Extended Decay Properties for Generalized BBM Equation

Abstract

In this note we show that all small solutions of the BBM equation must decay to zero as t → +∞ in large portions of the physical space, extending previous known results, and only assuming data in the energy space. Our results also include decay on the left portion of the physical line, unlike the standard KdV dynamics.

Appendix: About the Proof of (2.18)

In this section we estimate the nonlinear term

$$\displaystyle \begin{aligned} \begin{aligned} \mathcal N(t) =& ~{}\frac{\alpha}2\int \varphi' \left( 2f (1-\partial_x^{2})^{-1}(u^p) + ( (1-\partial_x^{2})^{-1}u^p)^2 \right) \\ & ~{} - \frac{\alpha}2\int \varphi' \left( 2f_x (1-\partial_x^{2})^{-1}(u^p)_x + (\partial_x (1-\partial_x^{2})^{-1}u^p)^2 \right)\\ &~ {} -\frac 1{p+1}(\alpha \sigma -1) \int \varphi' u^{p+1} + \int \varphi' u (1-\partial_x^2)^{-1}\left(u^p\right). \end{aligned} \end{aligned}$$

Clearly,

$$\displaystyle \begin{aligned} \left| \frac 1{p+1}(\alpha \sigma -1) \int \varphi' u^{p+1} \right| \lesssim \varepsilon^{p-1}\int |\varphi' |u^2, \end{aligned}$$

which is enough. Now, recall the following results.

Lemma A.1 ([15])

The operator $(1-\partial _x^2)^{-1}$ satisfies the following comparison principle: for any u, v ∈ H ¹,

$$\displaystyle \begin{aligned} v \le w \quad \Longrightarrow \quad (1-\partial_x^2)^{-1} v \le (1-\partial_x^2)^{-1} w. \end{aligned} $$

(A.1)

Also,

Lemma A.2 ([15, 23])

Suppose that ϕ = ϕ(x) is such that

$$\displaystyle \begin{aligned} (1-\partial_x^2)^{-1}\phi(x) \lesssim \phi(x), \quad x\in \mathbb R, \end{aligned} $$

(A.2)

for ϕ(x) > 0 satisfying $|\phi ^{(n)}(x)| \lesssim \phi (x)$, n ≥ 0. Then, for v, w ∈ H ¹, we have

$$\displaystyle \begin{aligned} \int\phi^{(n)} v (1-\partial_x^2)^{-1}(w^p) ~\lesssim ~\left\| v \right\|{}_{H^1}\left\| w \right\|{}_{H^1}^{p-2} \int \phi w^2 \end{aligned} $$

(A.3)

and

$$\displaystyle \begin{aligned} \int \phi v_x (1-\partial_x^2)^{-1} (w^p)_x \lesssim \left\| v \right\|{}_{H^1}\left\| w \right\|{}_{H^1}^{p-2} \int \phi (w^2 + w_x^2). \end{aligned} $$

(A.4)

Using (A.3) with n = 0,

$$\displaystyle \begin{aligned} \left| \int \varphi' u (1-\partial_x^2)^{-1}\left(u^p\right) \right| \lesssim \varepsilon \int |\varphi' | |u^p| \lesssim \varepsilon^{p-1} \int |\varphi' | u^2. \end{aligned}$$

Using (A.3) with n = 0 and (A.4), we also have from $\|f\|{ }_{L^{\infty }}, \|f_x\|{ }_{L^{\infty }} \lesssim \|u\|{ }_{H^1}$ that

$$\displaystyle \begin{aligned} \left| \int \varphi' f (1-\partial_x^{2})^{-1}(u^p) \right| \lesssim \varepsilon \int |\varphi'| |u^p| \lesssim \varepsilon^{p-1} \int |\varphi' | u^2 \end{aligned}$$

and

$$\displaystyle \begin{aligned} \left| \int \varphi' f_x (1-\partial_x^{2})^{-1}(u^p)_x \right| \lesssim \varepsilon \int |\varphi'| |(u^p)_x| \lesssim \varepsilon^{p-1} \int |\varphi' | (u^2 + u_x^2). \end{aligned}$$

For the rest terms, using (A.3) with n = 0 and (A.4),

$$\displaystyle \begin{aligned} \int |\varphi' | ((1-\partial_x^{2})^{-1}(u^p))^2 \lesssim \| (1-\partial_x^{2})^{-1}(u^p) \|{}_{H^1} \varepsilon^{p-2} \int |\varphi' | u^2 \end{aligned}$$

and

$$\displaystyle \begin{aligned} \int |\varphi' | ((1-\partial_x^{2})^{-1}\partial_x(u^p))^2 \lesssim \| (1-\partial_x^{2})^{-1}(u^p) \|{}_{H^1} \varepsilon^{p-2} \int |\varphi' | (u^2 + u_x^2). \end{aligned}$$

Finally, $\| (1-\partial _x^{2})^{-1}(u^p) \|{ }_{H^1} \lesssim \|u^p\|{ }_{H^{-1}} \lesssim \varepsilon ^{p}$. Gathering these estimates, we get for some δ small enough,

$$\displaystyle \begin{aligned} \left|\mathcal N(t) \right| ~\lesssim \delta \int |\varphi' | (u^2 + u_x^{2}). \end{aligned}$$