Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

Abstract

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., $λ | u |^{2} u$ . Our aim is to obtain the two results. One asserts that, if the $L^{2}$ -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ${(log t)}^{- 1 / 2}$ , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ${(log t)}^{- 1 / 2}$ in $L^{2}$ .

Keywords

Nonlinear Schrödinger equation decay estimate critical dissipative nonlinearity

1. Introduction and main results

We consider the Cauchy problem of 1D-nonlinear Schrödinger equation: $\begin{matrix} (1.1) & \{\begin{matrix} i \partial_{t} u + \frac{1}{2} \partial_{x}^{2} u = λ | u |^{2} u, \\ u (0, x) = u_{0} (x), \end{matrix} \end{matrix}$ where $t \in [0, \infty)$ , $x \in R$ , $λ = λ_{1} + i λ_{2}$ ( $λ_{1}, λ_{2} \in R$ ) and $λ_{2} < 0$ . By considering the associated nonlinear ODE: $i \partial_{t} u = λ | u |^{2} u$ , we see that $\partial_{t} | u |^{2} = λ_{2} | u |^{4} ⩽ 0$ . Therefore the condition $λ_{2} < 0$ implies the dissipation.

It is well known that the asymptotic behavior of solutions to (1.1) is different from that of solutions to the corresponding free equation (see [1]). There are some researches on the decay of the solutions. If $λ \in R$ , i.e., in the case that the $L^{2}$ -norm of the solution is conserved, it is easy to see that $‖ u (t) ‖_{L^{2}}$ does not decay except for $u_{0} \equiv 0$ . If $λ_{2} < 0$ , Shimomura [10] proved that $‖ u (t) ‖_{L^{2}} \to 0$ as $t \to \infty$ for small initial data in weighted Sobolev space (cf. Sunagawa [11] for the Klein–Gordon equation with a dissipative nonlinear term). The decay property is found to be caused by the dissipative nonlinearity. In fact, considering the free Schrödinger equation, we easily see that the $L^{2}$ -norm of the solution is conserved and never decays except for the trivial solution. We are next interested in specifying the decay rate of $‖ u (t) ‖_{L^{2}}$ . Hayashi–Li–Naumkin [3] proved $‖ u (t) ‖_{L^{2}} = O ({(log t)}^{- 1 / 3 + ε})$ , with $ε > 0$ , as $t \to \infty$ for $u_{0} \in H^{1} (R)$ and $x u_{0} \in L^{2} (R)$ . It was refined by Ogawa–Sato in [7], where $‖ u (t) ‖_{L^{2}} = O ({(log t)}^{- 1 / 2 + 1 / (4 m + 2) + ε})$ was proved for $u_{0} \in H^{m + 1} (R)$ and $x \partial_{x}^{m} u_{0} \in L^{2} (R)$ – it suggests that, if the regularity and spatial decay of $u_{0}$ are better, then the $L^{2}$ -norm of the solution decays more rapidly. Furthermore Sato [9] obtained $‖ u (t) ‖_{L^{2}} = O ({(log t)}^{- 1 / 2} {(log log t)}^{s / 2})$ for $u_{0}$ in weighted Gevery class with $s ⩾ 1$ . Viewing these recent progress, we notice that the decay rate of $‖ u (t) ‖_{L^{2}}$ is closer and closer to $O ({(log t)}^{- 1 / 2})$ .

The authors have a conjecture that the optimal $L^{2}$ -decay rate of the solution is $O ({(log t)}^{- 1 / 2})$ . Precisely speaking, we expect that, if $‖ u (t) ‖_{L^{2}}$ decays more rapidly than ${(log t)}^{- 1 / 2}$ , then $u_{0} \equiv 0$ . In addition, we infer that, if an initial datum is suitably selected, then $‖ u (t) ‖_{L^{2}}$ decays just at the rate of ${(log t)}^{- 1 / 2}$ . Our purpose in this paper is to verify this conjecture.

It is interesting to see the decay rate of $u (t)$ not only in $L^{2}$ but also in $L^{\infty}$ . As far as the cubic nonlinearity is concerned, Hayashi–Naumkin [4] detected the asymptotic profile of small amplitude data for $λ \in R$ – they proved that the profiles of solutions include modifications in phase, which is caused by the critical nonlinearity, and $‖ u (t) ‖_{L^{\infty}} = O (t^{- 1 / 2})$ where the nonlinear effect is invisible in the decay rate. Shimomura [10] considered the case of $λ_{2} < 0$ and proved $‖ u (t) ‖_{L^{\infty}} = O (t^{- 1 / 2} {(log t)}^{- 1 / 2})$ – this time, the nonlinearly dissipative effect is visible in the decay rate. It was shown that the nonlinear dissipative property holds true without size-restriction of $u_{0}$ if so-called the strong dissipative conditions, i.e., $λ_{2} < 0$ and $| λ_{1} | ⩽ \sqrt{3} | λ_{2} |$ are assumed (see [6]). The n-dimensional version (without upper bound of n) for both $λ \in R$ and $λ_{2} < 0$ was considered by Cazenave–Naumkin [2]. According to [5], the optimal decay rate in $L^{\infty}$ is $t^{- 1 / 2} {(log t)}^{- 1 / 2}$ . Namely, if a solution to (1.1) admits $‖ u (t) ‖_{L^{\infty}} = o (t^{- 1 / 2} {(log t)}^{- 1 / 2})$ , then $u_{0} \equiv 0$ . As a suggestion from the free Schrödinger equation, the optimal decay rate in $L^{2}$ is obtained by multiplying the decay rate in $L^{\infty}$ with $t^{1 / 2}$ . That is why we have the conjecture that the optimal $L^{2}$ -decay rate is possibly $O ({(log t)}^{- 1 / 2})$ .

To state our results, we define the weighted Sobolev space as follows. Let $S (R)$ be the set of rapidly decreasing smooth functions. For any $f \in S (R)$ , we define the Fourier transform and the Fourier inverse transform in such a way that $\begin{matrix} F [f] (ξ) = \hat{f} (ξ) \equiv {(2 π)}^{- 1 / 2} \int_{R} e^{- i x \cdot ξ} f (x) d x, \\ F^{- 1} [f] (x) \equiv {(2 π)}^{- 1 / 2} \int_{R} e^{i x \cdot ξ} f (ξ) d ξ . \end{matrix}$ For $s > 0$ and $r > 0$ , let $\begin{matrix} H^{s, r} \equiv {f \in L^{2} (R); ‖ f ‖_{H^{s, r}} \equiv {‖ {⟨ x ⟩}^{r} {⟨ \partial_{x} ⟩}^{s} f ‖}_{L^{2}} < \infty}, \end{matrix}$ where $⟨ x ⟩ \equiv {(1 + | x |^{2})}^{1 / 2}$ and ${⟨ \partial_{x} ⟩}^{s} f \equiv F^{- 1} [{⟨ ξ ⟩}^{s} \hat{f}]$ , and we denote $H^{s, 0} = H^{s}$ .

Our main results are the followings. The first result concerns the lower bound of the time-global solution.

Theorem 1.1.
Let $λ_{2} < 0$ and $u_{0} \in H^{1} \cap H^{0, 1}$ with $‖ u_{0} ‖_{H^{1} \cap H^{0, 1}} \neq 0$ sufficiently small. Then, there exists a unique time-global solution to (1.1) in $C ([0, \infty); H^{1} \cap H^{0, 1}) \cap C^{1} ([0, \infty); H^{- 1})$ . Furthermore there exist some $C > 0$ and $T > 0$ such that $\begin{matrix} (1.2) & {‖ u (t) ‖}_{L^{2}} ⩾ C {(log t)}^{- 1 / 2} {‖ u (T) ‖}_{L^{2}} \end{matrix}$ holds true for any $t > T$ .

Theorem 1.1 implies that the solution to (1.1) does not decay faster than ${(log t)}^{- 1 / 2}$ in $L^{2}$ -topology except for $u_{0} \equiv 0$ . We find that, if the solution admits more rapid decay than ${(log t)}^{- 1 / 2}$ , such a solution must be trivial. Indeed, as a byproduct of Theorem 1.1, we can easily show the next corollary.
Corollary 1.2.
Suppose the assumptions in Theorem 1.1 . If the time-global solution admits a decay estimate like $\begin{matrix} (1.3) & {‖ u (t) ‖}_{L^{2}} = o ({(log t)}^{- 1 / 2}) \end{matrix}$ as $t \to \infty$ , then $u_{0} \equiv 0$ .

By Corollary 1.2, one expects that the logarithmic decay order is optimal. The next result asserts that ${(log t)}^{- 1 / 2}$ is truly the optimal decay rate of solutions.
Theorem 1.3.
Let $λ_{2} < 0$ . Then there exists some $u_{0} \in H^{1} \cap H^{0, 1}$ such that the corresponding solution $u (t)$ of (1.1) admits $\begin{matrix} {‖ u (t) ‖}_{L^{2}} ⩽ C {(log t)}^{- 1 / 2} \end{matrix}$ for sufficiently large $t > 0$ .

Before closing this section, let us introduce some more notation. We define the dilation operator by $\begin{matrix} (1.4) & (D ϕ) (x) = {(i t)}^{- 1 / 2} ϕ (x / t) \end{matrix}$ and define $M = e^{i x^{2} / 2 t}$ for $t \neq 0$ . Then the Schrödinger group $U (t) = exp (i t \partial_{x}^{2} / 2)$ possesses a factorization formula such as $\begin{matrix} U (t) = M D F M . \end{matrix}$ Since $U (- t) = U {(t)}^{- 1}$ , we also have $\begin{matrix} U (- t) = M^{- 1} F^{- 1} D^{- 1} M^{- 1} . \end{matrix}$ The standard generator of the Galilei transformation is given as $\begin{matrix} J (t) = U (t) x U (- t) = x + i t \partial_{x} . \end{matrix}$ We use the same letter C for various positive constants.
2. Proof of Theorem 1.1 & Corollary 1.2

The $L^{\infty}$ estimate of the solution plays an important role. According to [6, 10], we have already had $‖ u (t) ‖_{L^{\infty}} ⩽ K t^{- 1 / 2} {(log t)}^{- 1 / 2}$ for large $t > 0$ . To prove Theorem 1.1, however, the explicit form of the constant K is required.

Lemma 2.1.
Under the assumptions in Theorem 1.1 , there exists some $C > 0$ and $T > 0$ such that $\begin{matrix} (2.1) & {‖ u (t) ‖}_{L^{\infty}}^{2} ⩽ {(2 | λ_{2} |)}^{- 1} t^{- 1} {(log t)}^{- 1} + C t^{- 1} {(log t)}^{- 2} \end{matrix}$ holds for any $t > T$ .
Proof of Lemma 2.1.
For the small initial datum $u_{0}$ , the time-global existence of the solution was proved in [10]. In particular, $‖ J u (t) ‖_{L^{2}} = ‖ U (t) x U (- x) u (t) ‖_{L^{2}}$ , which contributes to the estimate of error terms, admits an upper bound such as $\begin{matrix} (2.2) & {‖ J u (t) ‖}_{L^{2}} ⩽ C {(1 + t)}^{δ}, \end{matrix}$ where the growth order $δ > 0$ is taken as small as we like if the size of the initial datum in $H^{1} \cap H^{0, 1}$ is assumed to be small enough. Let $v (t) = U (- t) u (t)$ , where $U (t) = exp (i t \partial_{x}^{2} / 2)$ is the Schrödinger group, then the equation (1.1) can be rewritten in such a way that $\begin{matrix} (2.3) & i \partial_{t} v = λ U (- t) N (U (t) v), \end{matrix}$ where $N (u) = | u |^{2} u$ . Applying the factorization $U (t) = M D F M$ and $U (- t) = M^{- 1} F^{- 1} D^{- 1} M^{- 1}$ to the right hand side of (2.3), we see that $\begin{matrix} (2.4) & i \partial_{t} v = λ M^{- 1} F^{- 1} D^{- 1} M^{- 1} N (M D F M v) . \end{matrix}$ Since the nonlinearity is gauge-invariant, i.e., $N (e^{i θ} u) = e^{i θ} N (u)$ , $M^{- 1}$ and $M = e^{i x^{2} / 2 t}$ in neighbor of $N$ are canceled. From $D^{- 1}$ and D (inside $N$ ) which is the dilation operator defined by (1.4), the decaying factor $t^{- 1}$ is extracted. Taking the Fourier transform on both hand sides of (2.3), we have $\begin{array}{rcl} i \partial_{t} \hat{v} & = & λ t^{- 1} F M^{- 1} F^{- 1} N (F M v) \\ (2.5) & = & λ t^{- 1} N (\hat{v}) + R (t), \end{array}$ where the remainder term is described as $\begin{array}{rcl} R (t) & = & λ t^{- 1} {F M^{- 1} F^{- 1} N (F M v) - N (F v)} \\ = & λ t^{- 1} F (M^{- 1} - 1) F^{- 1} N (F M v) \\ (2.6) & + λ t^{- 1} {N (F M v) - N (F v)} . \end{array}$ For the purpose of the estimate of $‖ R (t) ‖_{L^{\infty}}$ , we apply the Sobolev embedding $‖ f ‖_{L^{\infty}} ⩽ C ‖ f ‖_{L^{2}}^{1 / 2} \cdot ‖ \partial_{x} f ‖_{L^{2}}^{1 / 2}$ , Plancerell’s identity, $| M^{- 1} - 1 | ⩽ | x | / \sqrt{2 t}$ and $| x (M^{- 1} - 1) | ⩽ 2 | x |$ etc. to (2.6). Then we have $\begin{array}{l} {‖ R (t) ‖}_{L^{\infty}} \\ ⩽ C t^{- 1} {‖ F (M^{- 1} - 1) F^{- 1} N (F M v) ‖}_{L^{\infty}} \\ + C t^{- 1} (‖ F M v ‖_{L^{\infty}}^{2} + ‖ F v ‖_{L^{\infty}}^{2}) {‖ F (M - 1) v ‖}_{L^{\infty}} \\ ⩽ C t^{- 1} {‖ F (M^{- 1} - 1) F^{- 1} N (F M v) ‖}_{L^{2}}^{1 / 2} \\ \times {‖ \partial_{ξ} F (M^{- 1} - 1) F^{- 1} N (F M v) ‖}_{L^{2}}^{1 / 2} \\ + C t^{- 1} ‖ v ‖_{L^{2}} ‖ x v ‖_{L^{2}} {‖ F (M - 1) v ‖}_{L^{2}}^{1 / 2} \cdot {‖ \partial_{ξ} F (M - 1) v ‖}_{L^{2}}^{1 / 2} \\ (2.7) & ⩽ C t^{- 5 / 4} ‖ v ‖_{L^{2}} ‖ x v ‖_{L^{2}}^{2} . \end{array}$ Note that $‖ v (t) ‖_{L^{2}} = ‖ u (t) ‖_{L^{2}} ⩽ ‖ u_{0} ‖$ which follows from the dissipative effect of nonlinearity, and invoke $‖ x v (t) ‖_{L^{2}} = ‖ J u (t) ‖_{L^{2}} ⩽ C t^{δ}$ as in (2.2). Then (2.6) and (2.7) yield $\begin{matrix} (2.8) & {‖ R (t) ‖}_{L^{\infty}} ⩽ C t^{- 5 / 4 + 2 δ} . \end{matrix}$ Keep (2.8) in our mind, and we proceed in the estimate of $| \hat{v} (t, ξ) |^{2}$ . From the equation (2.5), estimate (2.8) and $‖ \hat{v} (t) ‖_{L^{\infty}} ⩽ C ‖ u (t) ‖_{L^{2}}^{1 / 2} ‖ J u (t) ‖_{L^{2}}^{1 / 2}$ , it follows that $\begin{matrix} (2.9) & \partial_{t} {| \hat{v} (t) |}^{2} = 2 λ_{2} t^{- 1} {| \hat{v} (t) |}^{4} + O (t^{- 5 / 4 + 5 δ / 2}) . \end{matrix}$ From (2.9), we see that $\begin{matrix} (2.10) & \begin{matrix} \partial_{t} ({(log t)}^{2} {| \hat{v} (t) |}^{2}) = & 2 t^{- 1} (log t) {| \hat{v} (t) |}^{2} \\ + {(log t)}^{2} (- 2 | λ_{2} | t^{- 1} {| \hat{v} (t) |}^{4} + O (t^{- 5 / 4 + 5 δ / 2})) \\ = & 2 t^{- 1} (log t) {| \hat{v} (t) |}^{2} - 2 | λ_{2} | {(log t)}^{2} t^{- 1} {| \hat{v} (t) |}^{4} \\ + O (t^{- 5 / 4 + 3 δ}) . \end{matrix} \end{matrix}$ By Young’s inequality, it follows that $\begin{matrix} {| \hat{v} (t) |}^{2} ⩽ \frac{1}{4} | λ_{2} |^{- 1} {(log t)}^{- 1} + | λ_{2} | (log t) {| \hat{v} (t) |}^{4} \end{matrix}$ and hence (2.10) yields $\begin{matrix} (2.11) & \partial_{t} ({(log t)}^{2} {| \hat{v} (t) |}^{2}) ⩽ {(2 | λ_{2} |)}^{- 1} t^{- 1} + O (t^{- 5 / 4 + 3 δ}) . \end{matrix}$ By integrating (2.11) over $[1, t]$ , one can obtain $\begin{matrix} (2.12) & {‖ \hat{v} (t) ‖}_{L^{\infty}}^{2} ⩽ {(2 | λ_{2} |)}^{- 1} {(log t)}^{- 1} + O ({(log t)}^{- 2}) \end{matrix}$ for any $t ⩾ 1$ . Since we have $\begin{array}{rcl} {‖ u (t) ‖}_{L^{\infty}} & = & {‖ U (t) v ‖}_{L^{\infty}} \\ = & ‖ M D F M v ‖_{L^{\infty}} \\ ⩽ & t^{- 1 / 2} ‖ \hat{v} ‖_{L^{\infty}} + t^{- 1 / 2} {‖ F (M - 1) v ‖}_{L^{\infty}} \end{array}$ and $‖ F (M - 1) v ‖_{L^{\infty}} ⩽ C t^{- 1 / 4} ‖ J u (t) ‖_{L^{2}} ⩽ C t^{- 1 / 4 + δ}$ , (2.12) yields $\begin{array}{rcl} {‖ u (t) ‖}_{L^{\infty}}^{2} & ⩽ & t^{- 1} {‖ \hat{v} (t) ‖}_{L^{\infty}}^{2} + O (t^{- 5 / 4 + δ}) \\ ⩽ & {(2 | λ_{2} |)}^{- 1} t^{- 1} {(log t)}^{- 1} + O (t^{- 1} {(log t)}^{- 2}) . \end{array}$ Hence, we obtain the desired estimate (2.1). □

Now we are ready for the proof of Theorem 1.1.
Proof of Theorem 1.1.
Multiplying $2 \overline{u (t)}$ on both hand sides of (1.1), taking the integration and next taking the imaginary part, we have $\begin{matrix} \frac{d}{d t} {‖ u (t) ‖}_{L^{2}}^{2} = - 2 | λ_{2} | {‖ u (t) ‖}_{L^{4}}^{4} . \end{matrix}$ Since $‖ u (t) ‖_{L^{4}}^{4} ⩽ ‖ u (t) ‖_{L^{\infty}}^{2} ‖ u (t) ‖_{L^{2}}^{2}$ , we have $\begin{matrix} \frac{d}{d t} {‖ u (t) ‖}_{L^{2}}^{2} ⩾ - 2 | λ_{2} | {‖ u (t) ‖}_{L^{\infty}}^{2} {‖ u (t) ‖}_{L^{2}}^{2} . \end{matrix}$ Apply Lemma 2.1 to the above. Then we see that, for sufficiently large $t > 0$ , $\begin{matrix} \frac{d}{d t} {‖ u (t) ‖}_{L^{2}}^{2} ⩾ - (t^{- 1} {(log t)}^{- 1} + C t^{- 1} {(log t)}^{- 2}) {‖ u (t) ‖}_{L^{2}}^{2} . \end{matrix}$ Take $T > 0$ sufficiently large. Then, for $t > T$ , Gronwall’s inequality yields $\begin{matrix} {‖ u (t) ‖}_{L^{2}}^{2} ⩾ {‖ u (T) ‖}_{L^{2}}^{2} \frac{log T}{log t} e^{C ({(log t)}^{- 1} - {(log T)}^{- 1})}, \end{matrix}$ and it follows that $\begin{matrix} {‖ u (t) ‖}_{L^{2}} ⩾ C {‖ u (T) ‖}_{L^{2}} {(log t)}^{- 1 / 2} . \end{matrix}$ □

We next prove Corollary 1.2. Proof of Corollary 1.2.
By the assumption (1.3), we see that $\begin{matrix} {(log t)}^{1 / 2} {‖ u (t) ‖}_{L^{2}} \to 0 \end{matrix}$ as $t \to \infty$ . According to (1.2), we have $‖ u (T) ‖_{L^{2}} = 0$ for some large $T > 0$ . Solving (1.1) with $u (T) \equiv 0$ as an initial datum, we easily find that the solution is trivial and $u_{0} \equiv 0$ . □

3. Proof of Theorem 1.3

We will apply the pseudo-conformal transform of the unknown variable. For reader’s convenience, the way of this transform is heuristically introduced – the idea of our proof relies on the asymptotic analysis due to Ozawa [8], where a final value problem was considered for the $L^{2}$ -conserving nonlinear Schrödinger equations so that solutions approaching to some designated profiles are obtained. Let $v (t) = U (- t) u (t)$ , where $U (t) = exp (i t \partial_{x}^{2} / 2)$ . Then, following the derivation of (2.5), we have $\begin{matrix} i \partial_{t} v = λ t^{- 1} M^{- 1} F^{- 1} N (F M v) . \end{matrix}$ Applying $F M$ , we see that $\begin{matrix} i F M \partial_{t} v = λ t^{- 1} N (F M v) . \end{matrix}$ Since $M \partial_{t} = \partial_{t} M + (i x^{2} / 2 t^{2}) M$ , we see that $\begin{matrix} i \partial_{t} F M v + \frac{1}{2 t^{2}} \partial_{ξ}^{2} F M v = λ t^{- 1} N (F M v) . \end{matrix}$ Let $v_{1} (t, ξ) = F M v$ . Subsequently we replace t by $1 / s$ so that the problem for large $t > 0$ is reduced into the problem for small $s > 0$ . For this purpose, we let $w (s, ξ) = v_{1} (1 / s, ξ)$ or $w (1 / t, ξ) = v_{1} (t, ξ)$ . Some computation yields $\begin{matrix} (3.1) & i \partial_{s} w - \frac{1}{2} \partial_{ξ}^{2} w = - λ s^{- 1} N (w) . \end{matrix}$ We here remark that w is called “the pseudo-conformal transform” of u. Hereafter we will solve (3.1) near $s = 0$ , and obtain the desired optimally decaying solution.

A profile of decaying solution. As the decaying solution is caused by the nonlinearity, the profile (denoted by $φ = φ (s, ξ)$ ) is expected to be obtained by the ODE, i.e., $\begin{matrix} (3.2) & i \partial_{s} φ = - λ s^{- 1} N (φ) . \end{matrix}$ This is, of course, derived from (3.1) by dropping out the dispersive term. We will find, in a heuristic way, a solution to (3.2) such as ${lim}_{s ↓ 0} φ (s, ξ) = 0$ .

Multiply $2 \bar{φ}$ on both hand sides of (3.2), and take the imaginary part. Then we have $\begin{matrix} \partial_{s} | φ | = - λ_{2} s^{- 1} | φ |^{3} . \end{matrix}$ Dividing both hand sides with $| φ |^{3}$ , we have $\begin{matrix} (3.3) & \partial_{s} | φ |^{- 2} = 2 λ_{2} s^{- 1} . \end{matrix}$ Let $φ (1, ξ) = φ_{1} (ξ)$ . Then, integrating (3.3) from 1 to s, we see that $\begin{matrix} {| φ (s, ξ) |}^{- 2} - {| φ_{1} (ξ) |}^{- 2} = 2 λ_{2} log s . \end{matrix}$ Thus we have $\begin{matrix} (3.4) & {| φ (s, ξ) |}^{2} = \frac{| φ_{1} (ξ) |^{2}}{1 - 2 | λ_{2} | | φ_{1} (ξ) |^{2} log s} . \end{matrix}$ Substitute (3.4) into (3.2). Then we see that $\begin{array}{rcl} i \partial_{s} φ & = & - λ \frac{s^{- 1} | φ_{1} (ξ) |^{2}}{1 - 2 | λ_{2} | | φ_{1} (ξ) |^{2} log s} φ \\ = & \frac{λ}{2 | λ_{2} |} \frac{- 2 | λ_{2} | s^{- 1} | φ_{1} (ξ) |^{2}}{1 - 2 | λ_{2} | | φ_{1} (ξ) |^{2} log s} φ \\ = & \frac{λ}{2 | λ_{2} |} {(log (1 - 2 | λ_{2} | {| φ_{1} (ξ) |}^{2} log s))}^{'} φ . \end{array}$ Solving this ODE, we obtain $\begin{matrix} (3.5) & φ (s, ξ) = φ_{1} (ξ) {(1 - 2 | λ_{2} | {| φ_{1} (ξ) |}^{2} log s)}^{- i λ / 2 | λ_{2} |} . \end{matrix}$ Note that, if $φ_{1}$ is compactly supported, $‖ φ (s, \cdot) ‖_{L^{2}} = O (| log s |^{- 1 / 2})$ as $s ↓ 0$ .

The existence of a solution to ( 3.1 ). We will find a solution by adding a perturbation $\tilde{w}$ to the profile (3.5). Let $w = φ + \tilde{w}$ , and substitute it into (3.1). Then the equation of $\tilde{w}$ is found to be $\begin{matrix} (3.6) & i \partial_{s} \tilde{w} - \frac{1}{2} \partial_{ξ}^{2} \tilde{w} = \frac{1}{2} \partial_{ξ}^{2} φ - λ s^{- 1} {N (φ + \tilde{w}) - N (φ)} . \end{matrix}$ As an initial datum, it is natural to impose $\begin{matrix} (3.7) & \tilde{w} (0, ξ) \equiv 0 \end{matrix}$ for any $ξ \in R$ . For by the dissipative setting $λ_{2} < 0$ , ${lim}_{s ↓ 0} ‖ w (s) ‖_{L^{2}} = 0$ is now expected. We solve (3.6)–(3.7), and want to obtain a solution $\tilde{w}$ vanishing more rapidly in comparison with φ as $s ↓ 0$ . To solve (3.6)–(3.7), we consider the associated integral equation: $\begin{array}{rcl} \tilde{w} & = & Φ (\tilde{w}) \\ \equiv & \frac{i}{2} \int_{0}^{s} U (σ - s) \partial_{ξ}^{2} φ d σ \\ (3.8) & + i λ \int_{0}^{s} U (σ - s) σ^{- 1} {N (φ + \tilde{w}) - N (φ)} d σ, \end{array}$ and the contraction mapping principle will be applied in $C ([0, S]; H^{1} \cap H^{0, 1})$ . Let us provide conditions on $φ_{1} (ξ)$ in the form of the profile φ (see (3.5)). We assume that $\begin{matrix} (3.9) & φ_{1} (ξ) = \{\begin{matrix} e^{- \frac{1}{1 - ξ^{2}}} & (ξ \in (- 1, 1)) \\ 0 & (ξ \in (- \infty, - 1] \cup [1, \infty)) . \end{matrix} \end{matrix}$ Then we see that $φ_{1} \in C_{0}^{\infty} (R)$ and $φ_{1} (ξ) \in R$ .

Proposition 3.1.

Let $φ_{1}$ satisfy (3.9). Then, if $S > 0$ is sufficiently small, there exists a unique solution $\tilde{w}$ to the integral equation (3.8) in $C ([0, S]; H^{1} \cap H^{0, 1}) \cap C^{1} ([0, \infty); H^{- 1})$ . Furthermore, for some constant $C > 0$ , the solution satisfies $\begin{matrix} (3.10) & {‖ \tilde{w} (s) ‖}_{H^{1}} + {‖ ξ \tilde{w} (s) ‖}_{L^{2}} ⩽ C s \end{matrix}$ for $s \in [0, S]$ .

Proof of Proposition 3.1.

We define the function space X such as $X = {\tilde{w}; | | | \tilde{w} | | | < \infty}$ , where $\begin{matrix} | | | \tilde{w} | | | = sup_{0 ⩽ s ⩽ S} ({‖ s^{- 1} \tilde{w} (s) ‖}_{H^{1}} + {‖ s^{- 1} ξ \tilde{w} (s) ‖}_{L^{2}}) . \end{matrix}$ Let us consider the map Φ of (3.8) on a closed unit ball ${\bar{B}}_{1} (X) = {\tilde{w}; | | | \tilde{w} | | | ⩽ 1}$ . For $\tilde{w} \in {\bar{B}}_{1} (X)$ , we claim $\begin{array}{rcl} | | | Φ (\tilde{w}) | | | & ⩽ & C_{0, S} + C_{1, S} (| | | \tilde{w} | | | + | | | \tilde{w} | | |^{3}) \\ (3.11) & ⩽ & C_{0, S} + 2 C_{1, S}, \end{array}$ where $C_{0, S}$ and $C_{1, S}$ are positive constants which tends to 0 as $S ↓ 0$ . Thus, taking $S > 0$ sufficiently small, we see that Φ maps ${\bar{B}}_{1} (X)$ into itself. We show (3.11) as follows. By ${(1 - ξ^{2})}^{- 1} ⩾ 1$ and $| φ (ξ) | ⩽ 1$ for $| ξ | < 1$ , we see that, for any $k ⩾ 1$ , $\begin{matrix} {(1 - ξ^{2})}^{- 1} ⩽ {(1 - ξ^{2})}^{- k}, {| φ (ξ) |}^{k} ⩽ | φ (ξ) | . \end{matrix}$ Thus, by differentiating $φ_{1} (ξ) = \exp {- {(1 - ξ^{2})}^{- 1}}$ , there exists some constant $C > 0$ such that $\begin{matrix} (3.12) & {‖ \partial_{ξ}^{3} φ (s) ‖}_{L^{\infty}} ⩽ C sup_{- 1 < ξ < 1} | \frac{P (ξ)}{{(1 - ξ^{2})}^{6}} \frac{| φ_{1} (ξ) |}{| 1 - 2 | λ_{2} | | φ_{1} (ξ) |^{2} log s |^{1 / 2}} |, \end{matrix}$ where $P (ξ)$ is a polynomial of ξ. Note that $\begin{matrix} lim_{ξ \to \pm 1} {(1 - ξ^{2})}^{- 6} exp (- a {(1 - ξ^{2})}^{- 1}) = 0 \end{matrix}$ for any $a \in (0, 1)$ . Also we see that $\begin{array}{rcl} \frac{| φ_{1} (ξ) |^{1 - a}}{| 1 - 2 | λ_{2} | | φ_{1} (ξ) |^{2} log s |^{1 / 2}} & ⩽ & \frac{1}{{(2 | λ_{2} | | log s |)}^{(1 - a) / 2}} \cdot sup_{0 ⩽ η} \frac{η^{(1 - a) / 2}}{\sqrt{1 + η}} \\ ⩽ & C | log s |^{- (1 - a) / 2} . \end{array}$ Then it follows from (3.12) that, for some $θ \in (0, 1 / 2)$ and $C > 0$ , we have $\begin{matrix} (3.13) & {‖ \partial_{ξ}^{j} φ (s) ‖}_{L^{\infty}} ⩽ C | log s |^{- θ} (0 ⩽ j ⩽ 3) . \end{matrix}$ Similarly we are able to show that, for any $s > 0$ , $\begin{matrix} (3.14) & {‖ ξ \partial_{ξ}^{2} φ (s) ‖}_{L^{\infty}} ⩽ C | log s |^{- θ} . \end{matrix}$

We next estimate the nonlinear term in (3.8). By the Gagliardo–Nirenberg inequality, we have $\begin{matrix} (3.15) & \begin{matrix} {‖ \partial_{ξ} {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} \\ ⩽ C {‖ (| φ | + | \tilde{w} |) \tilde{w} \partial_{ξ} φ ‖}_{L^{2}} + C {‖ (| φ |^{2} + | \tilde{w} |^{2}) \partial_{ξ} \tilde{w} ‖}_{L^{2}} \\ ⩽ C (‖ φ ‖_{L^{\infty}} ‖ \partial_{ξ} φ ‖_{L^{\infty}} ‖ \tilde{w} ‖_{L^{2}} + ‖ φ ‖_{L^{\infty}} ‖ \partial_{ξ} φ ‖_{L^{\infty}} ‖ \tilde{w} ‖_{L^{2}}^{3 / 2} ‖ \partial_{ξ} \tilde{w} ‖_{L^{2}}^{1 / 2}) \\ + C (‖ φ ‖_{L^{\infty}}^{2} ‖ \partial_{ξ} \tilde{w} ‖_{L^{2}} + ‖ \tilde{w} ‖_{L^{2}} ‖ \partial_{ξ} \tilde{w} ‖_{L^{2}}^{2}) . \end{matrix} \end{matrix}$ Since $\tilde{w} \in {\bar{B}}_{1} (X)$ , (3.13) yields $\begin{matrix} (3.16) & {‖ \partial_{ξ} {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} ⩽ C (σ | log σ |^{- 2 θ} + σ^{2} | log σ |^{- 2 θ} + σ^{3}) . \end{matrix}$ Similarly to (3.15) and (3.16), the weighted norm of the nonlinearity is estimated as $\begin{matrix} (3.17) & {‖ ξ {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} ⩽ C (σ | log σ |^{- 2 θ} + σ^{3}) . \end{matrix}$ Combining (3.13), (3.14), (3.16), we have $\begin{matrix} (3.18) & \begin{matrix} s^{- 1} {‖ \partial_{ξ} Φ (\tilde{w}) ‖}_{L^{2}} ⩽ & \frac{s^{- 1}}{2} \int_{0}^{s} {‖ \partial_{ξ}^{3} φ ‖}_{L^{2}} d σ \\ + | λ | s^{- 1} \int_{0}^{s} σ^{- 1} {‖ \partial_{ξ} {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} d σ \\ ⩽ & C s^{- 1} \int_{0}^{s} | log σ |^{- θ} d σ \\ + C s^{- 1} \int_{0}^{s} {| log σ |^{- 2 θ} + σ | log σ |^{- 2 θ} + σ^{2}} d σ \\ ⩽ & C (| log s |^{- θ} + | log s |^{- 2 θ}) . \end{matrix} \end{matrix}$ We also have the estimate of $Φ (\tilde{w})$ in the weighted $L^{2}$ -norm in such a way that $\begin{matrix} (3.19) & \begin{matrix} s^{- 1} {‖ ξ Φ (\tilde{w}) ‖}_{L^{2}} \\ ⩽ \frac{s^{- 1}}{2} \int_{0}^{s} {‖ ξ \partial_{ξ}^{2} φ ‖}_{L^{2}} d σ \\ + | λ | s^{- 1} \int_{0}^{s} σ^{- 1} {‖ ξ U (σ - s) {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} d σ . \end{matrix} \end{matrix}$ Since $U (- (σ - s)) ξ U (σ - s) = ξ + i (s - σ) \partial_{ξ}$ , we see that $\begin{matrix} {‖ ξ U (σ - s) {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} \\ ⩽ {‖ (ξ + 2 i (s - σ) \partial_{ξ}) {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} \\ ⩽ C ({‖ ξ {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} \\ + | s | {‖ \partial_{ξ} {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}}) . \end{matrix}$ By applying (3.16) and (3.17) to the above, we deduce $\begin{array}{l} {‖ ξ U (σ - s) {N (φ + \tilde{w}) - N (φ)} ‖}_{L^{2}} \\ (3.20) & ⩽ C (σ | log σ |^{- 2 θ} + σ^{3}) + C s (σ | log σ |^{- 2 θ} + σ^{2} | log σ |^{- 2 θ} + σ^{3}) . \end{array}$ Applying (3.20) to (3.19), we have $\begin{matrix} (3.21) & s^{- 1} {‖ ξ Φ (\tilde{w}) ‖}_{L^{2}} ⩽ C (| log s |^{- θ} + | log s |^{- 2 θ}) . \end{matrix}$ Therefore, by (3.18) and (3.21), we see that $\begin{matrix} (3.22) & | | | Φ (\tilde{w}) | | | ⩽ C (| log S |^{- θ} + | log S |^{- 2 θ}), \end{matrix}$ and thus the boundedness (3.11) holds for small $S > 0$ .

We next show that Φ is a contraction. In fact, for ${\tilde{w}}_{1}$ , ${\tilde{w}}_{2} \in {\bar{B}}_{X} (1)$ , we have $\begin{matrix} (3.23) & Φ ({\tilde{w}}_{1}) - Φ ({\tilde{w}}_{2}) = i λ \int_{0}^{s} U (s - σ) σ^{- 1} {N (φ + {\tilde{w}}_{1}) - N (φ + {\tilde{w}}_{2})} d σ . \end{matrix}$ Since the nonlinear term in (3.23) is estimated as $\begin{matrix} {‖ \partial_{ξ} {N (φ + {\tilde{w}}_{1}) - N (φ + {\tilde{w}}_{2})} ‖}_{L^{2}} \\ ⩽ C {‖ (| φ | + | \tilde{w_{1}} | + | \tilde{w_{2}} |) \partial_{ξ} φ (\tilde{w_{1}} - \tilde{w_{2}}) ‖}_{L^{2}} \\ + C {‖ (| φ |^{2} + | \tilde{w_{1}} |^{2} + | \tilde{w_{1}} |^{2}) (\partial_{ξ} \tilde{w_{1}} - \partial_{ξ} \tilde{w_{2}}) ‖}_{L^{2}}, \end{matrix}$ one can apply the analogy in the estimate of (3.11) and it follows that $\begin{array}{rcl} | | | Φ ({\tilde{w}}_{1}) - Φ ({\tilde{w}}_{2}) | | | & ⩽ & C | log S |^{- θ} (1 + | | | {\tilde{w}}_{1} | | |^{2} + | | | {\tilde{w}}_{2} | | |^{2}) | | | {\tilde{w}}_{1} - {\tilde{w}}_{2} | | | \\ ⩽ & 3 C | log S |^{- θ} | | | {\tilde{w}}_{1} - {\tilde{w}}_{2} | | | . \end{array}$ Taking $S > 0$ sufficiently small, we see that Φ is a contraction. Accordingly there exists a unique solution to (3.8), and Proposition 3.1 follows. □

Now we are ready for the proof of Theorem 1.3.

Proof of Theorem 1.3.

Using φ and $\tilde{w}$ of Proposition 3.1 and substituting $s = 1 / t$ , we have $\begin{matrix} F M v (t, ξ) = φ (1 / t, ξ) + \tilde{w} (1 / t, ξ) . \end{matrix}$ Apply $U (t) M^{- 1} F^{- 1}$ to the above identity. Then we see that $\begin{matrix} u (t, x) = U (t) M^{- 1} F^{- 1} (φ (1 / t, \cdot) + \tilde{w} (1 / t, \cdot)) . \end{matrix}$ Plancherell’s identity yields $\begin{array}{rcl} {‖ u (t) ‖}_{L^{2}} & = & {‖ φ (1 / t, \cdot) + \tilde{w} (1 / t, \cdot) ‖}_{L^{2}} \\ ⩽ & {‖ φ (1 / t, \cdot) ‖}_{L^{2}} + {‖ \tilde{w} (1 / t, \cdot) ‖}_{L^{2}} . \end{array}$ Since $‖ φ (s, \cdot) ‖_{L^{2}} ⩽ C | log s |^{- 1 / 2}$ and $‖ \tilde{w} (s, \cdot) ‖_{L^{2}} ⩽ C s$ due to Proposition 3.1, we obtain $\begin{matrix} {‖ u (t) ‖}_{L^{2}} ⩽ C {(log t)}^{- 1 / 2} + C t^{- 1} \end{matrix}$ for large $t > 0$ . Hence the solution admits the best possible decay rate. The datum $u (T, x)$ belongs to $H^{1} \cap H^{0, 1}$ for some $T = 1 / S$ , and so, taking $u_{0} = u (T)$ , we conclude that the solution generates the optimal decay. □

Footnotes

Acknowledgements

This work is supported by JSPS Grant-in-Aid for Scientific Research (C) No. 17K05305.

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