Strong unique continuation for second order hyperbolic equations with time independent coefficients

TL;DR

This paper establishes a strong unique continuation property for second order hyperbolic equations with time-independent coefficients, showing that flatness on a segment implies local vanishing of solutions.

Contribution

It proves a novel strong unique continuation result for hyperbolic equations with variable coefficients, extending previous understanding of solution behavior.

Findings

Solutions flat on a segment vanish nearby

Unique continuation holds for second order hyperbolic equations

Results apply to equations with variable coefficients

Abstract

In this paper we prove that if $u$ is a solution to second order hyperbolic equation $\partial^2_tu+a(x)\partial_tu-(div_x\left(A(x)\nabla_x u\right)+b(x)\cdot\nabla_x u+c(x)u)=0$ and $u$ is flat on a segment $\{x_0\}\times (-T,T)$ then $u$ vanishes in a neighborhood of $\{x_0\}\times (-T,T)$.

Equations

$${\partial }_t^{2}u+a(x){\partial }_{t}u-L(u)=0,in{B}_{{\rho }_0}\times (-T,T),$$

$$L(u)={\text{div}}_x\left(A(x){\nabla }_{x}u\right)+b(x)\cdot {\nabla }_{x}u+c(x)u,$$

$${\Vert u\Vert }_{L^2\left(B_r\times (-T,T)\right)}=\mathcal{O}(r^N)\text{mbox},\forall N\in \mathbb{N}\text{mbox},asr\to 0,⟹u=0,\text{mbox}in\mathcal{U}.$$

$$v_k(x,y)={\int }_{-T}^{T}u(x,t){\phi }_k(t+iy)dt,\text{mbox}\forall k\in \mathbb{N},$$

$${\partial }_y^2{v}_k-ia(x){\partial }_y{v}_k+L(v_k)=F_k(x,y),in{B}_1\times \mathbb{R},$$

$${\partial }_t^{2}u+a(x,t){\partial }_{t}u-{\Delta }_{x}u=0,$$

$${\Vert v(x)\Vert }_{H^{\ell }(\Omega )}={\left(\sum _{0\le j\le \ell }{\int }_{\Omega }{\left|D^{j}v(x)\right|}^{2}dx\right)}^{1/2}.$$

$$\mathcal{W}\left(J;\Omega \right)=\left\{u\in C^0\left(J;H^2\left(\Omega \right)\right):{\partial }_t^{\ell }u\in C^0\left(J;H^{2-\ell }\left(\Omega \right)\right),\ell =1,2\right\}.$$

$$\lambda {\left|\xi \right|}^2\le A(x)\xi \cdot \xi \le {\lambda }^{-1}{\left|\xi \right|}^2,for\; everyx,\xi \in {\mathbb{R}}^n,$$

$$\left|A(x_{\ast })-A(x)\right|\le \frac{\Lambda }{{\rho }_0}\left|x-y\right|,for\; every{x}_{\ast },x\in {\mathbb{R}}^n.$$

$$T\left|a(x)\right|+T^2{\rho }_0^{-1}\left|b(x)\right|+T^2\left|c(x)\right|\le {\Lambda }_1,for\; almost\; everyx\in {\mathbb{R}}^n,$$

$$L(u)={\text{div}}_x\left(A(x){\nabla }_{x}u\right)+b(x)\cdot {\nabla }_{x}u+c(x)u.$$

$${\partial }_t^{2}u+a(x){\partial }_{t}u-L(u)=0,a.e.\; in{B}_{{\rho }_0}\times (-T,T).$$

$${\rho }_0^{-n}{T}^{-1}{\int }_{-T}^T{\int }_{B_{r_0}}{\left|u(x,t)\right|}^{2}dxdt\le {\epsilon }^2$$

$$\underset{t\in [-T,T]}{\max }\left({\rho }_0^{-n}{\int }_{B_{{\rho }_0}}{\left|u(x,t)\right|}^{2}dx+{\rho }_0^{-n+1}{\int }_{B_{{\rho }_0}}{\left|{\partial }_{t}u(x,t)\right|}^{2}dx\right)\le H^2.$$

$${\rho }_0^{-n}{\int }_{B_{(1-|t_0|T^{-1})\rho }}{\left|u(x,t_0)\right|}^{2}dx\le \frac{C({\rho }_0{\rho }^{-1}{)}^C(H+e\epsilon {)}^2}{{\left(\theta \log \left(\frac{H+e\epsilon }{\epsilon }\right)\right)}^{\alpha }},$$

$${\theta }_0=\frac{\log ({\rho }_0/C\rho )}{\log ({\rho }_0/r_0)}.$$

$${\left({\rho }_0^{-n}{T}^{-1}{\int }_{-T}^T{\int }_{B_{r_0}}{\left|u(x,t)\right|}^{2}dxdt\right)}^{1/2}=O(r_0^N)\text{mbox},\forall N\in \mathbb{N}\text{mbox},as{r}_0\to 0,$$

$$u(\cdot ,t)=0\text{mbox},for|x|+\frac{{\rho }_0|t|}{T}\le s_0{\rho }_0.$$

$${\Vert u(\cdot ,0)\Vert }_{L^2\left(B_{s_0{\rho }_0}\right)}>0$$

$$\underset{t\in [-T,T]}{\max }\left({\rho }_0^{-n}{\int }_{B_{{\rho }_0}}{\left|u(x,t)\right|}^{2}dx+{\rho }_0^{-n+1}{\int }_{B_{{\rho }_0}}{\left|{\partial }_{t}u(x,t)\right|}^{2}dx\right)=1.$$

$${\Vert u(\cdot ,0)\Vert }_{L^2\left(B_{s_0{\rho }_0}\right)}\le C{s}_0^{-C}{N}^{-\alpha /2}\text{mbox},\forall N\in \mathbb{N},$$

$$\mathcal{P}(w):={\partial }_y^{2}w+L(w)-ia(x){\partial }_{y}w.$$

$$\varrho (x,y)={\left(A^{-1}(0)x\cdot x+y^2\right)}^{1/2},$$

$${\tilde{B}}_r^{\varrho }=\left\{(x,y)\in {\mathbb{R}}^{n+1}:\varrho (x,y)<r\right\},r>0.$$

$${\tilde{B}}_{\sqrt{\lambda }r}^{\varrho }\subset {\tilde{B}}_r\subset {\tilde{B}}_{r/\sqrt{\lambda }}^{\varrho },\forall r>0.$$

$$\Psi (r)=r\exp \left({\int }_0^r\frac{e^{-C_{\ast }\eta }-1}{\eta }d\eta \right),$$

$$\psi (x,y)=\Psi \left(\varrho (x,y)/2\sqrt{\lambda }\right),$$

$${\int }_{{\mathbb{R}}^{n+1}}\left(\tau {\psi }^{1-2\tau }{\left|{\nabla }_{x,y}w\right|}^2+{\tau }^3{\psi }^{-1-2\tau }{\left|w\right|}^2\right)dxdy\le C_{\ast }{\int }_{{\mathbb{R}}^{n+1}}{\psi }^{2-2\tau }{\left|\mathcal{P}(w)\right|}^{2}dxdy.$$

$$\Psi (r)\simeq r,asr\to 0.$$

$$C^{-1}r\le \Psi \left(r\right)\le r,\forall r\in (0,1].$$

Full text

1. Introduction

In this paper we study strong unique continuation property for the equation

[TABLE]

where $\rho_{0},T$ are given positive numbers, $B_{\rho_{0}}$ is the ball of $\mathbb{R}^{n}$, $n\geq 2$, of radius $\rho_{0}$ and center at 0, $a\in L^{\infty}(\mathbb{R}^{n})$, $L$ is the second order elliptic operator

[TABLE]

$b\in L^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n})$, $c\in L^{\infty}(\mathbb{R}^{n})$ and $A(x)$ is a real-valued symmetric $n\times n$ matrix that satisfies a uniform ellipticity condition and entries of $A(x)$ are functions of Lipschitz class.

We say that the equation (1.1) has the strong unique continuation property (SUCP) if there exists a neighborhood $\mathcal{U}$ of $(-T,T)$ such that for every solution, $u$, to equation (1.1) we have

[TABLE]

Property (1.3) was proved (if the matrix $A$ belongs to $C^{2}$), , under the additional condition $T=+\infty$ and is $u$ bounded, by Masuda in 1968, [Ma]. Later on, in 1978, Baouendi and Zachmanoglou, [Bao-Z], proved the SUCP whenever the coefficients of equation (1.1) are analytic functions. In 1999, Lebeau, [Le], proved the SUCP for solution to (1.1) when $a=0$. The proof, rather intricate, of [Le] does not seem to adapt to the case of nonvanishing $a$. We also refer to [Si-Ve], [Ve2] where the SUCP at the boundary and the quantitative estimate of unique continuation related to property was proved when $a=0$.

It is worth noting that SUCP and the related quantitative estimates, has been extensively studied and today well understood in the context of second order elliptic and parabolic equation. Among the extensive literature on the subject here we mention, for the elliptic equations, [A-K-S], [Hö1], [Ko-Ta1], and, for the parabolic equations, [Al-Ve], [Es-Fe], [Ko-Ta2]. In the context of elliptic and parabolic equations the quantitative estimates of unique continuation appear in the form of three sphere inequalities [La], doubling inequalities [Ga-Li], or two-sphere one cylinder inequality [Es-Fe-Ve]. We refer to [Al-R-Ro-Ve] and [Ve1] for a more extensive literature concerning the elliptic context and the parabolic context respectively.

In the present paper we prove (Theorem 2.1) a quantitative estimate of unique continuation from which we derive (Corollary 2.2) property (1.3) for equation (1.1). The crucial step of the proof is Proposition 3.1, in such a Proposition 3.1 we exploit in a suitable way the simple and classical idea of converting a hyperbolic equation into an elliptic equation, see for instance [G, Ch. 6].

More precisely we define the function

[TABLE]

where $\varphi_{k}(t+iy)$ is a polynomial with the following property:

a) $\varphi_{k}(t+i0)$ is an approximation of Dirac’s $\delta$-function,

b) $\varphi_{k}(\pm T+iy)=\mathcal{O}((C|y|)^{k})$ as $k\rightarrow\infty$ and $|y|\leq 1$, where $C$ is a constant.

In this way functions $v_{k}$ turn out solutions to the elliptic equation

[TABLE]

where $F_{k}=\mathcal{O}((C|y|)^{k})$, as $k\rightarrow\infty$ and $|y|\leq 1$. This behavior of $F_{k}$ allows us to handle in a suitable way a Carleman estimate with singular weight for second order elliptic operators, see Section 2.3 below, in such a way to get $u(x,0)=0$ for $x\in B_{\rho}$, where $\rho\leq\rho_{0}/C$. Similarly we prove for every $t\in(-T,T)$, $u(\cdot,t)=0$ in $B_{\rho(t)}$, where $\rho(t)=(1-tT^{-1})\rho$. So that we obtain (1.3) with $\mathcal{U}=\bigcup_{t\in(-T,T)}(B_{\rho(t)}\times\{t\})$. As a consequence of this result and using the weak unique continuation property proved in [Hö2], [Ro-Zu] and [Ta], see also [Is1], and [Bo-K-L], for the related quantitative estimates, we have that $u=0$ in the domain of dependence of $\mathcal{U}$.

The quantitative estimate of unique continuation that we prove in Theorem 2.1 can be read, roughly speaking, as a continuous dependence estimate of $u_{{}_{|\mathcal{U}}}$ from $u_{{}_{|{B_{r_{0}}\times(-T,T)}}}$, where $r_{0}$ is arbitrarily small. The sharp character of such a continuous dependence result is related to the logarithmic character of this estimate, that, at the light of counterexample of John [Jo], cannot be improved and to the fact that the quantitative estimate implies the SUCP property. The quantitative estimate of strong unique continuation (at the interior and at the boundary) was a crucial tool, see [Ve3], to prove sharp stability estimate for inverse problems with unknown boundaries for wave equation $\partial^{2}_{t}-\text{div}_{x}\left(A(x)\nabla_{x}u\right)=0$.

Before concluding this Introduction we mention an open question(to the author knowledge). Such an open question concerns the SUCP, (1.3), for the second order hyperbolic equation with coefficients that are analytic in variable $t$ and smooth enough (but not analytic) in variables $x$. This is, for instance, the case of the equation

[TABLE]

where $a(x,t)$ is smooth enough w.r.t $x$ and analytic w.r.t. $t$. Concerning this topic we mention [Lu] in which it is proved that if $u$ satisfies the conditions: (a) $(-T,T)\times\{0\}\cap{\rm supp}u$ is compact and (b $D^{j}u(x,t)=\mathcal{O}\left(e^{-k/|x|}\right)$, $j=1,2$, for every $k$ as $x\rightarrow 0$, $t\in(-T,T)$, then $u$ vanishes in a neighborhood of $(-T,T)$.

The plan of the paper is as follows. In Section 2 we state the main result of this paper, in Section 3 we prove the main theorem.

2. The main results

2.1. Notation and Definition

Let $n\in\mathbb{N}$, $n\geq 2$. For any $x\in\mathbb{R}^{n}$, we will denote $x=(x^{\prime},x_{n})$, where $x^{\prime}=(x_{1},\ldots,x_{n-1})\in\mathbb{R}^{n-1}$, $x_{n}\in\mathbb{R}$ and $|x|=\left(\sum_{j=1}^{n}x_{j}^{2}\right)^{1/2}$. Given $r>0$, we will denote by $B_{r}$, $B^{\prime}_{r}$ $\widetilde{B}_{r}$ the ball of $\mathbb{R}^{n}$, $\mathbb{R}^{n-1}$ and $\mathbb{R}^{n+1}$ of radius $r$ centered at 0. For any open set $\Omega\subset\mathbb{R}^{n}$ and any function (smooth enough) $u$ we denote by $\nabla_{x}u=(\partial_{x_{1}}u,\cdots,\partial_{x_{n}}u)$ the gradient of $u$. Also, for the gradient of $u$ we use the notation $D_{x}u$. If $j=0,1,2$ we denote by $D^{j}_{x}u$ the set of the derivatives of $u$ of order $j$, so $D^{0}_{x}u=u$, $D^{1}_{x}u=\nabla_{x}u$ and $D^{2}_{x}u$ is the hessian matrix $\{\partial_{x_{i}x_{j}}u\}_{i,j=1}^{n}$. Similar notation are used whenever other variables occur and $\Omega$ is an open subset of $\mathbb{R}^{n-1}$ or a subset $\mathbb{R}^{n+1}$. By $H^{\ell}(\Omega)$, $\ell=0,1,2$, we denote the usual Sobolev spaces of order $\ell$ (in particular, $H^{0}(\Omega)=L^{2}(\Omega)$), with the standard norm

[TABLE]

For any interval $J\subset\mathbb{R}$ and $\Omega$ as above we denote

[TABLE]

We shall use the letters $c,C,C_{0},C_{1},\cdots$ to denote constants. The value of the constants may change from line to line, but we shall specified their dependence everywhere they appear. Generally we will omit the dependence of various constants by $n$.

2.2. Statements of the main results

Let $\rho_{0}>0$, $T$, $\lambda\in(0,1]$, $\Lambda>0$ and $\Lambda_{1}>0$ be given number. Let $A(x)=\left\{a^{ij}(x)\right\}^{n}_{i,j=1}$ be a real-valued symmetric $n\times n$ matrix whose entries are measurable functions and they satisfy the following conditions

[TABLE]

Let $b\in L^{\infty}(\mathbb{R}^{n};\mathbb{R}^{n})$ and $a,c\in L^{\infty}(\mathbb{R}^{n})$ satisfy

[TABLE]

Let

[TABLE]

Let $u\in\mathcal{W}\left([-T,T];B_{\rho_{0}}\right)$ be a solution to

[TABLE]

Let $\varepsilon$ and $H$ be given positive numbers and let $r_{0}\in(0,\rho_{0}]$. We assume

[TABLE]

and

[TABLE]

Theorem 2.1**.**

Let $u\in\mathcal{W}\left([-T,T];B_{\rho_{0}}\right)$ be a weak solution to (2.4) and let (2.1), (2.2), (2.5) and (2.6) be satisfied. For every $\alpha\in(0,1/2)$ there exist constants $s_{0}\in(0,1)$ and $C\geq 1$ depending on $\lambda$, $\Lambda$, $\Lambda_{1}$, $\alpha$ and $T\rho_{0}^{-1}$ only such that for every $t_{0}\in(-T,T)$ and every $0<r_{0}\leq\rho\leq s_{0}\rho_{0}$ the following inequality holds true

[TABLE]

where

[TABLE]

The proof of Theorem 2.1 is given in Section 3.

The proof of the following Corollary is standard (see, for instance, [Ve2, Remark 2.2]), but we give it for the reader convenience.

Corollary 2.2** (Strong Unique Continuation Property).**

Let $u\in\mathcal{W}\left([-T,T];B_{\rho_{0}}\right)$ be a weak solution to (2.4). Assume that (2.1) and (2.2) be satisfied. We have that, if

[TABLE]

then

[TABLE]

Proof.

We consider the case $t=0$, similarly we could proceed for $t\neq 0$. If $\left\|u(\cdot,0)\right\|_{L^{2}\left(B_{s_{0}\rho_{0}}\right)}=0$ there is nothing to proof, otherwise, if

[TABLE]

we argue by contradiction. By (2.10) it is not restrictive to assume that

[TABLE]

Now we apply inequality (2.7) with $\varepsilon=C_{N}r_{0}^{N}$, $H=1$ and passing to the limit as $r_{0}\rightarrow 0$ we derive

[TABLE]

by passing again to the limit as $N\rightarrow 0$, by (2.12), we obtain $\left\|u(\cdot,0)\right\|_{L^{2}\left(B_{s_{0}\rho_{0}}\right)}=0$ that contradicts (2.10). $\square$

2.3. Auxiliary result: Carleman estimate with singular weight

In order to prove Theorem 2.1 we need a Carleman estimate proved by several authors, here we recall [A-K-S], [Hö1]. In order to control the dependence of the various constants, we use here a version of such a Carleman estimate proved, in the context of parabolic operator, in [Es-Ve], see also [Bou-Ke, Section 8].

First we introduce some notation. Let $\mathcal{P}$ be the elliptic operator

[TABLE]

Denote

[TABLE]

[TABLE]

Notice that

[TABLE]

Theorem 2.3**.**

Let $\mathcal{P}$ be the operator (2.13) and assume that (2.1) is satisfied. There exists constants $C_{\ast}>1$ depending on $\lambda$, and $\Lambda$ only and $\tau_{0}>1$ depending on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only such that, denoting

[TABLE]

for every $\tau\geq\tau_{0}$ and $w\in C^{\infty}_{0}\left(\widetilde{B}^{\varrho}_{2\sqrt{\lambda}/C_{\ast}}\setminus\{0\}\right)$ we have

[TABLE]

Remark 2.4**.**

We emphasize that

[TABLE]

Moreover $\Psi$ is an increasing and concave function and there exists $C>1$ depending on $\lambda$, and $\Lambda$ such that

[TABLE]

3. Proof of Theorem 2.1

The primary step to achieve Theorem 2.1 consists in proving the following

Proposition 3.1**.**

Let us assume $\rho_{0}=1$ and $T=1$. Let $u\in\mathcal{W}\left([-1,1];B_{1}\right)$ be a weak solution to (2.4) and let (2.1), (2.2), (2.5) and (2.6) be satisfied. For every $\alpha\in(0,1/2)$ there exist constants $s_{0}\in(0,1)$ and $C\geq 1$ depending on $\lambda$, $\Lambda$, $\Lambda_{1}$ and $\alpha$ only such that for every $0<r_{0}\leq s\leq s_{0}$ the following inequality holds true

[TABLE]

where

[TABLE]

In order to prove Proposition 3.1 we define

[TABLE]

where

[TABLE]

and

[TABLE]

so that we have

[TABLE]

It is easy to check that

[TABLE]

We need some simple lemmas to state the properties of functions $v_{k}$.

Lemma 3.2**.**

We have

[TABLE]

where $c$ depends on $n$ only.

Proof.

By (3.3) and (3.6) we have

[TABLE]

hence, by Schwarz inequality and integrating over $B_{1}$ we have

[TABLE]

where $\gamma\in(0,1)$ is a number that we will choose. Now we have

[TABLE]

and, by (2.6),

[TABLE]

Hence, by (2.6), (3.7) and (3.9), we have

[TABLE]

where $c$ depends on $n$ only. Now, we choose $\gamma=k^{-1/2}\log k$ and we get (3.8). $\square$

Lemma 3.3**.**

Let $u$ be a solution to (2.4) and let (2.1) and (2.2) be satisfied, then $v_{k}\in H^{2}\left(B_{1}\times(-1,1)\right)$ is a solution to the equation

[TABLE]

where $F_{k}\in L^{\infty}(-1,1;L^{2}(B_{1}))$ and it satisfies

[TABLE]

$C$* depending on $\Lambda_{1}$ only.*

In addition, $v_{k}$ satisfies the following properties

[TABLE]

[TABLE]

where $C$ depends on $\lambda$ and $\Lambda_{1}$ only.

Proof.

The fact that $v_{k}$ belongs to $H^{2}\left(B_{1}\times(-1,1)\right)$ is an immediate consequence of differentiation under the integral sign. Actually we have

[TABLE]

hence by Schwarz inequality and taking into account that $u\in\mathcal{W}\left([-1,1];B_{1}\right)$ we have $v_{k}\in H^{2}\left(B_{1}\times(-1,1)\right)$.

Now we prove (3.11).

By integration by parts and taking into account that

[TABLE]

we have

[TABLE]

Hence we have

[TABLE]

Similarly we have

[TABLE]

Now, by (2.4), (3.15), (3.16) and (3.17) we have

[TABLE]

where

[TABLE]

and (3.11) is proved.

Now we prove (3.12).

It is easy to check that, for every $k\in\mathbb{N}$ we have

[TABLE]

In addition, since

[TABLE]

we have

[TABLE]

By (2.2), (2.6), (3.20a) and (3.20b) we have (3.12).

By Schwarz inequality and (3.21) we have, for any $R\in(0,1]$,

[TABLE]

hence, for $R=1$, taking into account (2.6), we obtain (3.13).

Finally, let us prove (3.14). To this purpose we firstly observe that applying (3.22) for $R=r_{0}$ and taking into account (2.5) we have

[TABLE]

Afterwards, since $v_{k}$ is solution to elliptic equation (3.11), the following Caccioppoli’s inequality, [Ca], [Gi], holds

[TABLE]

where $C$ depends on $\lambda$ only. Finally, by (3.7), (3.12), (3.23) and (3.24) we get (3.14). $\square$

Proof of Proposition 3.1

Set

[TABLE]

By (3.14) we have

[TABLE]

where $C$ depends on $\lambda$ and $\Lambda_{1}$ only and

[TABLE]

where $C_{1}=2\sqrt{5}\lambda^{-1/2}$.

Now we apply Theorem 2.3.

Denote

[TABLE]

and

[TABLE]

Let us define

[TABLE]

where $h$ belongs to $C^{2}_{0}\left(0,\psi_{0}\left(2R\right)\right)$ and satisfies

[TABLE]

where $c$ depends on $\lambda$ and $\Lambda$ only. Notice that if $2r_{1}\leq\varrho(x,y)\leq R$ then $\zeta(x,y)=1$ and if $\varrho(x,y)\geq 2R$ or $\varrho(x,y)\leq r_{1}$ then $\zeta(x,y)=0$.

By density, we can apply (2.18) to the function $w=\zeta v_{k}$ and we have, for every $\tau\geq\tau_{0}$,

[TABLE]

where $C$ depends on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only and

[TABLE]

Estimate of $I_{1}$.

By (2.19) we have

[TABLE]

where $C_{2}>2$ depends on $\lambda$ and $\Lambda$ only.

By (3.12), (3.29a) and (3.30) we have

[TABLE]

where $C$ depends on $\lambda$ and $\Lambda$ only.

Now let $k$ and $\tau$ satisfy

[TABLE]

By (3.31) and (3.32) we have

[TABLE]

Estimate of $I_{2}$.

By (3.13), (3.25) and (3.29b) we have

[TABLE]

hence, by (3.30) we have

[TABLE]

where $C$ depends on $\lambda$ and $\Lambda$ only.

Estimate of $I_{3}$.

By (3.29c) we have

[TABLE]

Now in order to estimate from above the righthand side of (3.35) we use the Caccioppoli inequality, (3.12), (3.13) and (3.25) and we get

[TABLE]

where $C$ depends on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only.

Let $r_{1}\leq\frac{R}{2}$ and let $s$ be such that $\frac{2r_{1}}{\sqrt{\lambda}}\leq s\leq\frac{R}{\sqrt{\lambda}}$. Denote

[TABLE]

By estimating from below trivially the left hand side of (3.28) and taking into account (3.36) we get

[TABLE]

where $C$ depends on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only.

Now, by (2.16), (3.25) and into account that $\psi_{0}(\widetilde{s})\geq\psi_{0}(r_{1})$ we have

[TABLE]

Now let us add at both the side of (3.37) the quantity

[TABLE]

and by (3.38) we have

[TABLE]

where $C$ depends on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only. Moreover, by (3.33), (3.34) and (3.36) we have

[TABLE]

Now by (3.30), (3.33), (3.34), (3.36) and (3.40) we have that, if (3.32) is satisfied then

[TABLE]

where $C$ depends on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only and

[TABLE]

By a standard trace inequality we have

[TABLE]

and Lemma (3.2) implies

[TABLE]

where $C$ depend on $\lambda$, $\Lambda$ and $\Lambda_{1}$ only.

Now, we choose $k=\tau$ in (3.44) and using trivial inequality we have that, for any $0<\alpha<\frac{1}{2}$ there exist constants $C_{3}>1$ and $k_{0}$ depending on $\lambda$, $\Lambda$, $\Lambda_{1}$ and $\alpha$ only such that for every $k\geq k_{0}$ we have

[TABLE]

where

[TABLE]

Let us denote

[TABLE]

If $k_{\ast}\geq k_{0}$ then we choose $k=k_{\ast}$ and by (3.45) we have, for

[TABLE]

[TABLE]

where

[TABLE]

Otherwise, if $k_{\ast}<k_{0}$ then $\frac{\log\varepsilon_{1}}{2\log r_{1}}<k_{0},$ hence

[TABLE]

This implies

[TABLE]

that, in turns, taking into account (2.6), gives trivially

[TABLE]

Finally, by (3.46) and (3.48) we obtain (3.1). $\Box$

Conclusion of the proof of Theorem 2.1.

Let $t_{0}\in(-T,T)$. It is not restrictive to assume $t_{0}\geq 0$. Denote

[TABLE]

and

[TABLE]

It is easy to check that $\widetilde{u}$ is a solution to

[TABLE]

where

[TABLE]

By (2.1a) and (2.1b) we have respectively

[TABLE]

where

[TABLE]

By (2.2) we have

[TABLE]

In addition, by (2.5), (2.6) we have respectively

[TABLE]

and

[TABLE]

Now we apply Proposition 3.1. Denoting $s=\rho\rho_{0}^{-1}$ we have $0<r_{0}\rho_{0}^{-1}<s\leq s_{0}$, therefore

[TABLE]

where

[TABLE]

Finally, come back to the variables $x$ and $t$ we get (2.7). $\Box$

Acknowledgment

The paper was partially supported by GNAMPA - INdAM.