2. Preliminaries

We shall use standard notations for the Lebesgue spaces $L^p, 1\leq p\leq \infty$. More generally, the Sobolev spaces $W^{k,p}(\mathbb R)$ for integer $k, 1 \lt p \lt \infty$, are defined via the norms

\begin{equation*}\|f\|_{W^{k,p}}=\|f\|_{L^p}+ \sum_{l=1}^k \|\partial_x^l f\|_{L^p}\end{equation*}

2.1. Fourier transform, fractional Burger’s and the Bernstein inequality

We take the Fourier transform and its inverse in the form

\begin{equation*} \hat{f}(\xi)=\int_{\mathbb R} f(x) e^{-2\pi i x\xi} dx, \ \ f(x)=\int_{\mathbb R} \hat{f}(\xi) e^{2\pi i x\xi} d\xi. \end{equation*}

In this notation, we have $\widehat{f''}(\xi)=-4\pi^2 \xi^2 \hat{f}(\xi)$. More generally, we introduce the fractional operators $(-\partial_{xx})^\alpha$ as acting on Schwartz functions via the symbol $(2\pi|\xi|)^{2\alpha}$. More precisely,

\begin{equation*} \widehat{(-\partial_{xx})^\alpha f}(\xi)= (2\pi|\xi|)^{2\alpha} \hat{f}(\xi). \end{equation*}

In fact, this allows one to introduce a companion spaces to $W^{s,p}$, where $s \gt 0, 1 \lt p \lt \infty$ is not necessarily an integer. More precisely, these are given by the norm

\begin{equation*} \|f\|_{W^{s,p}}=\|f\|_{L^p}+\| (-\partial_{xx})^{\frac{s}{2}} f\|_{L^p}. \end{equation*}

These spaces are equivalent to the previously defined $W^{k,p}$ for integer k. The Sobolev inequality in one spatial dimension takes the simple form

\begin{equation*} \|u\|_{L^q(\mathbb R)}\leq C \|u\|_{W^{s,p}} \ \ 1 \lt p \lt q \lt \infty, \ \ s\geq \frac{1}{p}-\frac{1}{q}. \end{equation*}

Next, we discuss Bernstein’s inequality, which is a relatively crude, but helpful form of Sobolev embedding. Consider a function $f: {\mathbf R}^n\to {\mathbf R}^n$ and a set of finite Lebesgue measure A. Let $f_A: \widehat{f_A}(\xi)=\hat{f}(\xi) \chi_A(\xi)$, where $\chi_A(x)=\left\{\begin{array}{cc} 1 & x\in A\\ 0 & x\notin A \end{array} \right..$ Then, for all $1\leq p\leq \infty$,

(2.1)\begin{equation} \|f_A\|_{L^q({\mathbf R}^n)}\leq |A|^{\frac{1}{p}-\frac{1}{q}} \|f\|_{L^p({\mathbf R}^n)}. \end{equation}

An easy estimate, which follows immediately from Plancherel’s identity, is

\begin{equation*} \|\partial_x f_A\|_q\leq C \min_{\xi\in A} |\xi| \|f_A\|_q, 1\leq q\leq \infty. \end{equation*}

We will use the rough Fourier cutoff projections

\begin{equation*} f_{ \lt \epsilon}: \widehat{f_{ \lt \epsilon}}(\xi):=\hat{f}(\xi)\chi_{(-\epsilon, \epsilon)}(\xi), \ \ f_{ \gt \epsilon}:=f- f_{ \lt \epsilon}. \end{equation*}

We also need the Gagliardo–Nirenberg estimate (or the log-convexity of the map $p\to \|f\|_p$). More precisely, for every $1\leq p \lt q \lt r\leq \infty: \frac{1}{q}=\frac{\theta}{p}+\frac{1-\theta}{r}$,

\begin{equation*} \|f\|_q\leq \|f\|_p^\theta \|f\|_r^{1-\theta}. \end{equation*}

Finally, there is the estimate, $ \|f\|_{L^\infty}^2 \leq C \|f'\|_{L^2} \|f\|_{L^2}, $ which will also be useful.

2.2. Diffusion semigroups

The notion of diffusive semigroups will play a role in our considerations. Shortly, we say that $\{T(t)\}_{t\geq 0}$ is a C ₀ semigroup on a Banach space, if $T(t)\in B(X)$, $T(t+s)=T(t)T(s), T(0)=Id$ and for every $f\in X: \lim_{t\to 0+} \|T(t) f - f\|_X=0$. It is standard that one can always associate with such objects a generator L, a generally unbounded operator, which can be introduced on a domain $D(L)=\{f\in X: \lim_{t\to 0+} \frac{T(t) f - f}{t} \ \ \textrm{exists}\}$ as

\begin{equation*} L f:=\lim_{t\to 0+} \frac{T(t) f - f}{t}, f\in D(L). \end{equation*}

Naturally, $T(t) f\in D(L)$ solves the X valued differential equation $x'=L x, x(0)=f$ for any $f\in X$, so we informally denote it by $T(t)=e^{t L}$.

Definition 2.1. We say that a C ₀ semigroup on $L^2({\mathbf R}^n)$, T(t), is a diffusive semigroup, if $T(t), t\geq 0$ is self-adjoint, and given by a probability density kernel, i.e.

\begin{equation*} T(t) f(x) = \int_{{\mathbf R}^n} p_t(x-y) f(y) dy, f\in L^2({\mathbf R}^n), p_t(x)\geq 0, \int_{{\mathbf R}^n} p_t(x) dx = 1. \end{equation*}

We observe that the generator L in such cases is necessarily a non-positive operator, since for all $t\geq 0$ and a test function f, we have

\begin{align*} \langle T(t) f,f \rangle &= \int p_t(x-y) f(y) \bar{f}(x) dx dy \leq \\ & \leq \left(\int p_t(x-y) |f (y)|^2 dx dy\right)^{\frac{1}{2}} \left(\int p_t(x-y) |f (x)|^2 dx dy\right)^{\frac{1}{2}}=\|f\|^2, \end{align*}

whence $\langle L f,f \rangle \leq \limsup_{t\to 0+} \frac{\langle T(t) f - f,f \rangle}{t}\leq 0$.

It is standard that the operator $L=\partial_{xx}$ is a generator of such semigroup. More generally, the fractional Burgers operators $L=-\sum_{j=1}^N a_j (-\partial_{xx})^{\alpha_j}, a_j\geq 0, 0 \lt \alpha_1 \lt \ldots \alpha_N\leq 1$ also generate diffusion semigroups,Footnote ⁶ see for example [Reference Chamorro and Lemarié-Rieusset6]. An important property of such semigroups is the following.

Proposition 2.2 (Reference Chamorro and Lemarié-Rieusset6, Theorem 3.2)

Assume that L generates a diffusive semigroup. Then, for all $1\leq p \lt \infty$, we have

(2.2)\begin{equation} \int_{{\mathbf R}^n} f|f|^{p-2} L(f) dx\leq 0. \end{equation}

If, in addition, $ 2\leq p \lt \infty$, then

(2.3)\begin{equation} -p \int f|f|^{p-2} L(f) dx \geq \int |\sqrt{-L}(f|f|^{\frac{p}{2}-1})|^2 dx \end{equation}

In particular, we have that for each $0 \lt \alpha\leq 1, 2\leq p \lt \infty$

(2.4)\begin{equation} \int_{{\mathbf R}^n} ((-\Delta)^\alpha f) f|f|^{p-2} dx \geq 0. \end{equation}

3. Attraction to fronts: the Burger-KdV model

We start by revisiting some of the results obtained in the proof of Theorem 1.1. Indeed, in the course of the proof, the authors set $\dot{x}_0(t)=\gamma \int_{-\infty}^\infty \phi_x(x) v(x) dx$, withFootnote ⁷ γ > 1. This is done to ensure that the rank-one perturbation $\mathcal H_\gamma=-\partial_{xx}+\frac{1}{2}\phi_x+\gamma \langle \phi_x,\cdot \rangle \phi_x$ of the important linearized operatorFootnote ⁸ $\mathcal H=-\partial_{xx}+\frac{1}{2}\phi_x$ has the property $\mathcal H_\gamma\geq 0$. Two important features of the construction are

(3.1)\begin{eqnarray} & & t\to \|v(t, \cdot)\|_{L^2}^2 \ \ \textrm{is monotonically decreasing} \end{eqnarray}

(3.2)\begin{eqnarray} & & \int_0^t |\dot{x}_0(s)|^2 ds+ \int_0^t \|v_x(s, \cdot)\|_{L^2}^2 ds \leq C \|v_0\|_{L^2}^2 \end{eqnarray}

We make substantial use of these two facts in our subsequent analysis. Next, we write the equation for the residual $v: u(x,t)=\phi(x-x_0(t))+v(t, x-x_0(t))$. This takes the form

(3.3)\begin{equation} v_t - v_{xx} -\dot{x}_0 v_x-\dot{x}_0 \phi_x+\phi_x v+\phi v_x+ v v_x+\nu v_{xxx}=0. \end{equation}

We setup an energy estimate for v in the space $L^2_1(\mathbb R)$. More precisely, we shall show that

\begin{equation*} \int_{-\infty}^\infty |v(t,x)|^2 |x| dx\leq C, t\geq 0. \end{equation*}

Of course, the global solution v produced in [Reference Barker, Bronski, Hur and Yang2] does not a priori live in the space $L^2_1(\mathbb R)$, so one begins by only formally setting the energy estimate for it. We will ignore the details of this justification, as this is standard by considering instead a quantity of the form $\int_{-\infty}^\infty |v(t,x)|^2 |x| \zeta(x/N)dx$ for $N \gt \gt 1$ and some smooth cutoff function ζ. With these caveats, we proceed to bound the $L^2_1(\mathbb R)$ norm. Taking a dot product of (3.3) with $v |x|$, we have

\begin{align*} &\frac{1}{2} \partial_t \int v^2(t,x) |x| dx -\int v_{xx} v |x| dx -\dot{x}_0 \int v_x v |x| dx - \dot{x}_0 \int \phi_x v |x| dx+\int \phi_x v v |x| dx \,+\\ & \quad +\int \phi v_x v |x| dx + \int v v_x v |x| dx +\nu \int v_{xxx} v |x| dx =0. \end{align*}

We now proceed to estimate various terms, by keeping in mind the relation (3.2), which guides us about the acceptable error terms. After integration by parts and algebraic manipulations, we have

\begin{align*} -\int v_{xx} v |x| dx &= \int (v_x)^2 |x| dx + \int_{-\infty}^\infty v_x v sgn(x) dx \geq -v^2(t,0)\geq -\|v(t, \cdot)\|_{L^\infty}^2,\\ \left| \dot{x}_0 \int v_x v |x| dx \right| &= \left| \frac{\dot{x}_0}{2} \int v^2 sgn(x) dx \right| \leq C |\dot{x}_0 (t)| \|v(t, \cdot)\|_{L^2}^2\leq C |\dot{x}_0 (t)| \|v(t, \cdot)\|_{L^2}\\ \left| \dot{x}_0 \int \phi_x v |x| dx\right|&\leq C |\dot{x}_0 (t)| \|v(t, \cdot)\|_{L^2}, \\ \left|\int v v_x v |x| dx\right|&=\frac{1}{3} \left|\int v^3 sgn(x)dx\right| \leq C \|v(t, \cdot)\|_{L^3}^3\leq C \|v\|_{H^{\frac{1}{6}}}^3\leq C \|v_x(t, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(t, \cdot)\|_{L^2}^{\frac{5}{2}}\\ &\leq C \|v_x(t, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(t, \cdot)\|_{L^2}^{\frac{3}{2}} \end{align*}

Also,

\begin{equation*} \int \phi_x v^2 |x| dx\geq -\|v(t, \cdot)\|_{L^\infty}^2 \int |\phi_x(x)| |x| dx =-C \|v(t, \cdot)\|_{L^\infty}^2. \end{equation*}

Here, in addition to Sobolev embedding and Gagliardo–Nirenberg estimates, we used the bound $\|v(t, \cdot)\|_{L^2}\leq C$. Next,

\begin{eqnarray*} & & \int \phi v_x v |x| dx=-\frac{1}{2} \left( \int \phi_x(x) v^2(x) |x| dx + \int \phi(x) v^2(t,x) sgn(x) dx \right)\geq \\ & & \geq - C \|v(t, \cdot)\|_{L^\infty}^2-\frac{1}{2}\int \phi(x) v^2(t,x) sgn(x) dx \end{eqnarray*}

Now, note that $-\phi(x) sgn(x) \gt 0$ for all x and also there exists A, so that $-\phi(x) sgn(x) \gt \frac{1}{2}$ for $|x| \gt A$. This last statement follows from $\lim_{x\to \pm \infty} \phi(x) = \mp 1$. So, we further estimate

\begin{eqnarray*} & & -\frac{1}{2}\int \phi(x) v^2(t,x) sgn(x) dx \gt -\frac{1}{2}\int_{|x| \gt A} \phi(x) v^2(t,x) sgn(x) dx \gt \frac{1}{4} \int_{|x| \gt A} v^2(t,x) dx = \\ & & = \frac{1}{4} \int v^2(t,x) dx -\frac{1}{4} \int_{-A}^A v^2(t,x) dx \geq \frac{1}{4} \int v^2(t,x) dx - C \|v(t, \cdot)\|_{L^\infty}^2, \end{eqnarray*}

Finally, for the term $\int v_x'' v |x| dx$, we have

\begin{equation*} \left|\int v_x'' v |x| dx \right|= \left| \frac{1}{2} \int (v_x)^2 sgn(x) - \int v_x' v sgn(x) dx \right| \leq 2 |v(0)| |v_x(0)| + C \|v_x(t, \cdot)\|_{L^2}^2. \end{equation*}

Putting it all together and estimating $-\|v(t, \cdot)\|_{L^\infty}^2 \geq - C\|v_x(t, \cdot)\|_{L^2} \|v(t, \cdot)\|_{L^2}$, we obtain

\begin{eqnarray*} & & \frac{1}{2} \partial_t \int v^2(t,x) |x| dx + \frac{1}{4} \int v^2(t,x) dx \leq C( |\dot{x}_0 (t)| \|v(t, \cdot)\|_{L^2}+ |v(t,0)| |v_x(t,0)| )+ \\ & & + C( \|v_x(t, \cdot)\|_{L^2}^2+\|v_x(t, \cdot)\|_{L^2} \|v(t, \cdot)\|_{L^2}+ \|v_x(t, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(t, \cdot)\|_{L^2}^{\frac{3}{2}}). \end{eqnarray*}

We are almost ready to derive an estimate on the desired norms, except that it is hard to control $|v_x(t,0)| $. Indeed, the easiest way out would be to use the norm $\|v_x(t, \cdot)\|_{L^\infty}$, for which we unfortunately lack a priori estimates. Instead, we employ a simple averaging argument to reduce to the quantity $\|v_x(t, \cdot)\|_{L^2}$, for which we do have a priori bound, see (3.2).

Indeed, instead of finding the bound for $ \int v^2(t,x) |x| dx$, we might employ an estimate for $ \int v^2(t,x) |x-a| dx$, where $a\in (-1,1)$. Running through identical estimates, we arrive at

\begin{eqnarray*} & & \frac{1}{2} \partial_t \int v^2(t,x) |x-a| dx + \frac{1}{4} \int v^2(t,x) dx \leq C( |\dot{x}_0 (t)| \|v(t, \cdot)\|_{L^2}+|v(t,a)| |v_x(t,a)| )+\\ & & + C(\|v_x(t, \cdot)\|_{L^2}^2+\|v_x(t, \cdot)\|_{L^2} \|v(t, \cdot)\|_{L^2}+ \|v_x(t, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(t, \cdot)\|_{L^2}^{\frac{3}{2}}). \end{eqnarray*}

Integrating on the time interval $(0,t)$ brings about

\begin{eqnarray*} & & \frac{1}{2} \int v^2(t,x) |x-a| dx + \frac{1}{4} \int_0^t \int v^2(s,x) dx ds \leq \int v_0^2(x) |x-a| dx + \int_0^t |v(s,a)| |v_x(s,a)| ds +\\ & & + C( \int_0^t |\dot{x}_0 (s)| \|v(s, \cdot)\|_{L^2} + \|v_x(s, \cdot)\|_{L^2}^2 +\|v_x(s, \cdot)\|_{L^2} \|v(s, \cdot)\|_{L^2}+ \|v_x(s, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(s, \cdot)\|_{L^2}^{\frac{3}{2}} ) ds. \end{eqnarray*}

Taking an average of the previous estimate, i.e. integrating the previous estimate $\int_{-1}^1 \ldots da$, and in doing so, recall the formula

\begin{equation*} \int_{-1}^1 |x-a| da=\left\{ \begin{array}{cc} 2|x| & |x| \gt 1 \\ x^2+1 & |x|\leq 1 \end{array} \right. \in (2|x|, 2|x|+1). \end{equation*}

This yields the estimate

\begin{align*} & \frac{1}{2} \int v^2(t,x) |x| dx + \frac{1}{2} \int_0^t \int v^2(s,x) dx ds \leq \int v_0^2(x) (2|x|+1) dx + \\ & \quad+ \int_{-1}^1 \int_0^t |v(s,a)| |v_x(s,a)| ) ds da+ \\ & \quad+ C \int_0^t [|\dot{x}_0 (s)| \|v(s, \cdot)\|_{L^2} + \|v_x(s, \cdot)\|_{L^2}^2+ \|v_x(s, \cdot)\|_{L^2} \|v(s, \cdot)\|_{L^2}+ \|v_x(s, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(s, \cdot)\|_{L^2}^{\frac{3}{2}}] ds . \end{align*}

Now, based on (3.2) and Hölder’s inequality

\begin{eqnarray*} & & \int_0^t |\dot{x}_0 (s)| \|v(s, \cdot)\|_{L^2} ds \leq \left( \int_0^t |\dot{x}_0 (s)|^2 ds \right)^{\frac{1}{2}} \left(\int_0^t \|v(s, \cdot)\|_{L^2}^2 ds\right)^{\frac{1}{2}} \leq C \|v\|_{L^2_{t,x}}, \\ & & \int_0^t \|v_x(s, \cdot)\|_{L^2}^2 ds \leq C, \\ & & \int_0^t \|v_x(s, \cdot)\|_{L^2} \|v(s, \cdot)\|_{L^2} ds\leq C \left(\int_0^t \|v_x(s, \cdot)\|_{L^2}^2\right)^{\frac{1}{2}} \left(\int_0^t \|v(s, \cdot)\|_{L^2}^2\right)^{\frac{1}{2}} \leq C \|v\|_{L^2_{tx}}, \\ & & \int_0^t \|v_x(s, \cdot)\|_{L^2}^{\frac{1}{2}} \|v(s, \cdot)\|_{L^2}^{\frac{3}{2}} ds \leq \left(\int_0^t \|v_x(s, \cdot)\|_{L^2}^2 ds\right)^{\frac{1}{4}} \left(\int_0^t \|v(s, \cdot)\|_{L^2}^2 ds\right)^{\frac{3}{4}}\leq C \|v\|_{L^2_{t,x}}^{\frac{3}{2}}, \\ & & \int_{-1}^1 \int_0^t |v(s,a)| |v_x(s,a)| da \leq \left(\int_0^t \|v_x(s, \cdot)\|_{L^2}^2 ds\right)^{\frac{1}{2}} \left(\int_0^t \|v(s, \cdot)\|_{L^2}^2 ds\right)^{\frac{1}{2}} \leq C \|v\|_{L^2_{t,x}}. \end{eqnarray*}

All in all, we conclude that

(3.4)\begin{equation} \int v^2(t,x) |x| dx+ \frac{1}{2} \|v\|_{L^2_{t x}}^2 \leq C \|v\|_{L^2_{t,x}}^{\frac{3}{2}}+C. \end{equation}

We can hide $\|v\|_{L^2_{t,x}}^{\frac{3}{2}}$ on the right-hand side behind $\frac{1}{2} \|v\|_{L^2_{t,x}}^2$. More precisely, by Young’s inequality, for any choice of ϵ > 0,

\begin{equation*} \|v\|_{L^2_{t,x}}^{\frac{3}{2}}\leq \frac{3}{4} \epsilon \|v\|_{L^2_{t,x}}^2+ \frac{1}{4} \epsilon^{-3}. \end{equation*}

With an appropriately small choice of ϵ, we enter this on the right-hand side of (3.4), which leads us to the bound

(3.5)\begin{equation} \sup_{0 \lt t \lt T} \int v^2(t,x) |x| dx+ \int_0^T \|v(t, \cdot)\|_{L^2_{x}}^2 dt \leq C, \end{equation}

for each T > 0. This bound has two important consequences. Firstly, using the Hölder inequality, we have that for each $1 \lt p \lt 2$,

\begin{eqnarray*} & & \int |v(x)|^p dx = \int_{|x| \lt 1} |v(x)|^p dx + \int_{|x|\geq 1} |v(x)|^p dx\leq \\ & & \leq C \|v\|_{L^2}^p+ \left(\int_{|x| \gt 1} v^2(x) |x| dx \right)^{\frac{p}{2}} \left(\int_{|x|\geq 1} |x|^{-\frac{p}{2-p}}dx \right)^{1-\frac{p}{2}}\leq C_p \left(\int |v|^2 (1+|x|) dx \right)^{\frac{p}{2}}, \end{eqnarray*}

for which we have good bounds from (3.5). It follows that for any p > 1, $\sup_{0 \lt t \lt T} \int |v(x)|^p\leq C_p$. Note, however, that the constant C_p blows up as $p\to 1+$, due to $\lim_{p\to 1+} \left(\int_{|x|\geq 1} |x|^{-\frac{p}{2-p}}dx \right)=+\infty$.

Secondly, by taking into account (3.1) and (3.5), we have that for each t > 0,

\begin{equation*} t \|v(t, \cdot)\|_{L^2}^2\leq \int_{0}^{t} \|v(s, \cdot)\|_{L^2}^2 ds \leq \int_0^{\infty} \|v(s, \cdot)\|_{L^2}^2 ds\leq C, \end{equation*}

whence it follows that v obeys the bound $\|v(t, \cdot)\|_{L^2}\leq C t^{-\frac{1}{2}}$. Interpolating this with the uniform bounds for $\|v(t, \cdot)\|_{L^p}, 1 \lt p \lt 2$, obtained just before, we have that for every δ > 0, there is $C=C(\delta)$, so that

\begin{equation*} \|v(t, \cdot)\|_{L^p}\leq C t^{-(1-\frac{1}{p})+\delta}, 1 \lt p \lt 2. \end{equation*}

4. Asymptotic attraction to odd fronts for the generalized Burgers model

In this section, we present an extension of the results in [Reference Barker, Bronski, Hur and Yang2], when the fronts and the respective perturbations are odd.

Now that ${\mathcal L}=-\sum_{j=1}^N a_j (-\partial_{xx})^{\alpha_j} $ preserves the parity of the inputs, it is clear that if u ₀ is odd, so is $v_0=u_0-\phi$. Clearly, the evolution preserves the odd solutions, whence $x\to v(t,x)$ is odd for all $t\geq 0$. Recall that by the construction in [Reference Barker, Bronski, Hur and Yang2], the translation function x ₀ was selected (see formula (20) there) via the formula $ \dot{x}_0(t)=\gamma \langle \phi_x,v \rangle=0, $ since ϕ_x is even, while v is odd. It follows by Theorem 1.1 that for this particular front and odd perturbations u ₀, $u(t,x)=\phi(x)+v(t,x)$, where

(4.1)\begin{equation} \lim_{t\to \infty} \|v(t, \cdot)\|_{L^p}=0, 2 \lt p\leq \infty. \end{equation}

Recall that v also satisfies the energy estimate (1.13).

We now aim to establish uniform estimates for the norm $\|v(t, \cdot)\|_{L^1(\mathbb R)}$. To this end, take arbitrary large T and assume $0\leq t \lt T$. Note that v satisfies

(4.2)\begin{equation} v_t - v_{xx} - {\mathcal L} v +\phi_x v+\phi v_x+ v v_x=0. \end{equation}

For each p > 1, multiply (4.2) by $v |v|^{p-2}$ and integrate in x. We obtain,

\begin{equation*} \int v_t v |v|^{p-2} dx = \frac{1}{p} \partial_t \int |v|^p dx, \end{equation*}

Integration by parts yields for each p > 1,

\begin{eqnarray*} & & \int v v_xv |v|^{p-2} dx = \int v_x |v|^p dx = - \int v (|v|^p)' dx =- p\int v v_xv |v|^{p-2} dx, \end{eqnarray*}

whence

\begin{equation*} \int v v_xv |v|^{p-2} dx=0. \end{equation*}

By (2.4), we have the coercivity estimates

\begin{equation*} -\int v_{xx} v |v|^{p-2} dx \geq 0, -\int ({\mathcal L} v) v |v|^{p-2} dx =\sum_{j=1}^N a_j \int ((-\partial_{xx})^{\alpha_j} v) v |v|^{p-2} dx \geq 0. \end{equation*}

Next,

\begin{equation*} \int \phi v_x v |v|^{p-2} dx = \frac{1}{p} \int \phi (|v|^p)' dx = - \frac{1}{p} \int \phi_x |v|^p dx. \end{equation*}

All in all, we get

\begin{equation*} \partial_t \int |v|^p dx + (p-1) \int \phi_x |v|^{p} dx \leq 0. \end{equation*}

Denoting $M_T:=\sup_{0 \lt t \lt T} \|v(t, \cdot)\|_{L^\infty}$, and integrating the above expression in $(0,t)$, we obtain

(4.3)\begin{equation} \int |v(t, x)|^p dx\leq \|v_0\|_{L^p}^p+ (p-1) \int_0^t \int_{\mathbb R} |\phi_x(x)| |v(t,x)|^{p} dx\leq \|v_0\|_{L^p}^p+(p-1) \|\phi_x\|_{L^1}T M_T^p. \end{equation}

Note that all the constants in (4.3) are written out explicitly. This allows us to take a limit $p\to 1+$. We obtain

(4.4)\begin{equation} \|v(t, \cdot)\|_{L^1}\leq \|v_0\|_{L^1}, \end{equation}

and this is valid for all T > 0, so (4.4) is a global estimate.

This provides another important ingredient to the estimates (4.1), which we have borrowed from Theorem 1.1. Fix ϵ > 0. By Plancherel’s identity

\begin{equation*} I(t):=\|v(t, \cdot)\|_{L^2}^2=\|v_{ \lt \epsilon}(t, \cdot)\|_{L^2}^2+\|v_{ \gt \epsilon}(t, \cdot)\|_{L^2}^2=:I_{ \lt \epsilon}(t)+I_{ \gt \epsilon}(t). \end{equation*}

We now use (1.13) to conclude

\begin{equation*} \partial_t(I_{ \lt \epsilon}(t)+I_{ \gt \epsilon}(t))\leq - C \|v_x(t)\|_{L^2}^2 \lt - C \|\partial_x v_{ \gt \epsilon}(t)\|_{L^2}^2\leq -C \epsilon^2 I_{ \gt \epsilon}(t). \end{equation*}

It follows that

\begin{equation*} I_{ \gt \epsilon}'(t)+C\epsilon^2 I_{ \gt \epsilon}(t)+ I_{ \lt \epsilon}'(t)\leq 0. \end{equation*}

Multiplying by $e^{C \epsilon^2 t}$ and integrating yields

(4.5)\begin{equation} e^{C \epsilon^2 t} I_{ \gt \epsilon}(t) - I_{ \gt \epsilon}(0) +\int_0^t e^{C \epsilon^2 s} I_{ \lt \epsilon}'(s) ds\leq 0. \end{equation}

However, integration by parts yields

\begin{eqnarray*} -\int_0^t e^{C \epsilon^2 s} I_{ \lt \epsilon}'(s) ds=-I_{ \lt \epsilon} (s)e^{C \epsilon^2 s}|_0^t + C\epsilon^2 \int_0^t e^{C \epsilon^2 s} I_{ \lt \epsilon}(s) ds\leq I_{ \lt \epsilon} (0)+C\epsilon^2 \int_0^t e^{C \epsilon^2 s} I_{ \lt \epsilon}(s) ds. \end{eqnarray*}

Plugging this back in (4.5) and multiplying by $e^{-C \epsilon^2 t}$, yields

(4.6)\begin{equation} I_{ \gt \epsilon}(t)\leq I(0) e^{-C \epsilon^2 t}+ C\epsilon^2 \int_0^t e^{C \epsilon^2 (s-t)} I_{ \lt \epsilon}(s) ds\leq I(0) e^{-C \epsilon^2 t}+\max_{0 \lt s \lt t} I_{ \lt \epsilon}(s). \end{equation}

Now, we apply the Bernstein’s inequality for $A=(-\epsilon, \epsilon)$, and (4.4) to obtain

(4.7)\begin{equation} I_{ \lt \epsilon}(s)= \|v_{ \lt \epsilon}(s, \cdot)\|_{L^2(\mathbb R)}^2\leq C \epsilon \|v(s, \cdot)\|_{L^1(\mathbb R)}^2 \leq C\epsilon \|v_0\|_{L^1(\mathbb R)}^2. \end{equation}

Adding $I_{ \lt \epsilon}(t)$ on both sides of (4.6) and using (4.7), we obtain

(4.8)\begin{equation} \|v(t, \cdot)\|_{L^2}^2=I(t)\leq e^{- C \epsilon^2 t} \|v_0\|_{L^2}^2+ C\epsilon \|v_0\|_{L^1}^2\leq C_1 (e^{- C_1 \epsilon^2 t} +\epsilon)\|v_0\|_{L^2\cap L^1}^2, \end{equation}

for some absolute constant C ₁, i.e. independent on $\epsilon, t$.

This inequality is true for every fixed ϵ > 0, and every t, but since clearly the best strategy is to optimize (4.8) in ϵ given t, we should proceed with some care. To this end, fix large $t \gt \gt 1$, and then select the unique $\epsilon \lt \lt 1: e^{- C_1 \epsilon^2 t}=\epsilon$. Clearly, such choice of ϵ is possible, let us compute its asymptotic in terms of t. Since we need to solve,

\begin{equation*} \frac{\epsilon^2}{\ln(\epsilon)}=-\frac{1}{C_1 t}, \end{equation*}

its unique solution obeys the asymptotic

(4.9)\begin{equation} \epsilon\sim \sqrt{\frac{\ln(t)}{t}}. \end{equation}

It follows that

(4.10)\begin{equation} \|v(t, \cdot)\|_{L^2}^2 \leq C_3 \frac{\sqrt{\ln(t)}}{\sqrt{t}} \|v_0\|_{L^2\cap L^1}^2 \end{equation}

We conclude that $\|v(t, \cdot)\|_{L^2}\leq C \|v_0\|_{L^2\cap L^1} \left(\frac{\ln(t)}{t}\right)^{\frac{1}{4}}$, as stated. Interpolating between the bounds (4.10) and (4.4), the Gagliardo-Nirenberg’s inequality now yields

\begin{equation*} \|v(t, \cdot)\|_{L^q}\leq C \|v_0\|_{L^2\cap L^1} \left(\frac{\ln(t)}{t}\right)^{\frac{1-\frac{1}{q}}{2}}, 1 \lt q \lt 2, \end{equation*}

for intermediate values of q. While one can similarly interpolate with the bound (4.1), this will not be a particularly good decay, especially near $q=\infty$.

Instead, we use an alternative estimate based again on (1.13). Indeed from (3.2) and (1.13) , it is clear that for every T > 0

\begin{equation*} \int_T^\infty \|v_x(s, \cdot)\|_{L^2}^2 ds \leq \|v(T, \cdot)\|_{L^2}^2\lesssim \left(\frac{\ln(T)}{T}\right)^{\frac{1}{2}}, \end{equation*}

where the last estimate follows from the L ² bounds (4.10), Also, for every $T \gt \gt 1$, we have that there exists $T_0\in (T, 2T)$, so that

\begin{equation*} T \|v_x(T_0, \cdot)\|_{L^2}^2 \leq \int_T^{2T} \|v_x(s, \cdot)\|_{L^2}^2 ds \lt \int_T^\infty \|v_x(s, \cdot)\|_{L^2}^2 ds\lesssim \left(\frac{\ln(T)}{T}\right)^{\frac{1}{2}}. \end{equation*}

In other words,

(4.11)\begin{equation} \|v_x(T_0, \cdot)\|_{L^2}^2 \leq C \frac{\sqrt{\ln(T_0)}}{T_0^{\frac{3}{2}}}, \int_{T_0}^\infty \|v_x(s, \cdot)\|_{L^2}^2 ds \leq C \left(\frac{\ln(T_0)}{T_0}\right)^{\frac{1}{2}}. \end{equation}

For large T ₀, denote $\delta:=\left(\frac{\ln(T_0)}{T_0}\right)^{\frac{1}{2}} \gt \gt C \frac{\sqrt{\ln(T_0)}}{T_0^{\frac{3}{2}}}$. A this point, we refer to the argument in [Reference Barker, Bronski, Hur and Yang2], which establishes that starting with (4.11), we conclude

(4.12)\begin{equation} \sup_{t \gt T_0} \|v_x(t)\|_{L^2}\leq C \delta^{\frac{2}{3}}\sim \left(\frac{\ln(T_0)}{T_0}\right)^{\frac{1}{3}} \end{equation}

Since this can be done for at least one point $T_0\in (2^k, 2^{k+1}), k \gt \gt 1$, we obtain the bound

\begin{equation*} \|v_x(t)\|_{L^2}\leq C \left(\frac{\ln(t)}{t}\right)^{\frac{1}{3}} \end{equation*}

By Sobolev embedding, followed by the Gagliardo-Nirenberg’s inequality we conclude

\begin{equation*} \|v(t, \cdot)\|_{L^q}\leq \|v(t, \cdot)\|_{\dot{H}^{\frac{1}{2}-\frac{1}{q}}}\leq \|v(t, \cdot)\|_{L^2}^{\frac{1}{2}+\frac{1}{q}}\|v_x(t, \cdot)\|_{L^2}^{\frac{1}{2}-\frac{1}{q}}\leq C \left(\frac{\ln(t)}{t}\right)^{\frac{7}{24}-\frac{1}{12 q}}. \end{equation*}