On stabilization of solutions of nonlinear parabolic equations with a gradient term

TL;DR

This paper establishes conditions under which solutions to certain nonlinear parabolic equations with gradient terms decay to zero over time, contributing to the understanding of their long-term stabilization behavior.

Contribution

It provides new criteria ensuring the asymptotic decay of solutions for a class of nonlinear parabolic equations with gradient dependence.

Findings

Solutions tend to zero as time approaches infinity under specified conditions.

Derived sufficient conditions for stabilization of solutions.

Applicable to a broad class of nonlinear parabolic equations with gradient terms.

Abstract

For parabolic equations of the form $$ \frac{\partial u}{\partial t} - \sum_{i,j=1}^n a_{ij} (x, u) \frac{\partial^2 u}{\partial x_i \partial x_j} + f (x, u, D u) = 0 \quad \mbox{in } {\mathbb R}_+^{n+1}, $$ where ${\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty)$, $n \ge 1$, $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ is the gradient operator, and $f$ is some function, we obtain conditions guaranteeing that every solution tends to zero as $t \to \infty$.

Equations

$$\frac{\partial u}{\partial t}-\sum _{i,j=1}^n{a}_{ij}(x,u)\frac{{\partial }^{2}u}{\partial x_i\partial x_j}+f(x,u,Du)=0\text{mbox}in{\mathbb{R}}_{+}^{n+1},$$

$$\sum _{i,j=1}^n{a}_{ij}(x,\zeta ){\xi }_i{\xi }_j>0$$

$$\underset{K}{\inf }g>0\text{mbox}and\underset{K}{\inf }h>0$$

$$f(x,\zeta ,\lambda x)\mathrm{sign}\zeta \ge p(x)g(|\zeta |)\left(1+\sum _{i,j=1}^n|a_{ij}(x,\zeta )|\right)$$

$$\mathop{\rm sign}\nolimits\zeta=\left\{\begin{array}[]{rl}1,&\zeta>0,\\ 0,&\zeta=0,\\ -1,&\zeta<0.\end{array}\right.$$

$$q(r)=\underset{B_r}{\inf }p,r\in (0,\infty ).$$

$${\phi }_{\theta }(\zeta )=\underset{(\zeta /\theta ,\theta \zeta )}{\inf }\phi ,\zeta \in (0,\infty ).$$

$${\int }_1^{\infty }rq(r)dr=\infty$$

$${\int }_1^{\infty }(g_{\theta }(\zeta )\zeta {)}^{-1/2}d\zeta <\infty$$

$${\int }_1^{\infty }\frac{d\zeta }{h_{\theta }(\zeta )}<\infty$$

$$\underset{t\to \infty }{\lim }\underset{x\in K}{\sup }|u(x,t)|=0.$$

$$u_t=\Delta u+b(x,u,Du)-c(x)|u{|}^{\sigma -1}u\text{mbox}in{\mathbb{R}}_{+}^{n+1},$$

$$|b(x,\zeta ,\xi )|\le b_0(1+|x|{)}^k{\zeta }^{\mu }|\xi {|}^{\alpha },b_0=const>0,$$

$$c(x)\ge c_0(1+|x|{)}^l,c_0=const>0,$$

$$\min \{l-k+\alpha ,l+2\}\ge 0$$

$$\sigma >\max \{1,\alpha +\mu \},$$

$$p(x)=p_0(1+|x|{)}^{\min \{(l-k)/\alpha -1,l\}},$$

$$\Delta u+b(x,u,Du)\ge c(x)u^{\sigma }\text{mbox}in{\mathbb{R}}^n$$

$$|b(x,\zeta ,\xi )|\le b_0(1+|x|{)}^k{\log }^s(2+|x|){\zeta }^{\mu }|\xi {|}^{\alpha },b_0=const>0,$$

$$c(x)\ge c_0(1+|x|{)}^l{\log }^m(2+|x|),c_0=const>0,$$

$$\min \{l-k+\alpha ,l+2\}=0.$$

$$p(x)=p_0(1+|x|{)}^{-2}{\log }^{\gamma }(2+|x|),$$

$$\gamma =\{\begin{array}{rlrl}& m,& & l+2<l-k+\alpha ,\\ & \min \{(m-s)/\alpha ,m\},& & l+2=l-k+\alpha ,\\ & (m-s)/\alpha ,& & l+2>l-k+\alpha .\end{array}$$

$$\gamma \ge -1,$$

$$u_t=\Delta u+b(x,u,Du)-c(x)|u{|}^{\sigma -1}u{\log }^{\nu }(1+|u|)\text{mbox}in{\mathbb{R}}_{+}^{n+1},$$

$$\sigma =\max \{1,\alpha +\mu \},$$

$$\nu >\{\begin{array}{rlrl}& 2,& & \alpha +\mu <1,\\ & \max \{2,\alpha \},& & \alpha +\mu =1,\\ & \alpha ,& & \alpha +\mu >1,\end{array}$$

$$u_t=\Delta u+b(x,Du)-c(x,u)\text{mbox}in{\mathbb{R}}_{+}^{n+1},$$

$$|b(x,\xi )|\le \phi (|\xi |)$$

Full text

[TABLE]

where ${\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^{n}\times(0,\infty)$, $n\geq 1$, $D=(\partial/\partial x_{1},\ldots,\partial/\partial x_{n})$ is the gradient operator, and $f$ is some function, we obtain conditions guaranteeing that every solution tends to zero as $t\to\infty$.

The research was supported by RFBR, grant 11-01-12018-ofi-m-2011.

1. Introduction

We study solutions of the equations

[TABLE]

where ${\mathbb{R}}_{+}^{n+1}={\mathbb{R}}^{n}\times(0,\infty)$, $n\geq 1$, and $D=(\partial/\partial x_{1},\ldots,\partial/\partial x_{n})$ is the gradient operator. We assume that

[TABLE]

for all $x\in{\mathbb{R}}^{n}$, $\zeta\in{\mathbb{R}}\setminus\{0\}$, and $\xi=(\xi_{1},\ldots,\xi_{n})\in{\mathbb{R}}^{n}\setminus\{0\}$. Also let there are locally bounded measurable functions $g:(0,\infty)\to(0,\infty)$, $h:(0,\infty)\to(0,\infty)$, and $p:{\mathbb{R}}^{n}\to[0,\infty)$ such that

[TABLE]

for any compact set $K\subset(0,\infty)$ and, moreover,

[TABLE]

for all $x\in{\mathbb{R}}^{n}$, $\zeta\in{\mathbb{R}}\setminus\{0\}$, and $0\leq\lambda\leq p(x)h(|\zeta|)$, where

[TABLE]

By a solution of (1.1) we mean a function $u$ that has two continuous derivatives with respect to $x$ and one continuous derivative with respect to $t$ and satisfies equation (1.1) in the classical sense [5].

No smoothness assumptions on $a_{ij}$ and $f$ are imposed, we do not even require these functions to be measurable.

Let us denote $B_{r}^{x}=\{y\in{\mathbb{R}}^{n}:|y-x|<r\}$, $S_{r}^{x}=\{y\in{\mathbb{R}}^{n}:|y-x|=r\}$, and $Q_{x,r}^{t_{1},t_{2}}=B_{r}^{x}\times(t_{1},t_{2})$. In the case of $x=0$, we write $B_{r}$, $S_{r}$, and $Q_{r}^{t_{1},t_{2}}$ instead of $B_{r}^{0}$, $S_{r}^{0}$, and $Q_{0,r}^{t_{1},t_{2}}$, respectively.

Put

[TABLE]

For any function $\varphi:(0,\infty)\to{\mathbb{R}}$ and a real number $\theta>1$ we also denote

[TABLE]

The questions treated in this paper were investigated earlier by a number of authors [1–7, 10, 11]. Below, we obtain conditions guaranteeing that every solution of (1.1) tends to zero as $t\to\infty$. These conditions take into account the dependence of the function $f$ on the gradient term $Du$. Our results are applicable to a wide class of nonlinear equations (see Examples 2.1–2.4).

2. Main results

Theorem 2.1**.**

Let

[TABLE]

and, moreover,

[TABLE]

and

[TABLE]

for some real number $\theta>1$. Then any solution of (1.1) stabilizes to zero uniformly on an arbitrary compact set $K\subset{\mathbb{R}}^{n}$ as $t\to\infty$, i.e.

[TABLE]

Theorem 2.1 will be proved later. Now, we demonstrate its application.

Example 2.1*.*

Consider the equation

[TABLE]

where

[TABLE]

and

[TABLE]

for all $x\in{\mathbb{R}}^{n}$, $\zeta\in{\mathbb{R}}\setminus\{0\}$, and $\xi\in{\mathbb{R}}^{n}$. We assume that $\alpha>0$ whereas $\mu$, $\sigma$, $k$, and $l$ can be arbitrary real numbers.

According to Theorem 2.1, if

[TABLE]

and

[TABLE]

then any solution of (2.4) stabilizes to zero uniformly on an arbitrary compact subset of ${\mathbb{R}}^{n}$ as $t\to\infty$. Really, taking $f(x,\zeta,\xi)=c(x)|\zeta|^{\sigma-1}\zeta-b(x,\zeta,\xi)$, $g(\zeta)=\zeta^{\sigma}$, and $h(\zeta)=\zeta^{(\sigma-\mu)/\alpha}$, we obtain that (1.2) is valid with

[TABLE]

where $p_{0}>0$ is a sufficiently small real number. In so doing, relation (2.1) is equivalent to (2.7) while (2.2) and (2.3) are equivalent to (2.8).

At the same time, if (2.7) is not fulfilled, then for some functions $b$ and $c$ satisfying (2.5) and (2.6) equation (2.4) has a positive solution which does not depend on the variable $t$. This solution obviously can not stabilize to zero as $t\to\infty$. In turn, if (2.8) is not fulfilled, then for all real numbers $k$ and $l$ there exist functions $b$ and $c$ such that (2.5) and (2.6) are valid and equation (2.4) has a positive solution independent of $t$. In this sense, conditions (2.7) and (2.8) are the best possible.

We also note that (2.7) and (2.8) correspond to the blow-up conditions for non-negative solutions of the elliptic inequalities

[TABLE]

considered in  [8, Example 2.1] for $\mu=0$ and $1\leq\alpha\leq 2$.

Example 2.2*.*

In (2.4), let the functions $b$ and $c$ satisfy the relations

[TABLE]

and

[TABLE]

for all $x\in{\mathbb{R}}^{n}$, $\zeta\in{\mathbb{R}}\setminus\{0\}$, and $\xi\in{\mathbb{R}}^{n}$, where $\alpha$, $k$, $s$, $l$, and $m$ are real numbers with $\alpha>0$ and

[TABLE]

In other words, we consider the case of the critical exponents $l$ and $k$ in (2.7).

As in the previous example, we take $f(x,\zeta,\xi)=c(x)|\zeta|^{\sigma-1}\zeta-b(x,\zeta,\xi)$, $g(\zeta)=\zeta^{\sigma}$, and $h(\zeta)=\zeta^{(\sigma-\mu)/\alpha}$. It can be verified that (1.2) is valid for

[TABLE]

where $p_{0}>0$ is a sufficiently small real number and

[TABLE]

Thus, by Theorem 2.1, if (2.8) holds and

[TABLE]

then any solution of (2.4) stabilizes to zero uniformly on an arbitrary compact subset of ${\mathbb{R}}^{n}$ as $t\to\infty$.

If (2.12) is not fulfilled and $n\geq 3$, then there exist functions $b$ and $c$ such that (2.10) and (2.11) are valid and (2.4) has a positive solution independent of $t$. Condition (2.8) is also the best possible.

Example 2.3*.*

Consider the equation

[TABLE]

where the functions $b$ and $c$ satisfy (2.5) and (2.6) with $\alpha>0$ and

[TABLE]

i.e. we are interested in the case of the critical exponent $\sigma$ in (2.8).

Taking $f(x,\zeta,\xi)=c(x)|\zeta|^{\sigma-1}\zeta\log^{\nu}(1+\zeta)-b(x,\zeta,\xi)$, $g(\zeta)=\zeta^{\sigma}\log^{\nu}(1+\zeta)$, and $h(\zeta)=\zeta^{(\sigma-\mu)/\alpha}\log^{\nu/\alpha}(1+\zeta)$, one can see that (1.2) holds with some function $p$ of the form (2.9). Thus, in accordance with Theorem 2.1 if (2.7) is valid and

[TABLE]

then any solution of (2.13) stabilizes to zero uniformly on an arbitrary compact subset of ${\mathbb{R}}^{n}$ as $t\to\infty$.

If (2.7) is not fulfilled, then there are functions $b$ and $c$ such that (2.5) and (2.6) hold and equation (2.13) has a positive solution independent of the variable $t$. In turn, if (2.14) is not fulfilled, then for all real numbers $k$ and $l$ there exist functions $b$ and $c$ satisfying (2.5) and (2.6) for which (2.13) has a positive solution independent of $t$.

Example 2.4*.*

In the equation

[TABLE]

let

[TABLE]

and

[TABLE]

for all $x\in{\mathbb{R}}^{n}$, $\zeta\in{\mathbb{R}}\setminus\{0\}$, and $\xi\in{\mathbb{R}}^{n}$, where $\varphi:[0,\infty)\to[0,\infty)$ and $\psi:(0,\infty)\to(0,\infty)$ are non-decreasing continuous functions. We also assume that $\varphi:[0,\infty)\to[0,\infty)$ is a bijection and

[TABLE]

where $\varphi^{-1}$ is the inverse function of $\varphi$.

According to Theorem 2.1, if

[TABLE]

and

[TABLE]

then any solution of (2.15) stabilizes to zero uniformly on an arbitrary compact subset of ${\mathbb{R}}^{n}$ as $t\to\infty$. Indeed, we put $f(x,\zeta,\xi)=c(x,\zeta)-b(x,\xi).$ It does not present any particular problem to verify that (1.2) is valid with $g(\zeta)=\psi(\zeta),$ $h(\zeta)=\varphi^{-1}(\varepsilon\psi(\zeta)),$ and $p(x)=\varepsilon,$ where $\varepsilon>0$ is a sufficiently small real number. In so doing, condition (2.1) is certainly satisfied while (2.2) and (2.3) are equivalent to (2.18) and (2.19), respectively.

We can show that, in the case where at least one of conditions (2.18), (2.19) is not fulfilled, there are functions $b$ and $c$ such that (2.16) and (2.17) hold and equation (2.15) has a positive solution which does not depend on $t$.

Proof of Theorem 2.1 relies on the following assertion.

Theorem 2.2**.**

Suppose that $u$ is a solution of the inequality

[TABLE]

Also let there exist a real number $\theta>1$ such that (2.1), (2.2), and (2.3) hold. Then

[TABLE]

for any compact set $K\subset{\mathbb{R}}^{n}$, where

[TABLE]

Proof of Theorem 2.2 is given in Section 3. As for equation (1.1), solutions of (2.20) are understood in the classical sense.

Proof of Theorem 2.1.

We apply Theorem 2.2 to the functions $u$ and $-u$. ∎

3. Proof of Theorem 2.2

In this section we assume that hypotheses of Theorem 2.2 are satisfied. Let us denote

[TABLE]

By $C$ we mean various positive constants which can depend only on $n$ and $\theta$.

Definition 3.1** ([5]).**

A point $(x,t)$ belongs to the upper lid $\gamma(\Omega)$ of an open set $\Omega\subset{\mathbb{R}}_{+}^{n+1}$ if there exist real numbers $r>0$ and $\varepsilon>0$ such that $Q_{x,r}^{t-\varepsilon,t}\subset\Omega$ and $Q_{x,r}^{t,t+\varepsilon}\cap\Omega=\emptyset$. The set $\Gamma(\Omega)=\partial\Omega\setminus\gamma(\Omega)$ is called the proper (or parabolic) boundary of $\Omega$.

Lemma 3.1**.**

Let $v$ and $w$ satisfy the inequalities

[TABLE]

and

[TABLE]

where $\omega\neq\emptyset$ is a bounded open subset of ${\mathbb{R}}_{+}^{n}$ such that $u$ is a positive function on $\omega\cup\gamma(\omega)$. Then

[TABLE]

Proof.

Lemma 3.1 is the standard maximum principle for parabolic inequalities in bounded domains [5]. The only subtlety is that (3.1) contains the function $f$. However, this fact can not affect the proof in a significant way. Not wanting to be unfounded, we give this proof in detail.

It can obviously be assumed that, in formula (3.1), the inequality is strong; otherwise we replace $v$ by $v-\varepsilon t$ and pass to the limit as $\varepsilon\to+0$.

Denote

[TABLE]

If (3.3) is not valid, then there exists a real number $\mu>0$ for which the set $\omega_{\mu}=\{(x,t)\in\omega:\varphi(x,t)>\mu\}$ is not empty.

According to (3.2), the closure $\overline{\omega}_{\mu}$ of the set $\omega_{\mu}$ is contained in $\omega\cup\gamma(\omega)$. Let us take a point $(x^{\prime},t^{\prime})\in\overline{\omega}_{\mu}$ such that

[TABLE]

We have $D\varphi(x^{\prime},t^{\prime})=0$ and $\varphi_{t}(x^{\prime},t^{\prime})\geq 0$ or, in other words, $Dv(x^{\prime},t^{\prime})=Dw(x^{\prime},t^{\prime})$ and $v_{t}(x^{\prime},t^{\prime})\geq w_{t}(x^{\prime},t^{\prime})$. It can easily be seen that

[TABLE]

otherwise, introducing the new coordinates $y=y(x)$ such that

[TABLE]

we obtain

[TABLE]

for some $1\leq i\leq n$. This contradicts (3.4).

At the same time, (3.5) is equivalent to the relation

[TABLE]

therefore, we arrive at a contradiction with our assumption that the inequality in (3.1) is strong.

The proof is completed. ∎

Corollary 3.1**.**

Let $v$ satisfy the inequality

[TABLE]

where $\omega\neq\emptyset$ is a bounded open subset of ${\mathbb{R}}_{+}^{n}$ such that $u$ is a positive function on $\omega\cup\gamma(\omega)$. Then

[TABLE]

for all $(x,t)\in\omega\cup\gamma(\omega)$.

Proof.

In Lemma 3.1, we take

[TABLE]

∎

Lemma 3.2**.**

Let $0<r_{1}<r_{2}$ and $0<\tau_{1}<\tau_{2}<\tau$ be real numbers with

[TABLE]

Then at least one of the following three estimates is valid:

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

Proof.

Assume that $\mu$ is a real number satisfying the condition $0<\mu<M_{1}.$ Also let $r_{0}=(r_{1}+r_{2})/2$ and $\varphi\in C^{\infty}({\mathbb{R}})$ be a non-decreasing function such that

[TABLE]

We denote

[TABLE]

where

[TABLE]

and

[TABLE]

Further, let $\Omega=\{(x,t)\in{\mathbb{R}}_{+}^{n}:u(x,t)>\mu\}$ and $\omega=\Omega\cap Q_{r_{2}}^{\tau-\tau_{2},\tau}$. By direct calculation, it can be shown that

[TABLE]

for all $(x,t)\in\omega\cup\gamma(\omega)$. In so doing, we obviously have

[TABLE]

and

[TABLE]

for all $(x,t)\in\omega\cup\gamma(\omega)$, whence in accordance with (1.2) it follows that

[TABLE]

for all $(x,t)\in\omega\cup\gamma(\omega)$. Combining the last inequality with (3.6), we obtain

[TABLE]

Let us show that

[TABLE]

Really, if (3.8) is not valid, then

[TABLE]

for some $\varepsilon>0$. Without loss of generality, it can also be assumed that $\mu<M_{1}-\varepsilon.$ We denote

[TABLE]

From (2.20), it follows that

[TABLE]

Combining this with (3.7), we immediately obtain (3.1). Let us now establish the validity of inequality (3.2). We have

[TABLE]

and

[TABLE]

Relations (3.10) and (3.12) imply that

[TABLE]

In so doing,

[TABLE]

since $\Omega\cap Q_{r_{1}}^{\tau-\tau_{1},\tau}\subset\omega$ and $w$ is equal to zero on $Q_{r_{1}}^{\tau-\tau_{1},\tau}$; therefore, we obtain

[TABLE]

The last formula and the fact that $w$ is a non-negative function yield

[TABLE]

At the same time, taking into account (3.9) and (3.10), we have

[TABLE]

Consequently, one can assert that (3.2) is fulfilled. Thus, by Lemma 3.1, inequality (3.3) holds or, in other words,

[TABLE]

whence it follows that

[TABLE]

This contradiction proves (3.8).

Since $w$ is equal to zero on $Q_{r_{1}}^{\tau-\tau_{1},\tau}$, formula (3.8) implies the estimate

[TABLE]

from which, by the relations

[TABLE]

and

[TABLE]

we obtain

[TABLE]

From Corollary 3.1, it follows that

[TABLE]

In addition, (3.12) implies the inequality

[TABLE]

therefore, inclusion (3.11) allows us to assert that

[TABLE]

Combining this with (3.13), we obtain

[TABLE]

To complete the proof, it remains to note that

[TABLE]

∎

Lemma 3.3**.**

Let $r>0$ and $t>0$ be real numbers such that $4r^{2}<t$ and

[TABLE]

Then at least one of the following two estimates is valid:

[TABLE]

[TABLE]

where

[TABLE]

Proof.

Let $k$ be the maximal positive integer for which $\theta^{k/2}M_{1}\leq M_{2}$. We put $m_{i}=\theta^{i/2}M_{1},$ $i=0,\ldots,k-1,$ and $m_{k}=M_{2}$. It can easily be seen that

[TABLE]

Let us further take an increasing sequence of real numbers $\{r_{i}\}_{i=0}^{k}$ such that $r_{0}=r$, $r_{k}=2r$, and

[TABLE]

Since $u$ is a continuous function in ${\mathbb{R}}_{+}^{n}$, such a sequence obviously exists.

By Lemma 3.2, for any $i\in\{0,\ldots,k-1\}$ at least one of the following three inequalities is valid:

[TABLE]

[TABLE]

[TABLE]

If (3.18) is valid, then we have

[TABLE]

Since

[TABLE]

this implies the estimate

[TABLE]

In turn, if (3.19) holds, then

[TABLE]

whence in accordance with the inequality

[TABLE]

we obtain

[TABLE]

Let us also note that (3.20) implies (3.18); therefore, in this case, we again arrive at (3.21). Thus, for any $i\in\{0,\ldots,k-1\}$ at least one of estimates (3.21), (3.22) is valid. We denote by $\Xi_{1}$ the set of integers $i\in\{0,\ldots,k-1\}$ for which (3.21) holds. In so doing, let $\Xi_{2}=\{0,\ldots,k-1\}\setminus\Xi_{1}$.

At first assume that

[TABLE]

Then, summing (3.21) over all $i\in\Xi_{1}$, we have

[TABLE]

This implies (3.15).

Now, let (3.23) is not valid. Then

[TABLE]

therefore, summing (3.22) over all $i\in\Xi_{2}$, we conclude that

[TABLE]

whence (3.16) immediately follows.

The proof is completed. ∎

Lemma 3.4**.**

In the hypotheses of Lemma 3.3, let the inequality

[TABLE]

be fulfilled instead of (3.14). Then at least one of the following two estimates is valid:

[TABLE]

[TABLE]

where $M_{1}$ and $M_{2}$ are defined by (3.17).

Proof.

By Lemma 3.2, we obviously obtain either

[TABLE]

or

[TABLE]

If (3.26) holds, then

[TABLE]

Thus, taking into account the inequality

[TABLE]

we can assert that

[TABLE]

whence (3.24) follows at once.

Analogously, (3.27) implies the estimate

[TABLE]

from which (3.25) can be obtained.

The proof is completed. ∎

Lemma 3.5**.**

Let $0<\alpha\leq 1$, $\sigma>1$, $\nu>1$, $M_{1}>0$, and $M_{2}>0$ be real numbers with $M_{2}\geq\nu M_{1}$. Then

[TABLE]

for any measurable function $\psi:(0,\infty)\to(0,\infty)$ such that $\psi_{\sigma}(s)>0$ for all $s\in(0,\infty)$, where $A>0$ is a constant depending only on $\alpha$, $\nu$, and $\sigma$.

Lemma 3.6**.**

Let $0<\alpha\leq 1$, $\sigma>1$, $\nu>1$, $r_{1}>0$, and $r_{2}>0$ be real numbers with $r_{2}\geq\nu r_{1}$. Then

[TABLE]

for any measurable function $\varphi:[r_{1},r_{2}]\to[0,\infty)$, where $A>0$ is a constant depending only on $\alpha$, $\nu$, and $\sigma$.

Lemmas 3.5 and 3.6 are proved in [9, Lemmas 2.3 and 2.6].

Proof of Theorem 2.2.

Let $K$ be a compact subset of ${\mathbb{R}}^{n}$ and, moreover, $r>0$ and $t>0$ be real numbers such that $K\subset B_{r}$ and $t>r^{2}$. If

[TABLE]

then

[TABLE]

therefore, we can assume that

[TABLE]

Let us take the maximal integer $k$ satisfying the condition $4^{k}r^{2}<t$. Also put $r_{i}=2^{i}r$ and

[TABLE]

We show that

[TABLE]

Really, by Lemmas 3.3 and 3.4, for any $i\in\{0,\ldots,k-1\}$ at least one of the following three inequalities is valid:

[TABLE]

[TABLE]

[TABLE]

By $\Xi_{1}$, $\Xi_{2}$, and $\Xi_{3}$ we denote the sets of integers $i\in\{0,\ldots,k-1\}$ satisfying relations (3.29), (3.30), and (3.31), respectively.

At first, let

[TABLE]

Summing (3.29) over all $i\in\Xi_{1}$, we obtain

[TABLE]

By Lemma 3.6, this implies the estimate

[TABLE]

from which (3.28) immediately follows.

Now, assume that (3.32) is not valid. Then

[TABLE]

In this case, summing (3.30) over all $i\in\Xi_{2}$, we have

[TABLE]

Analogously, (3.31) allows us to assert that

[TABLE]

Since

[TABLE]

in view of Lemma 3.5, this yields the inequality

[TABLE]

Combining (3.33), (3.34), and (3.35), we again arrive at (3.28).

Further, assuming that $r$ is fixed, we obviously obtain $r_{k}\to\infty$ as $t\to\infty$; therefore, in accordance with (2.1), (2.2), and (2.3) formula (3.28) implies that $m_{0}\to 0$ as $t\to\infty$. Thus,

[TABLE]

The proof is completed. ∎