }_{{\Omega }^{\prime }}-{\phi }_{\Omega })(x)\le C^{\ast }(\mu -1)d^{-\frac{2-m}{m-1}}(x)(1+o(1))asx\to x_0,$$

$$-\Delta (\mu {\phi }_{{\Omega }^{\prime }})(x_{\mu })+|D(\mu {\phi }_{{\Omega }^{\prime }})(x_{\mu }){|}^m$$

$$-\Delta {\phi }_{\Omega }(x_{\mu })+|D{\phi }_{\Omega }(x_{\mu }){|}^m=f(x_{\mu })+c_{\Omega }.$$

$$|D{\phi }_{{\Omega }^{\prime }}{|}^m\ge |D{\phi }_{\Omega }{|}^m+m|D{\phi }_{\Omega }{|}^{m-2}D{\phi }_{\Omega }\cdot (D({\phi }_{{\Omega }^{\prime }}-{\phi }_{\Omega })).$$

Full text

1. 1 Introduction

2. 2 Preliminary Results on the Stationary Ergodic and Dirichlet Problems

3. 3 Asymptotic Behavior for the Parabolic Problem

4. 3.1 Convergence to the solution of the Stationary Dirichlet Problem

5. 3.2 Convergence of $\ \dfrac{u(x,t)}{t}$

6. 3.3 Convergence to the Stationary Ergodic Problem when $\frac{3}{2}<m\leq 2$ and $c>0$

7. 4 The Non-Convergence Cases: $c=0$ or $c>0$ and $1<m\leq 3/2$

8. 5 Appendix

1 Introduction

We are interested in this work in the behavior, when $t\to+\infty,$ of the solution of the following initial-boundary value problem

[TABLE]

where $\Omega$ is a $C^{2}$- bounded and connected subset of $\mathbb{R}^{N},$ $1<m\leq 2$ and $f,u_{0},g$ are real-valued continuous functions defined respectively on ${\overline{\Omega}},{\overline{\Omega}}$ and ${\partial\Omega}.$ The boundary and initial data satisfy the following compatibility condition

[TABLE]

It is standard to show that this problem has a unique solution $u:{\overline{\Omega}}\times[0,+\infty)\to\mathbb{R}$ and, as long as one does not need regularity properties, proofs of this fact are easy by using viscosity solutions theory (see Barles and Da Lio [4] and references therein).

In the superquadratic case ($m>2$) the study of the asymptotic behavior has been done by the third author in [16] where it is shown that the solution $u$ can have only two different behaviors whether the equation

[TABLE]

has bounded subsolutions or not. If (1.5) has a bounded subsolution, then there exists a solution $u_{\infty}$ of the stationary Dirichlet problem, i.e. of (1.5) together with the generalized Dirichlet boundary condition

[TABLE]

and $u(x,t)\to u_{\infty}(x)$ uniformly on ${\overline{\Omega}}.$ This is the most expected behavior. On the other hand, it can happen that (1.5) has no bounded subsolution and, in this case, one has to introduce the ergodic problem with state constraint boundary conditions, namely (the reader has to keep in mind that we are here in the superquadratic case)

[TABLE]

We recall that, in this type of problems, both the solution $\varphi$ and the constant $c$ (the ergodic constant) are unknown. The existence and uniqueness of solutions $(c,\varphi)$ for (1.7) is studied in Lasry and Lions [14] (see also [16] for a viscosity solutions approach): the constant $c$ is indeed unique while the solution $\varphi$ is continuous up to the boundary and unique up to an additive constant.

Concluding the reference when $m>2$, it is proved in [16] that, if (1.5) has no bounded subsolution, then $c>0$ and the function $u(x,t)+ct$ converges uniformly on ${\overline{\Omega}}$ to a solution $\varphi$ of (1.7) when $t\to+\infty.$ In fact, in the superquadratic case, even if the boundary condition reads $u(x,t)+ct=g(x)+ct$ on ${\partial\Omega}$ with $g(x)+ct\to+\infty$ as $t\to+\infty,$ there is a loss of boundary condition (cf. [4]) and $u(x,t)+ct$ remains bounded on ${\overline{\Omega}}.$

The first key difference in the subquadratic case is that there is no loss of boundary conditions and $u(x,t)+ct$ is actually equal to $g(x)+ct$ on the boundary. Formally this forces the limit of $u(x,t)+ct$ to tend to $+\infty$ on the boundary and, actually, the analogue of (1.7) is

[TABLE]

This problem was also studied in [14] where it is proved that there exists a unique constant $c$ such that (1.8) has a solution and, as in the superquadratic case, the solution is unique up to an additive constant.

Coming back to the asymptotic behavior of $u,$ it can be thought, at first glance, that it is essentially the same as in the superquadratic case, with problem (1.7) being replaced by problem (1.8). Surprisingly this is not true for any $1<m\leq 2$ nor for any $c.$ Part of the explanations concerning $c$ is that the equation

[TABLE]

has subsolutions which are bounded on ${\overline{\Omega}}$ for any $\tilde{c}\geq c$ in the superquadratic case but only for $\tilde{c}>c$ in the subquadratic one.

In this article we prove the following: first, if the stationary Dirichlet problem (1.5)-(1.6) has a solution, then $u$ converges uniformly in ${\overline{\Omega}}$ to this solution. Otherwise we show that necessarily $c\geq 0$ and $u$ can have different types of behavior. Observe that a curious feature in the subquadratic case (which does not occur in the superquadratic case) is that $c$ can be equal to [math] even if the stationary Dirichlet problem has no solution; this can be seen as a corollary of the above remark. The different behaviors of $u$ can be described as follows.

• (i)

If $3/2<m\leq 2$ and $c>0$ then $u(x,t)+ct$ converges locally uniformly in $\Omega$ to $\varphi$ where $\varphi$ is a solution of (1.8).

• (ii)

If $1<m\leq 3/2$ and $c>0$ then $\displaystyle\frac{u(x,t)}{t}\to-c$ locally uniformly in $\Omega$ but it can happen that $u(x,t)+ct\to-\infty$ in $\Omega$.

• (iii)

If $c=0$, then $\displaystyle\frac{u(x,t)}{t}\to 0$ locally uniformly in $\Omega$ but it can happen that $u(x,t)\to-\infty$ in $\Omega$.

The behaviors (ii) and (iii) are striking differences with the superquadratic case. This is related to the blow-up rate of $u(x,t)$ which we estimate in Theorem 3.2, and which is influenced by the behavior of $\varphi$ (solution of (1.8)) near the boundary ${\partial\Omega}$ (changing according to the values of $m$) and eventually by the case that $c=0$. We prove the optimality of such estimates in star-shaped domains in cases when $f+c<0$ on ${\overline{\Omega}}$: in such situations, we show that it can actually happen that $u(x,t)+ct\to-\infty.$

This article is organized as follows. In Section 2, we recall the main results about the problem (1.8); these results are mainly borrowed from Lasry and Lions [14] and Porretta and Véron [15] where the precise behavior of $\varphi$ near ${\partial\Omega}$ is described. Section 3.1 is devoted to the easy case when (1.5)-(1.6) has a solution; in such situation $u(\cdot,t)$ is bounded and converges to the (unique) solution of the stationary problem. We recall that the existence of a stationary solution corresponds, for the ergodic constant $c$ of (1.8), to the case $c<0$ (see Proposition 2.2). In Section 3.2 we consider the case $c\geq 0$ and we state the estimates on the blow-up rate of $u$ proving, as a consequence, that $\frac{u(x,t)}{t}\to-c$ locally uniformly in $\Omega$, in the whole range $1<m\leq 2$. In the Section 3.3, we prove the result (i) above. Finally, in Section 4 we show that a similar result cannot hold in general in the range $m\geq\frac{3}{2}$ or if $c=0$ proving that, in these cases, we have $u+ct\to-\infty$ at least in some circumstances. We leave to the Appendix the construction of sub and supersolutions, for the problem (1.1)–(1.2)–(1.3), which we use to obtain the crucial estimates on the blow-up rate of $u$ when $c\geq 0$. Throughout this paper, we assume that $\Omega$ is a domain of class $C^{2}$. We denote by $d$ the signed-distance to ${\partial\Omega}$, which is positive in $\Omega$, i.e. $d(x):=\inf\{|x-y|:y\in\partial\Omega\}$ if $x\in\Omega$ and negative in the complementary of $\Omega$ in $\mathbb{R}^{N}.$ As a consequence of the regularity of ${\partial\Omega},$ $d$ is a $C^{2}$- function in a neighborhood $\mathcal{W}$ of ${\partial\Omega}.$ We also denote by $\nu$ the $C^{1}$- function defined by $\nu(x)=-Dd(x)\hbox{ in }\mathcal{W};$ if $x\in{\partial\Omega},$ then $\nu(x)$ is just the unit outward normal vector to ${\partial\Omega}$ at $x.$

2 Preliminary Results on the Stationary Ergodic and Dirichlet Problems

In this section, we deal with properties of the pair $(c,\varphi)$ solution of (1.8). We start by recalling the main results concerning $(c,\varphi)$ which were proved in [14] and [15]. Next, we provide some useful properties of $c$ giving a relationship between the stationary ergodic problem (1.8) and the stationary Dirichlet problem (1.5)-(1.6). We end the section with an example to illustrate the possibly unsolvability of (1.5)-(1.6).

We start with

Theorem 2.1** (On the Stationary Ergodic Problem)**

Assume that $1<m\leq 2$ and that $f\in C({\overline{\Omega}}).$ There exists a unique constant $c\in\mathbb{R}$ such that the problem (1.8) has a solution $\varphi\in W_{loc}^{2,p}(\Omega)$ for every $p>1$. This solution is unique up to an additive constant and satisfies the following properties

• (i)

If $1<m<2,$ then

[TABLE]

• (ii)

If $m=2,$ then

[TABLE]

• (iii)

There exists a constant $\Lambda>0$ depending only on $m,$ ${\Omega}$ and $f$ such that

[TABLE]

• (iv)

We have

[TABLE]

where $c_{m}=(m-1)^{-\frac{1}{m-1}}.$

As far as the proof of Theorem 2.1 is concerned, we refer the reader to the proofs of [14, Theorems I.1, IV.1 and VI.1] for (i), (ii), (iii) and to [15, Theorem 1.1-B] for (iv). We continue with further estimates on the solutions of (1.8), which can be deduced by the previous ones.

Lemma 2.1

Assume $1<m\leq 2$, $f\in W^{1,\infty}(\Omega)$ and let $\varphi$ be a solution of (1.8) given by Theorem 2.1. Then $\varphi\in C^{2,\beta}_{loc}(\Omega)$ for all $\beta\in(0,1)$ and

(i) There exists a constant $K_{0}>0$ such that, for every $x\in\Omega,$ we have

[TABLE]

(ii) There exists $K_{1}>0$ and $\sigma>0$ such that, for every $x\in\Omega$ with $0\leq d(x)<\sigma$

[TABLE]

and

[TABLE]

Proof of Lemma 2.1. We first remark that the $C^{2,\beta}_{loc}$–regularity of the solutions of (1.8) comes from the additional assumption on $f$ ($f\in W^{1,\infty}(\Omega)$) and a standard bootstrap argument : indeed, by Theorem 2.1, $\varphi$ is in $W_{loc}^{2,p}(\Omega)$ for every $p>1$, hence in $C^{1,\beta}_{loc}(\Omega)$ for any $\beta\in(0,1)$, and therefore a standard regularity result implies that $\varphi\in C^{2,\beta}_{loc}(\Omega)$ for all $\beta\in(0,1)$ since $|D\varphi|^{m}$ and $f$ are in $C^{0,\beta}_{loc}(\Omega)$.

In order to prove (i), let $x_{0}\in\Omega,$ and $r=\frac{d(x_{0})}{2}$. We introduce the function $\psi(x)=r^{\alpha}\varphi(x_{0}+rx)$ and compute

[TABLE]

Since $m(\alpha+1)=\alpha+2,$ it is easy to see that $\psi$ is a viscosity solution of

[TABLE]

Since $r^{\alpha+2}f$ and $|Df|r^{\alpha+2}$ are in $L^{\infty}(\Omega),$ we can use the interior estimates available in [14, Theorem A.1] and obtain

[TABLE]

and then, by elliptic regularity (see e.g. [13]),

[TABLE]

which yields

[TABLE]

By taking $x=0$ and remembering the definition of $r,$ we obtain (2.5) with $K_{0}:=2^{\alpha+2}K^{\prime}.$ The estimate (2.6) is a consequence of (2.4) which implies

[TABLE]

for some constant $K>0$ and for $x$ in a suitable neighborhood of ${\partial\Omega}$. Since $\frac{1}{m-1}=\frac{\alpha+2}{m},$ combining (2.8) with (2.5) we obtain

[TABLE]

hence (2.6) for any $K_{1}>K_{0}/K^{m}$. As far as (2.7) is concerned, we use (2.1) and (2.8) if $m<2$ and (2.2) and (2.8) if $m=2$ obtaining that (2.7) holds true for some $K_{1}>0$.

$\Box$

We continue by showing monotonicity and stability properties of the ergodic constant $c$ with respect to the domain.

Proposition 2.1

Let $\Omega^{\prime}$ be an open bounded subset of $\mathbb{R}^{N}$ such that $\Omega\subseteq\Omega^{\prime}.$ Let $c_{\Omega}$ and $c_{\Omega^{\prime}}$ be the ergodic constants associated to (1.8) in $\Omega$ and $\Omega^{\prime}$ respectively. Then we have

[TABLE]

Let $\Omega^{\prime}$ be an open bounded subset of $\mathbb{R}^{N}$ such that $\Omega\subset\subset\Omega^{\prime}.$ The respective ergodic constants $c_{\Omega}$ and $c_{\Omega^{\prime}}$ of $\Omega$ and $\Omega^{\prime}$ satisfy

[TABLE]

Moreover, the ergodic constant $c$ depends continuously on $\Omega.$ Otherwise said, for $0<\eta<1,$ if $c_{\eta}$ is the ergodic constant in (1.8) set in $\Omega+B(0,\eta)$111$\Omega+B(0,\eta):=\{x\in\mathbb{R}^{N}:d(x)>-\eta\}.$ then

[TABLE]

where $B(0,\eta)$ is a ball of radius $\eta.$

Proof of Proposition 2.1. (i) Let $(c_{\Omega},\varphi_{\Omega})$ and $(c_{\Omega^{\prime}},\varphi_{\Omega^{\prime}})$ be the pair of solutions of the ergodic problem (1.8) in $\Omega$ and $\Omega^{\prime}$ respectively. From what we obtained above, the constants $c_{\Omega}$ and $c_{\Omega^{\prime}}$ are unique whereas the functions $\varphi_{\Omega}$ and $\varphi_{\Omega^{\prime}}$ are unique up to a constant. We study $\max_{{\overline{\Omega}}}(\mu\varphi_{\Omega^{\prime}}-\varphi_{\Omega})$ for some $\mu\in(0,1)$ close to $1$. Now, we need to show that the maximum is achieved inside $\Omega$.

Observe that, since $\Omega\subseteq\Omega^{\prime},$ it could happen that ${\partial\Omega}\cap\partial\Omega^{\prime}\neq\emptyset,$ meaning that the two domains $\Omega$ and $\Omega^{\prime}$ touch at some points. Due to the behavior of $\varphi_{\Omega^{\prime}}$ and $\varphi_{\Omega}$ near $\partial\Omega^{\prime}$ and $\partial\Omega$ respectively, we first deal with points on $\partial\Omega$ which do not belong to $\partial\Omega^{\prime}$ and next we treat common points of $\partial\Omega$ and $\partial\Omega^{\prime}.$

We pick any $x_{0}\in{\partial\Omega}$ such that $x_{0}\notin\partial\Omega^{\prime};$ in this case, since $\varphi_{\Omega^{\prime}}$ is bounded in ${\overline{\Omega}}\setminus({\partial\Omega}\cap\partial\Omega^{\prime}),$ it follows that

[TABLE]

On the other hand, if $x_{0}\in{\partial\Omega}\cap\partial\Omega^{\prime},$ using the asymptotic behavior (2.1) and that, if $d_{\partial\Omega^{\prime}}$ denotes the distance to $\partial\Omega^{\prime}$, $d_{\partial\Omega^{\prime}}(x)\geq d(x)$ we have, when $1<m<2$

[TABLE]

hence $(\mu\varphi_{\Omega^{\prime}}-\varphi_{\Omega})(x)\to-\infty$ in this case too. When $m=2$, the same conclusion holds by the use of (2.2). Therefore, in any case it follows that $(\mu\varphi_{\Omega^{\prime}}-\varphi_{\Omega})$ has a maximum point $x_{\mu}\in\Omega$. Going back to the equations solved by $\mu\varphi_{\Omega^{\prime}}$ and $\varphi_{\Omega},$ we obtain:

[TABLE]

and

[TABLE]

By subtracting (2.13) from (2.12) and using the following properties: $D(\mu\varphi_{\Omega^{\prime}})(x_{\mu})=D\varphi_{\Omega}(x_{\mu})$ and $\Delta(\mu\varphi_{\Omega^{\prime}}-\varphi_{\Omega})(x_{\mu})\leq 0,$ one gets: $c_{\Omega}\leq\mu c_{\Omega^{\prime}}+(\mu-1)f(x_{\mu})$ and (2.9) follows by sending $\mu$ to $1.$ (ii) Let $\Omega^{\prime}$ be such that $\Omega\subset\subset\Omega^{\prime}$; from (2.9) it follows that $c_{\Omega}\leq c_{\Omega^{\prime}}.$ Moreover, we would like to show that for this case, we have: $c_{\Omega}<c_{\Omega^{\prime}};$ to do so, we assume the contrary by setting $c_{\Omega}=c_{\Omega^{\prime}}.$ Using the boundedness of $\varphi_{\Omega^{\prime}}$ on ${\overline{\Omega}}$ and behavior of $\varphi_{\Omega}$ near ${\partial\Omega},$ it is easy to see that $\varphi_{\Omega^{\prime}}-\varphi_{\Omega}\to-\infty$ on ${\partial\Omega},$ meaning that $\varphi_{\Omega^{\prime}}-\varphi_{\Omega}$ achieves its global maximum on ${\overline{\Omega}}$ at some ${\bar{x}}$ inside $\Omega.$ On the other hand, the convexity of $p\mapsto|p|^{m}$ yields

[TABLE]

With this argument and using the local bounds on $|D\varphi_{\Omega}|$ in (2.3), we have that $\varphi_{\Omega^{\prime}}-\varphi_{\Omega}$ solves

[TABLE]

for some $C(x)>0$ which is bounded in any compact subset. Applying the Strong Maximum Principle (see [16, Lemma 2.1]) we find that $\varphi_{\Omega^{\prime}}-\varphi_{\Omega}$ is constant in ${\overline{\Omega}}$ which clearly leads to a contradiction since $\varphi_{\Omega^{\prime}}-\varphi_{\Omega}$ blows up on the boundary ${\partial\Omega},$ whence (2.10) holds.

(iii) Now, we turn to the proof of the continuous dependence of $c$ and $\varphi$ in $\Omega,$ namely (2.11). Let $0<\eta<\eta^{\prime}<1,$ since $\Omega+B(0,\eta)$ is a strict subset of $\Omega+B(0,\eta^{\prime})$, we find from (2.10) that $c_{\eta}<c_{\eta^{\prime}},$ meaning that the sequence $(c_{\eta})_{0<\eta<1}$ decreases as $\eta$