Decay of periodic entropy solutions to degenerate nonlinear parabolic equations

TL;DR

This paper proves that space-periodic entropy solutions to certain multidimensional degenerate nonlinear parabolic equations decay over time under specific nonlinearity-diffusivity conditions.

Contribution

It establishes decay results for entropy solutions in multidimensional degenerate nonlinear parabolic equations under a precise nonlinearity-diffusivity condition.

Findings

Entropy solutions decay over time

Decay holds under specific nonlinearity-diffusivity conditions

Results apply to multidimensional equations

Abstract

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation.

Equations

$$u_t+{\mathrm{div}}_x(\phi (u)-a(u){\nabla }_{x}u)=0,$$

$$u_t+{\mathrm{div}}_x\phi (u)=0.$$

$$u(0,x)=u_0(x).$$

$$T_g(f)(u)=g(u-)f(u)-{\int }_0^{u}f(s)dg(s),$$

$${\int }_0^{u}f(s)dg(s)=\mathrm{sign}u{\int }_{J(u)}f(s)dg(s),$$

$${\mathrm{div}}_x{B}_r(u(t,x))\in L_{loc}^2(\Pi ),$$

$${\mathrm{div}}_x{T}_g(B_r)(u(t,x))=g(u(t,x)){\mathrm{div}}_x{B}_r(u(t,x))\text{mbox}in{\mathcal{D}}^{\prime }(\Pi );$$

$$\eta (u{)}_t+{\mathrm{div}}_x(T_{{\eta }^{\prime }}(\phi )(u))-D^2\cdot T_{{\eta }^{\prime }}(A(u))+$$

$${\eta }^{\prime \prime }(u)\sum _{r=1}^n({\mathrm{div}}_x{B}_r(u){)}^2\le 0\text{mbox}in{\mathcal{D}}^{\prime }(\Pi );$$

$$D^2\cdot T_{{\eta }^{\prime }}(A(u))\doteq \sum _{i,j=1}^n\frac{{\partial }^2}{\partial x_i\partial x_j}{T}_{{\eta }^{\prime }}(A_{ij}(u)).$$

$${\int }_{\Pi }[\eta (u)f_t+T_{{\eta }^{\prime }}(\phi )(u)\cdot {\nabla }_{x}f+T_{{\eta }^{\prime }}(A(u))\cdot D_x^{2}f]dtdx\ge 0,$$

$$T_g(B_r)(u)={\int }_0^{u}g(v)B_r^{\prime }(v)dv.$$

$$g_l(u(t,x)){\mathrm{div}}_x{B}_r(u(t,x)){\to }_{l\to \infty }g(u(t,x)){\mathrm{div}}_x{B}_r(u(t,x))\text{mbox}in{L}_{loc}^2(\Pi ).$$

$$y_i=\sum _{j=1}^n{q}_{ij}{x}_j-c_{i}t,i=1,\dots ,n.$$

$$v_t+{\mathrm{div}}_y(\tilde{\phi }(v)-\tilde{a}(v){\nabla }_{y}v)=0,v(0,y)=v_0(y),$$

$$\tilde{\phi }(v)=q\phi (v)-cv,\tilde{a}(v)=qa(v)q^{\top },$$

$$\operatorname{div}_{y}(B(v)q^{\top})_{r}\big{|}_{y=y(t,x)}=\sum_{l,j=1}^{n}\partial_{y_{l}}B_{rj}(v)q_{lj}=\sum_{j=1}^{n}\partial_{x_{j}}B_{rj}(u)=\operatorname{div}_{x}B_{r}(u).$$

$$u_t+{\mathrm{div}}_x({\phi }_k(u)-a_k(u){\nabla }_{x}u)=0,$$

$$L^{\prime }=\{\xi \in {\mathbb{R}}^n|\xi \cdot e\in \mathbb{Z}\forall e\in L\}$$

$$I=\frac{1}{V}{\int }_{{\mathbb{T}}^n}{u}_0(x)dx$$

$$\underset{t\to +\infty }{\mathrm{esslim}}u(t,x)=I\text{mbox}in{L}^1({\mathbb{T}}^n).$$

$$\mathrm{meas}\{u\in \mathbb{R},|u|\le \Vert u_0{\Vert }_{\infty }|\tau u+\xi \cdot {\phi }^{\prime }(u)=a(u)\xi \cdot \xi =0\}=0.$$

$$\eta (u{)}_t+{\mathrm{div}}_x(T_{{\eta }^{\prime }}(\phi )(u))-D^2\cdot T_{{\eta }^{\prime }}(A(u))=-{\mu }_{\eta }\text{mbox}in{\mathcal{D}}^{\prime }(\Pi ).$$

$$\mu ((t_1,t_2)\times {\mathbb{T}}^n)\le {\int }_{{\mathbb{T}}^n}\eta (u(t_1,x))dx-{\int }_{{\mathbb{T}}^n}\eta (u(t_2,x))dx.$$

$$\eta (u{)}_t+{\mathrm{div}}_x(T_{{\eta }^{\prime }}(\phi )(u))-D^2\cdot T_{{\eta }^{\prime }}(A(u))\le 0\text{mbox}in{\mathcal{D}}^{\prime }(\Pi ).$$

$$⟨{\mu }_{\eta },\alpha (t)\beta (x/k)⟩\doteq {\int }_{\Pi }\alpha (t)\beta (x/k)d{\mu }_{\eta }(t,x)={\int }_{\Pi }\eta (u(t,x)){\alpha }^{\prime }(t)\beta (x/k)dtdx+$$

$$\frac{1}{k}{\int }_{\Pi }{T}_{{\eta }^{\prime }}(\phi )(u)\cdot {\nabla }_y\beta (x/k)\alpha (t)dtdx+\frac{1}{k^2}{\int }_{\Pi }{T}_{{\eta }^{\prime }}(A)(u)\cdot D_y^2\beta (x/k)\alpha (t)dtdx.$$

$${\int }_{{\mathbb{R}}_{+}\times {\mathbb{T}}^n}\alpha (t)d{\mu }_{\eta }(t,x)={\int }_{{\mathbb{R}}_{+}\times {\mathbb{T}}^n}\eta (u(t,x)){\alpha }^{\prime }(t)dtdx,$$

$$\underset{k\to \infty }{\lim }{k}^{-n}⟨\mu ,\alpha (t)\beta (x/k)⟩={\int }_{{\mathbb{R}}_{+}\times {\mathbb{T}}^n}\alpha (t)d\mu (t,x){\int }_{{\mathbb{R}}^n}\beta (y)dy$$

$$\frac{d}{dt}{\int }_{{\mathbb{T}}^n}\eta (u(t,x))dx=-{\int }_{{\mathbb{T}}^n}d{\mu }_{\eta }(t,x)\text{mbox}in{\mathcal{D}}^{\prime }({\mathbb{R}}^{+}),$$

$$⟨{\mu }_{\eta }^t,\alpha (t)⟩={\int }_{{\mathbb{R}}_{+}\times {\mathbb{T}}^n}\alpha (t)d{\mu }_{\eta }(t,x),$$

Full text

1 Introduction

In the half-space $\Pi=\mathbb{R}_{+}\times\mathbb{R}^{n}$, $\mathbb{R}_{+}=(0,+\infty)$, we consider the nonlinear parabolic equation

[TABLE]

where the flux vector $\varphi(u)=(\varphi_{1}(u),\ldots,\varphi_{n}(u))$ is merely continuous: $\varphi_{i}(u)\in C(\mathbb{R})$, $i=1,\ldots,n$, and the diffusion matrix $a(u)=(a_{ij}(u))_{i,j=1}^{n}$ is Lebesgue measurable and bounded: $a_{ij}(u)\in L^{\infty}(\mathbb{R})$, $i,j=1,\ldots,n$. We also assume that the matrix ${a(u)\geq 0}$ (nonnegative definite). This matrix may have nontrivial kernel. Hence (1.1) is a degenerate (hyperbolic-parabolic) equation. In particular case $a\equiv 0$ it reduces to a first order conservation law

[TABLE]

Equation (1.1) is endowed with the initial condition

[TABLE]

Let $g(u)\in BV_{loc}(\mathbb{R})$ be a function of bounded variation on any segment in $\mathbb{R}$. We will need the bounded linear operator $T_{g}:C(\mathbb{R})/C\to C(\mathbb{R})/C$, where $C$ is the space of constants. This operator is defined up to an additive constant by the relation

[TABLE]

where $\displaystyle g(u-)=\lim_{v\to u-}g(v)$ is the left limit of $g$ at the point $u$, and the integral in (1.4) is understood in accordance with the formula

[TABLE]

where $\operatorname{sign}u=1$, $J(u)$ is the interval $[0,u)$ if $u>0$, and $\operatorname{sign}u=-1$, $J(u)=[u,0)$ if $u\leq 0$. Observe that $T_{g}(f)(u)$ is continuous even in the case of discontinuous $g(u)$. For instance, if $g(u)=\operatorname{sign}(u-k)$ then $T_{g}(f)(u)=\operatorname{sign}(u-k)(f(u)-f(k))$. Notice also that for $f\in C^{1}(\mathbb{R})$ the operator $T_{g}$ is uniquely determined by the identity $T_{g}(f)^{\prime}(u)=g(u)f^{\prime}(u)$ (in $\mathcal{D}^{\prime}(\mathbb{R})$). In the case $f^{\prime}(u),g(u)\in L^{2}_{loc}(\mathbb{R})$ the function $T_{g}(f)\in C(\mathbb{R})/C$ can be defined by the identity $T_{g}(f)^{\prime}(u)=g(u)f^{\prime}(u)\in L^{1}_{loc}(\mathbb{R})$. As is easy to see, in this case the correspondence $g\to T_{g}(f)$ is a linear continuous map from $L^{2}_{loc}(\mathbb{R})$ into $C(\mathbb{R})/C$.

We fix some representation of the diffusion matrix $a(u)$ in the form $a(u)=b^{\top}(u)b(u)$, where $b(u)=(b_{ij}(u))_{i,j=1}^{n}$ is matrix-valued function with measurable and bounded entries, $b_{ij}(u)\in L^{\infty}(\mathbb{R})$. We recall the notion of entropy solution of the Cauchy problem (1.1), (1.3) introduced in [7].

Definition 1.1**.**

A function $u=u(t,x)\in L^{\infty}(\Pi)$ is called an entropy solution (e.s. for short) of (1.1), (1.3) if the following conditions hold:

(i) for each $r=1,\ldots,n$ the distributions

[TABLE]

where vectors $B_{r}(u)=(B_{r1}(u),\ldots,B_{rn}(u))\in C(\mathbb{R},\mathbb{R}^{n})$, and $B_{ri}^{\prime}(u)=b_{ri}(u)$, $r,i=1,\ldots,n$;

(ii) for every $g(u)\in C^{1}(R)$, $r=1,\ldots,n$

[TABLE]

(iii) for any convex function $\eta(u)\in C^{2}(\mathbb{R})$

[TABLE]

(iv) $\displaystyle\operatornamewithlimits{ess\,lim}_{t\to 0}u(t,\cdot)=u_{0}$ in $L^{1}_{loc}(\mathbb{R}^{n})$.

In (1.1) the operator

[TABLE]

Relation (1.1) means that for any non-negative test function $f=f(t,x)\in C_{0}^{\infty}(\Pi)$

[TABLE]

where $D^{2}_{x}f$ is the symmetric matrix of second order derivatives of $f$, and ”$\cdot$” denotes the standard scalar multiplications of vectors or matrices.

Remark 1.1**.**

The chain rule postulated in (ii) actually holds for arbitrary locally bounded Borel function $g(u)\in L^{\infty}_{loc}(\mathbb{R})$. First of all, observe that since $B^{\prime}_{r}(u)\in L^{\infty}(\mathbb{R},\mathbb{R}^{n})$ then

[TABLE]

For $R>0$ we consider the set $F_{R}$ consisting of bounded Borel functions $g(u)$ such that $\sup|g(u)|\leq R$ and the chain rule (1.6) holds. By (ii) $F_{R}$ contains all bounded functions $g(u)\in C^{1}(\mathbb{R})$ such that $\sup|g(u)|\leq R$. Let $g_{l}(u)\in F_{R}$, $l\in\mathbb{N}$, be a sequence which converges pointwise to a function $g(u)$ as $l\to\infty$. It is clear that $g(u)$ is a Borel function and $\sup|g(u)|\leq R$. Observe that the functions $g_{l}(u(t,x))$ are bounded and measurable (because $g_{l}$ are Borel functions), the sequence $g_{l}(u(t,x))\to g(u(t,x))$ pointwise as $l\to\infty$, and $\|g_{l}(u(t,x))\|_{\infty}\leq R$. This implies that

[TABLE]

From the other hand, it follows from (1.8) that $T_{g_{l}}(B_{r})(u(t,x))$ converges to $T_{g}(B_{r})(u(t,x))$ as $l\to\infty$ uniformly on $\Pi$. Therefore, we can pass to the limit as $l\to\infty$ in relation (1.6) with $g=g_{l}$ and derive that (1.6) holds for our limit function $g(u)$. Thus, $g(u)\in F_{R}$. We see that $F_{R}$ is closed under pointwise convergence and by Lebesgue theorem $F_{R}$ contains all Borel functions $g$ such that $\sup|g|\leq R$. Since $R>0$ is arbitrary, we find that (1.6) holds for all bounded Borel functions. It only remains to notice that the behavior of $g(u)$ out of the segment $[-M,M]$, where $M=\|u\|_{\infty}$, does not matter. Therefore, (1.6) holds for any Borel function bounded on $[-M,M]$, in particular, for each locally bounded Borel function.

Remark also that for correctness of (1.6) we have to choose the Borel representative for $g(u)$ (recall that $g(u)$ is defined up to equality on a set of full measure and such the representative exists). Notice that $T_{g}(B_{r})(u)$ does not depend on the choice of a Borel representative of $g$. Hence, the right-hand side of (1.6) does not depend on this choice either.

If to be precise, in [7] the representation $a=b^{\top}b$ was used with $b=a(u)^{1/2}$. In order to make the definition invariant under linear changes of the independent variables, we have to extend the class of admissible representations. For instance, let us introduce the change $y=y(t,x)=qx-tc$, where $q:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a nondegenerate linear map and $c\in\mathbb{R}^{n}$. This can be written in the coordinate form as

[TABLE]

As is easy to verify, the function $u=v(t,y(t,x))$ is an e.s. of (1.1), (1.3) with initial data $u_{0}=v_{0}(y(0,x))$ if and only if $v(t,y)$ is an e.s. of the problem

[TABLE]

where

[TABLE]

corresponding to the representation $\tilde{a}(v)=(b(v)q^{\top})^{\top}(b(v)q^{\top})$, $b(v)q^{\top}\in L^{\infty}(\mathbb{R})$, of the diffusion matrix $\tilde{a}(v)$. We remark that

[TABLE]

Recall that $B_{r}^{\prime}(u)=b_{r}(u)=(b_{r1}(u),\ldots,b_{rn}(u))$, $r=1,\ldots,n$.

We underline that the matrix $bq^{\top}$ is not necessarily symmetric and therefore differs from $\tilde{a}(v)^{1/2}$.

Now, we suppose that the initial function $u_{0}$ is periodic with a lattice of periods $L$: $u_{0}(x+e)=u_{0}(x)$ a.e. in $\mathbb{R}^{n}$ for all $e\in L$. Denote $\mathbb{T}^{n}=\mathbb{R}^{n}/L$ the corresponding torus (which can be identified with a fundamental parallelepiped), $dx$ is the Lebesgue measure on $\mathbb{T}^{n}$, normalized by the condition $dx(\mathbb{T}^{n})=V$, $V$ being volume of a fundamental parallelepiped (observe that $V$ does not depend on its choice). Then there exists a space-periodic e.s. $u=u(t,x)$ of the problem (1.1), (1.3), $u(t,x+e)=u(t,x)$ a.e. in $\Pi$ for all $e\in L$. This can be written as $u\in L^{\infty}((0,+\infty)\times\mathbb{T}^{n})$. This solution can be constructed as a limit of the sequence $u_{k}$ of solutions to the regularized problem

[TABLE]

with smooth flux vectors $\varphi_{k}(u)$, and smooth and strictly positive definite diffusion matrices $a_{k}(u)$, which approximate $\varphi(u)$ and $a(u)$, respectively. For details, we refer to [7, 8]. It is known [7, Theorem 1.2] that e.s. $u(t,x)$ satisfies the maximum principle: $|u(t,x)|\leq\|u_{0}\|_{\infty}$ for a.e. $(t,x)\in\Pi$.

Our main result is the long time decay property of entropy solutions. Let

[TABLE]

be a dual lattice to the lattice of periods $L$,

[TABLE]

be the mean value of initial data.

Theorem 1.1**.**

Assume that the following nonlinearity-diffusivity condition holds: for all $\xi\in L^{\prime}$, $\xi\not=0$ there is no vicinity of $I$, where simultaneously the function $\xi\cdot\varphi(u)$ is affine and the function $a(u)\xi\cdot\xi\equiv 0$. Then

[TABLE]

The decay property was established in [9] under the more restrictive version of the nonlinearity-diffusivity condition, in the case of locally Lipschitz flux vector when $\varphi^{\prime}(u)\in L^{\infty}_{loc}(\mathbb{R},\mathbb{R}^{n})$. This conditions reads: for all $\tau\in\mathbb{R}$, $\xi\in L^{\prime}$, $\xi\not=0$

[TABLE]

In the hyperbolic case $a\equiv 0$ decay property (1.11) was proved in [18], see also the previous papers [6, 10, 17].

Let us show that our nonlinearity-diffusivity condition is exact. In fact, if it fails, we can find an interval $(I-\delta,I+\delta)$, a nonzero vector $\xi\in L^{\prime}$, and a constant $c\in\mathbb{R}$ such that $\xi\cdot\varphi(u)-cu\equiv\mathrm{const}$, $a(u)\xi\cdot\xi\equiv 0$ on this interval. Then, as is easily verified, the function $u(t,x)=I+\delta\sin(2\pi(\xi\cdot x-ct))$ is an $x$-periodic (with the lattice of periods $L$) e.s. of (1.1), (1.3) with the periodic initial data $u_{0}(x)=I+\delta\sin(2\pi(\xi\cdot x))$, which has the mean value $I$. It is clear that $u(t,x)$ does not converges as $t\to+\infty$ in $L^{1}(\mathbb{T}^{n})$, and the decay property fails.

2 Some properties of periodic entropy solutions

Lemma 2.1**.**

For each convex $\eta(u)$ there exists a locally finite nonnegative $x$-periodic Borel measure $\mu_{\eta}$ on $\Pi$ such that

[TABLE]

This measure can be identified with a finite measure on $\mathbb{R}_{+}\times\mathbb{T}^{n}$. Moreover, for almost each (a.e.) $t_{1},t_{2}$, $0<t_{1}<t_{2}$,

[TABLE]

Proof.

It follows from (1.1) that for every convex $\eta(u)$ the distribution

[TABLE]

By the Schwartz theorem on the representation of nonnegative distributions we conclude that (2.1) holds for some locally finite nonnegative Borel measure $\mu_{\eta}$ on $\Pi$. Ii is clear that $\mu_{\eta}$ is periodic with respect to $x$ and can be treated as a measure on $\mathbb{R}_{+}\times\mathbb{T}^{n}$. Let $\alpha(t)\in C_{0}^{1}(\mathbb{R}_{+})$, $\beta(y)\in C_{0}^{2}(\mathbb{R}^{n})$, $\alpha(t),\beta(y)\geq 0$, $\displaystyle\int_{\mathbb{R}^{n}}\beta(y)dy=1$. Applying (2.1) to the test function $\alpha(t)\beta(x/k)$, with $k\in\mathbb{N}$, we arrive at the relation

[TABLE]

Multiplying this equality by $k^{-n}$ and passing to the limit as $k\to\infty$, we obtain

[TABLE]

where we use the known property

[TABLE]

for an arbitrary $x$-periodic locally finite measure on $\Pi$. Identity (2.3) means that

[TABLE]

where $\displaystyle\int_{\mathbb{T}^{n}}d\mu_{\eta}(t,x)$ is treated as the measure $\mu_{\eta}^{t}$ on $\mathbb{R}_{+}$ defined by the relation

[TABLE]

that is, $\mu_{\eta}^{t}$ is the projection of $\mu_{\eta}$ on the $t$-axis. It follows from (2.3) that for all $t_{1},t_{2}\in\mathbb{R}_{+}$, $t_{1}<t_{2}$, being Lebesgue points of the function $\displaystyle t\to\int_{\mathbb{T}^{n}}\eta(u(t,x))dx$ relation (2.2) holds. ∎

Corollary 2.1**.**

Let $\eta(u)$ be a convex function. Then

(i) The function $I_{\eta}(t)=\displaystyle\int_{\mathbb{T}^{n}}\eta(u(t,x))dx$ decreases: $I_{\eta}(t_{2})\leq I_{\eta}(t_{1})$ for a.e. $t_{1},t_{2}\geq 0$, $t_{2}>t_{1}$;

(ii) for a.e. $t>0$

[TABLE]

(iii) the measure $\mu_{\eta}$ is finite on $\mathbb{R}_{+}\times\mathbb{T}^{n}$. Moreover,

[TABLE]

where $M=\|u_{0}\|_{\infty}$.

Proof.

Assertion (i) readily follows from (2.2). Applying (i) to the entropies $\eta(u)=\pm u$, we obtain that for a.e. $t,\tau>0$

[TABLE]

and (2.4) follows in the limit as $\tau\to 0$ with the help of initial condition (iv) of Definition 1.1.

To prove (iii), we pass to the limits in (2.2) as $t_{1}\to 0$, $t_{2}\to\infty$, taking again into account the initial condition. ∎

Lemma 2.2**.**

In the notations of Definition 1.1 the distributions

[TABLE]

and for almost all $\tau>0$

[TABLE]

Proof.

In view of relation (1.1) with $\eta(u)=u^{2}/2$

[TABLE]

Applying this relation to the nonnegative test function $k^{-n}\alpha(t)\beta(x/k)$, with $\alpha(t)\in C_{0}^{1}(\mathbb{R}_{+})$, $\beta(y)\in C_{0}^{2}(\mathbb{R}^{n})$, $k\in\mathbb{N}$, and passing to the limit as $k\to\infty$, we obtain, like in the proof of Lemma 2.1, that

[TABLE]

This relation implies that for almost all $\tau,T>0$, $\tau<T$

[TABLE]

and (2.2) follows in the limit as $T\to+\infty$. We take also into account that for a.e. $t_{0}\in(0,\tau)$

[TABLE]

by Corollary 2.1(i), which implies in the limit as $t_{0}\to 0$ that

[TABLE]

∎

Lemma 2.3**.**

Let $u_{1}=u_{1}(t,x)$, $u_{2}=u_{2}(t,x)$ be $x$-periodic e.s. of (1.1), (1.3) with initial functions $u_{10}(x),u_{20}(x)\in L^{\infty}(\mathbb{T}^{n})$, respectively. Then for a.e. $t,\tau>0$ such that $t>\tau$

[TABLE]

Proof.

As was established in [7, 1] (for the isotropic case see earlier paper [3]) by application of a variant of Kruzhkov’s doubling variables method,

[TABLE]

Applying this relation to a nonnegative test function $k^{-n}\alpha(t)\beta(x/k)$ like in the proofs of Lemmas 2.1-2.2, and passing to the limit as $k\to\infty$, we arrive at the relation

[TABLE]

This means that

[TABLE]

Therefore, for a.e. $t,\tau,\delta\in\mathbb{R}_{+}$ such that $t>\tau>\delta$

[TABLE]

and (2.6) follows in the limit as $\delta\to 0$. ∎

Corollary 2.2**.**

Let $u(t,x)$ be a periodic e.s. of (1.1), (1.3). Then the family $u(t,\cdot)$, $t>0$, is pre-compact in $L^{2}(\mathbb{T}^{n})$ (after possible correction on a set of null measure of values $t$).

Proof.

Let

[TABLE]

It is known (see, for example, [19, Lemma 1.2]) that $E$ is a set of full measure and $t\in E$ is a common Lebesgue point of the functions $t\to\int u(t,x)g(x)dx$ for all $g(x)\in L^{1}(\mathbb{R}^{n})$. It is clear that for $t\in E$ and $\Delta x\in\mathbb{R}^{n}$ $(t,x)$ is a Lebesgue point of $|u(t,x+\Delta x)-u(t,x)|$ for a.e. $x\in\mathbb{R}^{n}$. By [19, Lemma 1.2] again, $t\in E$ is a Lebesgue point of the functions $\displaystyle t\to\int_{\mathbb{T}^{n}}|u(t,x+\Delta x)-u(t,x)|dx$ for all $\Delta x\in\mathbb{R}^{n}$.

Applying Lemma 2.3 to e.s. $u(t,x)$, $u(t,x+\Delta x)$, we find that for each $t\in E$

[TABLE]

and $|u(t,x)|\leq M=\|u_{0}\|_{\infty}$. Therefore, for all $t\in E$ and all $\Delta x\in\mathbb{R}^{n}$

[TABLE]

This means that the family $u(t,\cdot)$, $t\in E$, is uniformly bounded and equicontinuous in $L^{2}(\mathbb{T}^{n})$. By the Kolmogorov-Riesz criterion this implies the pre-compactness of the family $u(t,\cdot)$ in $L^{2}(\mathbb{T}^{n})$. ∎

3 Reduction of the problem

We will need the following algebraic statement.

Lemma 3.1**.**

Let $A$ be a lattice in $X=\mathbb{R}^{n}$, $A_{0}$ be a subgroup of $A$, $X_{0}$ be a linear span of $A_{0}$. Assume that $A_{0}=A\cap X_{0}$. Then any basis of $A_{0}$ can be completed to a basis of $A$.

Proof.

Let $\xi_{1},\ldots,\xi_{k}$ be a basis of $A_{0}$, that is, any element $\xi\in A_{0}$ is uniquely represented as $\displaystyle\xi=\sum_{i=1}^{k}n_{i}\xi_{i}$ with integer coefficients $n_{i}$. It is clear that $A_{0}$ is a lattice and therefore $\xi_{1},\ldots,\xi_{k}$ is a basis of the vector space $X_{0}$. We consider the natural projection ${\operatorname{pr}:X\to X/X_{0}}$. Then $B=\operatorname{pr}(A)$ is an additive subgroup of $X/X_{0}$. We will show that $B$ is a lattice, i.e., a discrete subgroup of $X/X_{0}$. It is sufficient to show that any ball $B_{R}=\{\ x\in X/X_{0}\ |\ p(x)\leq R\ \}$ contains only finite set of points of $B$. Here

[TABLE]

is the factor-norm. Here, and in the sequel, $|z|$ denotes the Euclidean norm of a finite-dimensional vector $z$. We assume that $x=\operatorname{pr}(\xi)\in B$, where $\xi\in A$, and that $p(x)\leq R$. We can choose $y\in X_{0}$ such that $|\xi-y|=p(x)\leq R$. Recall that $\xi_{i}$, $i=1,\ldots,k$ is a basis of $X_{0}$. Therefore, we can represent $\displaystyle y=\sum_{i=1}^{k}s_{i}\xi_{i}$, $s_{i}\in\mathbb{R}$. Let $n_{i}=[s_{i}]\in\mathbb{Z}$ be integer parts of $s_{i}$, so that $\alpha_{i}=s_{i}-n_{i}\in[0,1)$. Then

[TABLE]

Thus,

[TABLE]

Since $A$ is a lattice, the set $A_{R+C}$ is finite. Taking into account that $\displaystyle\sum_{i=1}^{k}n_{i}\xi_{i}\in A_{0}\subset X_{0}$, we find that

[TABLE]

Hence, $B\cap B_{R}\subset\operatorname{pr}(A_{R+C})$ is a finite set for each $R>0$. We conclude that $B$ is a lattice. Therefore, there exists a basis $x_{i}$, $i=k+1,\ldots,m$, of the free abelian group $B$. We can find such $\xi_{i}\in A$ that $x_{i}=\operatorname{pr}(\xi_{i})$. Then for every $\xi\in A$ there exist unique $n_{i}\in\mathbb{Z}$, $i=k+1,\ldots,m$, such that $\displaystyle\eta=\xi-\sum_{i=k+1}^{m}n_{i}\xi_{i}\in X_{0}$. We also observe that $\eta\in A$. Thus, $\eta\in A\cap X_{0}=A_{0}$. Since $\xi_{i}$, $i=1,\ldots,k$ is a basis of $A_{0}$, then there exist unique $n_{i}\in\mathbb{Z}$, $i=1,\ldots,k$, such that $\displaystyle\eta=\sum_{i=1}^{k}n_{i}\xi_{i}.$ We conclude that

[TABLE]

and this representation is unique. This means that $\xi_{i}$, $i=1,\ldots,m$ is a basis of $A$. The proof is complete. ∎

Remark 3.1**.**

The requirement $A\cap X_{0}=A_{0}$ is exact: if a basis of $A_{0}$ can be completed to a basis of $A$ then $A\cap X_{0}=A_{0}$. In fact, let $\xi_{1},\ldots,\xi_{k},\xi_{k+1},\ldots,\xi_{m}$ be a basis of $A$ completing a basis $\xi_{1},\ldots,\xi_{k}$ of $A_{0}$ and $\xi\in A\cap X_{0}$. Since $\xi_{1},\ldots,\xi_{k}$ is also a basis of the linear space $X_{0}$, we have two representation

[TABLE]

Taking into account that the vectors $\xi_{1},\ldots,\xi_{m}$ are linearly independent, we conclude that these representations must coincide. In particular, $n_{i}=0$ for $i>k$ and $\displaystyle\xi=\sum_{i=1}^{k}n_{i}\xi_{i}\in A_{0}$. Hence, $A\cap X_{0}\subset A_{0}$. Since the inverse inclusion $A_{0}\subset A\cap X_{0}$ is evident, we conclude that $A\cap X_{0}=A_{0}$.

We define

[TABLE]

It is clear that for any vector $\xi\in X_{0}$, where $X_{0}$ is the linear span of $L^{\prime}_{0}$, the function $\xi\cdot\varphi(u)$ is affine in some vicinity of $I$, this means that $\xi\cdot\varphi(u)=a(\xi)u+\mathrm{const}$ in this vicinity. Since the map $\xi\to a(\xi)$ is linear, there exist a unique vector $\bar{c}\in X_{0}$ such that $a(\xi)=\bar{c}\cdot\xi$.

Obviously, $L^{\prime}\cap X_{0}\subset L^{\prime}_{0}$ and $L^{\prime}_{0}$ is a subgroup of the lattice $L^{\prime}$ satisfying the condition of Lemma 3.1. Let $m$ be a rank of $L^{\prime}_{0}$, $d=n-m$, $\zeta_{i}$, $i=d+1,\ldots,n$, be a basis of $L^{\prime}_{0}$. By Lemma 3.1 this basis can be completed to a basis $\zeta_{i}$, $i=1,\ldots,n$, of the lattice $L^{\prime}$. Then the vectors $\zeta_{i}$, $i=1,\ldots,n$ forms a basis of $\mathbb{R}^{n}$ as well. We introduce the linear change $y=y(t,x)$, $y_{i}=\zeta_{i}\cdot x-c_{i}t$, $i=1,\ldots,n$, where $c_{i}=\bar{c}\cdot\zeta_{i}=a(\zeta_{i})$. After this change, we obtain entropy solution $v(t,y)$ of the problem

[TABLE]

where, in view of (1.10), $\tilde{\varphi}_{i}(v)=\zeta_{i}\cdot\varphi(v)-c_{i}v$, $\tilde{a}_{ij}(v)=a(v)\zeta_{i}\cdot\zeta_{j}$. By the construction the last $m$ flux component $\tilde{\varphi}_{i}(v)$, $i=d+1,\ldots,n$ are constant on some interval $(\alpha,\beta)\ni I$. We notice that $v(t,y)$ is periodic with respect to $y$ with the standard lattice of periods $\mathbb{Z}^{n}$, this follows from the fact that $y(0,x)$ is an isomorphism from $L$ into $\mathbb{Z}^{n}$. We underline that the mean value $I$ remains the same under the described change. The nonlinearity-diffusivity condition of Theorem 1.1 converts to the requirement: for all $\kappa\in\mathbb{Z}^{n}$, $\kappa\not=0$, there is no vicinity of $I$, where simultaneously the function $\kappa\cdot\tilde{\varphi}(u)$ is affine and the function $\tilde{a}(u)\kappa\cdot\kappa\equiv 0$. This directly follows from the relations

[TABLE]

where $\displaystyle\xi=\sum_{i=1}^{n}\kappa_{i}\zeta_{i}\in L^{\prime}$. The condition that the function $\xi\cdot\varphi(u)$ is affine in some vicinity of $I$ is equivalent to the inclusion $\xi\in L^{\prime}_{0}\Leftrightarrow\tilde{\kappa}=0$, where we denote by $\tilde{\kappa}$, $\bar{\kappa}$ the orthogonal projections of $\kappa$ on the spaces

[TABLE]

respectively. Therefore, the nonlinearity-diffusivity condition reduces to the requirement: $\forall\kappa=\bar{\kappa}\in\mathbb{Z}^{n}$, $\kappa\not=0$, the function $\tilde{a}(u)\kappa\cdot\kappa\not\equiv 0$ in any vicinity of $I$.

Hence, going back to the original notations of equation (1.1), we can suppose that

• (R1)

$L=\mathbb{Z}^{n}$ (and in particular the volume $V=1$);

• (R2)

the function $\xi\cdot\varphi(u)$, $\xi\in L^{\prime}=\mathbb{Z}^{n}$, is affine if and only if $\xi\in(\mathbb{R}^{d})^{\perp}$;

• (R3)

the function $\xi\cdot\varphi(u)\equiv c(\xi)=\mathrm{const}$ on an interval $(\alpha,\beta)\ni I$ for all $\xi\in\mathbb{Z}^{n}\cap(\mathbb{R}^{d})^{\perp}$;

• (R4)

the function $a(u)\xi\cdot\xi\not\equiv 0$ in any vicinity of $I$ for all $\xi\in\mathbb{Z}^{n}\cap(\mathbb{R}^{d})^{\perp}$.

4 Preliminaries

4.1 Measure valued functions and H-measures

Recall (see [4, 20]) that a measure-valued function on a domain $\Omega\subset\mathbb{R}^{N}$ is a weakly measurable map $x\mapsto\nu_{x}$ of $\Omega$ into the space $\operatorname{Prob}_{0}(\mathbb{R})$ of probability Borel measures with compact support in $\mathbb{R}$.

The weak measurability of $\nu_{x}$ means that for each continuous function $g(\lambda)$ the function $x\to\langle\nu_{x},g(\lambda)\rangle\doteq\int g(\lambda)d\nu_{x}(\lambda)$ is measurable on $\Omega$.

A measure-valued function $\nu_{x}$ is said to be bounded if there exists $M>0$ such that $\operatorname{supp}\nu_{x}\subset[-M,M]$ for almost all $x\in\Omega$.

Measure-valued functions of the kind $\nu_{x}(\lambda)=\delta(\lambda-u(x))$, where $u(x)\in L^{\infty}(\Omega)$ and $\delta(\lambda-u^{*})$ is the Dirac measure at $u^{*}\in\mathbb{R}$, are called regular. We identify these measure-valued functions and the corresponding functions $u(x)$, so that there is a natural embedding of $L^{\infty}(\Omega)$ into the set $\operatorname{MV}(\Omega)$ of bounded measure-valued functions on $\Omega$.

Measure-valued functions naturally arise as weak limits of bounded sequences in $L^{\infty}(\Omega)$ in the sense of the following theorem by L. Tartar [20].

Theorem 4.1**.**

Let $u_{k}(x)\in L^{\infty}(\Omega)$, $k\in\mathbb{N}$, be a bounded sequence. Then there exist a subsequence (we keep the notation $u_{k}(x)$ for this subsequence) and a bounded measure valued function $\nu_{x}\in\operatorname{MV}(\Omega)$ such that

[TABLE]

Besides, $\nu_{x}$ is regular, i.e., $\nu_{x}(\lambda)=\delta(\lambda-u(x))$ if and only if $u_{k}(x)\mathop{\to}\limits_{k\to\infty}u(x)$ in $L^{1}_{loc}(\Omega)$ (strongly).

Another useful tool for evaluation of weak convergence is Tartar’s H-measures and their variants. H-measures were introduced by L. Tartar [21] and independently by P. Gérard in [5]. We recall the notion of H-measure in the simple case of scalar sequences.

Let $S=\{\ \xi\in\mathbb{R}^{N}\ |\ |\xi|=1\ \}$ be the unit sphere in $\mathbb{R}^{N}$, $\pi:\mathbb{R}^{N}\setminus\{0\}\to S$, $\pi(\xi)=\xi/|\xi|$, be the orthogonal projection on the sphere. Let

[TABLE]

be the Fourier transformation extended as a unitary operator on the space $L^{2}(\mathbb{R}^{N})$, $u\to\overline{u}$, $u\in\mathbb{C}$, be the complex conjugation.

Now, we assume that $u_{k}=u_{k}(x)$ is a bounded sequence in $L^{\infty}(\Omega)$ (more generally, in $L^{2}_{loc}(\Omega)$) weakly convergent to [math],

Proposition 4.1** (see [21, 5]).**

There exists a nonnegative Borel measure $\mu$ in $\Omega\times S$ and a subsequence $u_{r}(x)=u_{k}(x),$ $k=k_{r},$ such that

[TABLE]

for all $\Phi_{1}(x),\Phi_{2}(x)\in C_{0}(\Omega)$ and $\psi(\xi)\in C(S)$.

Remark 4.1**.**

It follows from (4.2) and the Plancherel identity that $\operatorname{pr}_{\Omega}\mu\leq C\operatorname{meas}$, and that (4.2) remains valid for all $\Phi_{1}(x),\Phi_{2}(x)\in L^{2}(\Omega)$, cf. [15, Remark 2(a)]. Here we denote by $\operatorname{meas}$ the Lebesgue measure on $\Omega$.

The measure $\mu$ is called an H-measure corresponding to $u_{r}(x)$. It is clear that $\mu=0$ if and only if the sequence $u_{r}\to 0$ as $r\to\infty$ in $L^{2}_{loc}(\Omega)$. In some sense, the H-measure $\mu=\mu(x,\xi)$ indicates the strength of oscillations of the sequence $u_{r}$ at the point $x$ and in the direction $\xi$.

In the sequel we will use the special sequences obtained from a given function by rescaling of independent variables. Namely, let $u(t,x)\in L^{\infty}(\Pi)$ be a function periodic over the space variables $x$ (with the standard lattice of periods $\mathbb{Z}^{n}$), and $s_{k}\in\mathbb{R}_{+}$, $p_{k},q_{k}\in\mathbb{N}$ be sequences converging to infinity as $k\to\infty$. We consider the sequence $u_{k}=u_{k}(t,x)=u(s_{k}t,p_{k}\tilde{x}+q_{k}\bar{x})$, where $\tilde{x},\bar{x}$ are the orthogonal projection of $x\in\mathbb{R}^{n}$ into the spaces $\mathbb{R}^{d}=\{\ x=(x_{1},\ldots,x_{n})\ |\ x_{i}=0,i=d+1,\ldots,n\ \}$ and $(\mathbb{R}^{d})^{\perp}=\{\ x=(x_{1},\ldots,x_{n})\ |\ x_{i}=0,i=1,\ldots,d\ \}$, respectively. By the periodicity, the function $u$ and the functions $u_{k}$ can be treated as elements of $L^{\infty}(\mathbb{R}_{+}\times\mathbb{T}^{n})$, where $\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$ is the standard torus.

Lemma 4.1**.**

Suppose that the sequence $I_{0}(s_{k}t)=\int_{\mathbb{T}^{n}}u(s_{k}t,x)dx\rightharpoonup v(t)$ weakly-$*$ in $L^{\infty}(\mathbb{R}_{+})$ as $k\to\infty$. Then $u_{k}\rightharpoonup v(t)$ weakly-$*$ in $L^{\infty}(\Pi)$ as $k\to\infty$.

Proof.

For given $r\in\mathbb{Z}^{n}$, $r\not=0$, we choose $\bar{k}\in\mathbb{N}$ so large that $\min(p_{k},q_{k})>|r|^{2}$ for all $k\geq\bar{k}$. Observe that the projections $\tilde{r},\bar{r}\in\mathbb{Z}^{n}$ and, by the periodicity of $u(t,\cdot)$,

[TABLE]

Since either $\tilde{r}\not=0$ or $\bar{r}\not=0$ while $p_{k}>|r|^{2}\geq|\tilde{r}|^{2}$, $q_{k}>|r|^{2}\geq|\bar{r}|^{2}$, one of the factors $e^{2\pi i|\tilde{r}|^{2}/p_{k}}$, $e^{2\pi i|\bar{r}|^{2}/q_{k}}$ is different from $1$. Therefore, it follows from (4.1), (4.1) that for $k\geq\bar{k}$

[TABLE]

This implies that for each $a(t)\in L^{1}(\mathbb{R}_{+})$

[TABLE]

If $r=0$ we have

[TABLE]

because the map $x\to p_{k}\tilde{x}+q_{k}\bar{x}$ keeps the Lebesgue measure on $\mathbb{T}^{n}$. Since $I_{0}(s_{k}t)$ weakly-$*$ converges to the function $v(t)$, we conclude that for each $a(t)\in L^{1}(\mathbb{R}_{+})$

[TABLE]

From relations (4.5), (4.6) it follows that

[TABLE]

for every function $f(t,x)=\sum_{r\in I}a_{r}(t)e^{2\pi ir\cdot x}$, where the parameter $r$ runs over a finite set $I\subset\mathbb{Z}^{n}$ while $a_{r}(t)\in L^{1}(\mathbb{R}_{+})$. Since the space of such functions is dense in $L^{1}(\mathbb{R}_{+}\times\mathbb{T}^{n})$, relation (4.7) implies that $u_{k}\rightharpoonup v(t)$ as $k\to\infty$ weakly-$*$ in $L^{\infty}(\mathbb{R}_{+}\times\mathbb{T}^{n})$. By the periodicity, the above limit relation holds also weakly-$*$ in the space $L^{\infty}(\Pi)$. The proof is complete. ∎

Corollary 4.1**.**

Assume in addition that for a dense set of functions $p(u)\in C(\mathbb{R})$

[TABLE]

Then the sequence $u_{k}$ converges to a constant measure valued function ${\nu_{t,x}\equiv\nu}$ in the sense of relation (4.1).

Proof.

Since the set of functions $p(\lambda)$, for which relation (4.8) holds, is dense in $C([-M,M])$, where $M=\|u\|_{\infty}$, this relation remains valid for all $p(\lambda)\in C([-M,M])$. It is clear that the functional $p\to c_{p}$ is linear and continuous. By the Riesz-Markov representation theorem $c_{p}=\langle\nu,p(\lambda)\rangle=\int p(\lambda)d\nu(\lambda)$ with some Borel measure $\nu$ on $[-M,M]$. Evidently, $c_{p}\geq 0$ for $p(\lambda)\geq 0$ and $c_{p}=1$ for $p\equiv 1$, which implies that $\nu$ is a probability measure. By Lemma 4.1 we conclude that for each $p(\lambda)\in C(\mathbb{R})$

[TABLE]

which is exactly (4.1) with $\nu_{t,x}\equiv\nu$. ∎

Remark 4.2**.**

Condition (4.8) is always satisfied in the case when $u(t,x)$ is a periodic e.s. of (1.1), (1.3). In fact, any function $p(u)\in C^{2}(\mathbb{R})$ is a difference of two convex functions. By Corollary 2.1(i) the function $\displaystyle I_{p}(t)=\int_{\mathbb{T}^{n}}p(u(t,x))dx$ is a difference of two decreasing functions (after possible extraction of a set of null measure). Hence $I_{p}(t)$ is a function of bounded variation and there exists a limit $c_{p}$ of this function as $t\to+\infty$. This implies that the sequence $I_{p}(s_{k}t)$ converges in $L^{1}_{loc}(\mathbb{R}_{+})$ to the constant $c_{p}$. Hence, limit relation (4.8) holds, even in the stronger topology of $L^{1}_{loc}$.

Let $u_{k}=u(k^{2}t,k^{2}\tilde{x}+k\bar{x})$ be a sequence of the considered above kind (with $s_{k}=p_{k}=k^{2}$, $q_{k}=k$). Passing to a subsequence, we may assume that this sequence converges weakly-$*$ in $L^{\infty}(\Pi)$. Then the sequence

[TABLE]

converges weakly-$*$ in $L^{\infty}(\mathbb{R}_{+})$ to some function $v(t)$. By Lemma 4.1 the function $v(t)$ is a weak-$*$ limit of the sequence $u_{k}$.

We are going to investigate the H-measure corresponding to the sequence $u_{k}-v(t)$. We suppose that the family $u(t,\cdot)$, $t>0$, is pre-compact in $L^{2}(\mathbb{T}^{n})$. As was shown in [16, Lemma 3.1], this requirement implies that the Fourier series

[TABLE]

converge in $L^{2}(\mathbb{T}^{n})$ uniformly in $t\geq 0$.

By Proposition 4.1 there exists $\mu=\mu(t,x,\tau,\xi)$, an H-measure corresponding to some subsequence of $u_{k}-v$. Denote by

[TABLE]

where

[TABLE]

is the orthogonal projection on the sphere

[TABLE]

Proposition 4.2**.**

The support of H-measure $\mu$ is contained in $\Pi\times S_{0}$, that is, $\mu(\Pi\times(S\setminus S_{0}))=0$.

Proof.

For $m\in\mathbb{N}$ we introduce the sets

[TABLE]

It is clear that $S_{m}$ is a closed subset of $S$ (it is the union of the finite set of circles \{\ \bigl{(}p,q|\xi|^{-1}\xi\bigr{)}\ |\ p^{2}+q^{2}=1\ \}, where $\xi\in\mathbb{Z}^{n}$, $|\tilde{\xi}|\cdot|\bar{\xi}|=0$, $0<|\xi|\leq m$), and $\displaystyle S_{0}=\cup_{m=1}^{\infty}S_{m}$. Let

[TABLE]

be the Fourier series for $u(t,\cdot)$ in $L^{2}(\mathbb{T}^{n})$. Then

[TABLE]

We denote $b_{0,k}=a_{0}(k^{2}t)-v(t)$; $b_{\kappa,k}=a_{\kappa}(k^{2}t)$, where $\kappa\in\mathbb{Z}^{n}$, $\kappa\not=0$. Let $\alpha(t)\in C_{0}(\mathbb{R}_{+})$, and $\beta(x)\in L^{2}(\mathbb{R}^{n})\cap C^{\infty}(\mathbb{R}^{n})$ be such function that its Fourier transform is a continuous compactly supported function:

[TABLE]

We take $R=\max\limits_{\xi\in\operatorname{supp}\tilde{\beta}}|\xi|$. Let $\Phi(t,x)=\alpha(t)\beta(x)$. By (4.10) we find that

[TABLE]

Observe that the Fourier transform of $e^{2\pi i(k^{2}\kappa+k\bar{\kappa})\cdot x}\beta(x)$ in $\mathbb{R}^{n}$ coincides with ${\tilde{\beta}(\xi-(k^{2}\tilde{\kappa}+k\bar{\kappa}))}$. Since for $k>2R+1$ supports of these functions do not intersect, then for such $k$ the series

[TABLE]

is orthogonal in $L^{2}(\mathbb{R}^{n})$ for each $t>0$. Besides, by the Plancherel equality

[TABLE]

and

[TABLE]

Therefore, orthogonal series (4.13) converges in $L^{2}(\mathbb{R}^{n})$ for each $t>0$. Moreover, by the uniform converges of the Fourier series (4.9)

[TABLE]

uniformly with respect to $t>0$. Hence, series (4.13) converges in $L^{2}(\mathbb{R}^{n})$ uniformly with respect to $t$. Since the Fourier transformation is an isomorphism on $L^{2}(\mathbb{R}^{n})$, we conclude that series (4.12) also converges in $L^{2}(\mathbb{R}^{n})$ uniformly with respect to $t$. Since $\alpha(t)\in C_{0}(\mathbb{R})$, this implies that (4.12) converges in $L^{2}(\Pi)$, and

[TABLE]

where $F^{t}(h)(\tau)=\int_{\mathbb{R}}e^{-2\pi i\tau t}h(t)dt$ denotes the Fourier transform over the time variable (we extend functions $h(t)\in L^{2}(\mathbb{R}_{+})$ on the whole line $\mathbb{R}$, setting $h(t)=0$ for $t<0$). It follows from (4.15) that for $k>2R+1$

[TABLE]

where $\psi(\hat{\xi})\in C(S)$ is arbitrary. Now we fix $\varepsilon>0$. Recall that $b_{\kappa,k}=a_{\kappa}(k^{2}t)$ for $\kappa\not=0$, and by (4.14) there exists $m\in\mathbb{N}$ such that

[TABLE]

Then

[TABLE]

Now we suppose that $\|\psi\|_{\infty}\leq 1$ and $\psi(\hat{\xi})=0$ on the set $S_{m}$. By (4.1)

[TABLE]

Since continuous function $\psi(\hat{\xi})$ is uniformly continuous on the compact $S$ then we can find such $\delta>0$ that $|\psi(\hat{\xi}_{1})-\psi(\hat{\xi}_{2})|<\varepsilon$ whenever $\hat{\xi}_{1},\hat{\xi}_{2}\in S$, $|\hat{\xi}_{1}-\hat{\xi}_{2}|<\delta$. Suppose that $\kappa\not=0$, $\tilde{\beta}(\xi-(k^{2}\tilde{\kappa}+k\bar{\kappa}))\not=0$. Then $|\xi-(k^{2}\tilde{\kappa}+k\bar{\kappa})|\leq R$. For a fixed $\tau\in\mathbb{R}$ we denote $\hat{\xi}=(\tau,\xi)$, $\hat{\eta}=(\tau,k^{2}\tilde{\kappa}+k\bar{\kappa})$; $\hat{\eta}_{1}=(\tau,k^{2}\tilde{\kappa})$ if $\tilde{\kappa}\not=0$, $\hat{\eta}_{1}=(\tau,k\bar{\kappa})$ if $\tilde{\kappa}=0$. By the evident estimate $\displaystyle|\pi(x)-\pi(y)|\leq\frac{2|x-y|}{|y|}$, for each $\kappa\in\mathbb{Z}^{n}$, $\kappa\not=0$,

[TABLE]

In view of (4.19), (4.20), we have

[TABLE]

By (4.21) for all $k>k_{0}=2(R+m)/\delta$ and all $\kappa\in\mathbb{Z}^{n}$ such that $0<|\kappa|\leq m$ and $\tilde{\beta}(\xi-(k^{2}\tilde{\kappa}+k\bar{\kappa}))\not=0$ the inequality $|\pi(\hat{\xi})-\pi(\hat{\eta}_{1})|<\delta$ holds, which implies the estimate

[TABLE]

We use here that $\pi(\hat{\eta}_{1})\in S_{m}$ and, therefore, $\psi(\pi(\hat{\eta}_{1}))=0$. In view of (4.22), for all $k>k_{0}$

[TABLE]

where $C=4\|u\|_{\infty}^{2}$. Further, in the case $\kappa=0$ when $\hat{\eta}_{1}=\hat{\eta}=(\tau,0)\in S_{m}$, we have the estimate

[TABLE]

for $|\xi|\leq R$. Taking $|\tau|>T\doteq\frac{2R}{\delta}$, we find

[TABLE]

Therefore,

[TABLE]

Here \theta(r)=\left\{\begin{array}[]{lcr}1&,&r>0,\\ 0&,&r\leq 0\end{array}\right. is the Heaviside function.

For $|\tau|\leq T$ we are reasoning in the following way. Since $\alpha(t)b_{0,k}(t)={\alpha(t)(a_{0,k}(t)-v(t))\rightharpoonup 0}$ as $k\to\infty$, and $\|\alpha b_{0,k}\|_{1}\leq C_{1}=2\|u\|_{\infty}\|\alpha\|_{1}$, the Fourier transform $F^{t}(\alpha b_{0,k})(\tau)\mathop{\to}\limits_{k\to\infty}0$ for all $\tau\in\mathbb{R}$ and uniformly bounded: $|F^{t}(\alpha b_{0,k})(\tau)|\leq C_{1}$. By Lebesgue dominated convergence theorem

[TABLE]

Therefore (recall that $\|\psi\|_{\infty}\leq 1$),

[TABLE]

In view of (4.24), (4.1) we find

[TABLE]

Using (4.1), (4.18), (4.1) and (4.26), we arrive at the relation

[TABLE]

where $C_{2}$ is a constant. By the definition of H-measure and Remark 4.1

[TABLE]

and (4.27) implies that

[TABLE]

for all $\psi(\hat{\xi})\in C_{0}(S\setminus S_{m})$ such that $0\leq\psi(\hat{\xi})\leq 1$. Therefore, we can claim that

[TABLE]

and since $S\setminus S_{0}\subset S\setminus S_{m}$, we obtain the relation

[TABLE]

which holds for arbitrary positive $\varepsilon$. Therefore,

[TABLE]

Since for every point $(t_{0},x_{0})\in\Pi$ one can find functions $\alpha(t)$, $\beta(x)$ with the prescribed above properties in such a way that $\Phi(t,x)=\alpha(t)\beta(x)\not=0$ in a neighborhood of $(t_{0},x_{0})$, we derive from (4.28) the desired relation $\mu(\Pi\times(S\setminus S_{0}))=0$. ∎

4.2 Variants of H-measures

We will need the variant of H-measures with “continuous indexes” introduced in [12], see also subsequent papers [13, 14, 15, 18]. Let $u_{k}(x)$ be a bounded sequence in $L^{\infty}(\Omega)$. Passing to a subsequence if necessary, we can suppose that this sequence converges to a bounded measure valued function $\nu_{x}\in\operatorname{MV}(\Omega)$ in the sense of relation (4.1). We introduce the measures $\gamma_{x}^{k}(\lambda)=\delta(\lambda-u_{k}(x))-\nu_{x}(\lambda)$ and the corresponding distribution functions $U_{k}(x,p)=\gamma_{x}^{k}((p,+\infty))$, $u_{0}(x,p)=\nu_{x}((p,+\infty))$ on $\Omega\times\mathbb{R}$. Observe that $U_{k}(x,p),u_{0}(x,p)\in L^{\infty}(\Omega)$ for all $p\in\mathbb{R}$, see [12, Lemma 2]. We define the set

[TABLE]

As was shown in [12, Lemma 4], the complement $\mathbb{R}\setminus E$ is at most countable and if $p\in E$ then $U_{k}(x,p)\mathop{\rightharpoonup}\limits_{k\to\infty}\,0$ weakly-$*$ in $L^{\infty}(\Omega)$.

The following result was established in [12] (see also [13, 14, 15] ).

Proposition 4.3**.**

There exist a family of locally finite complex Borel measures $\{\mu^{pq}\}_{p,q\in E}$ on $\Omega\times S$ and a subsequence $U_{r}(x)=\{U_{k_{r}}(x,p)\}_{p\in E}$ such that for all $\Phi_{1}(x),\Phi_{2}(x)\in C_{0}(\Omega)$ and $\psi(\xi)\in C(S)$

[TABLE]

Moreover for every $p_{1},\ldots,p_{l}\in E$, $\zeta_{1},\ldots,\zeta_{l}\in\mathbb{C}$ the measure ${\displaystyle\sum_{i,j=1}^{l}\zeta_{i}\overline{\zeta_{j}}\mu^{p_{i}p_{j}}\geq 0}$.

Since $|U_{r}(x,p)|\leq 1$, the projection $\operatorname{pr}_{\Omega}|\mu^{pq}|\leq\operatorname{meas}$ and the measures $\mu^{pq}$ admits disintegration: $\mu^{pq}(x,\xi)=\mu^{pq}_{x}(\xi)dx$, where $\mu^{pq}_{x}(\xi)\in M(S)$, $p,q\in E$, $x\in\Omega$, is a family of finite Borel measures on $S$. This means that for every $f(x,\xi)\in C_{0}(\Omega\times S)$ the function

[TABLE]

is Lebesgue-measurable, bounded and

[TABLE]

Let $D\subset E$ be a countable dense subset of $E$. As was demonstrated in [18], for almost all $x\in\Omega$ and all $p_{1},q_{1},p_{2},q_{2}\in D$

[TABLE]

and these estimates implies that the maps $D^{2}\ni(p,q)\to\mu^{pq}_{x}$ admit right and left continuous in $M(S)$ extensions $\mu^{pq+}_{x}$ and $\mu^{pq-}_{x}$, respectively, on the space of all pairs $(p,q)\in\mathbb{R}^{2}$, where $x$ runs over some set of full measure in $\Omega$. Actually, these extensions do not depend on the choice of a set $D$. Namely, for different sets $D$ the corresponding measures $\mu^{pq\pm}_{x}$ coincide for all $x\in\Omega^{\prime}$, where $\Omega^{\prime}\subset\Omega$ is a set of full Lebesgue measure, independent of $p,q$. The measures $\mu^{pq+}_{x}$ inherits the nonnegative definiteness property: for every $p_{1},\ldots,p_{l}\in\mathbb{R}$, $\zeta_{1},\ldots,\zeta_{l}\in\mathbb{C}$ the measures $\displaystyle\sum_{i,j=1}^{l}\zeta_{i}\overline{\zeta_{j}}\mu^{p_{i}p_{j}\pm}_{x}\geq 0$. As was shown in [18, Corollary 1], this property implies that for each $p,q\in\mathbb{R}$ for every Borel set $A\subset S$

[TABLE]

where $|\mu|$ denotes the variation of a measure $\mu$, i.e., the smallest nonnegative measure $m$ with the property $|\mu(A)|\leq m(A)$ for every Borel set $A$.

The following property connecting the H-measure $\mu_{x}^{pp\pm}$ and the measure valued function $\nu_{x}$, corresponding to the same subsequence $u_{r}$, was proved in [18, Corollary 2].

Lemma 4.2**.**

For almost all $x\in\Omega$ (independent of $p$) for all $p$ belonging to the smallest segment $[a,b]$ containing $\operatorname{supp}\nu_{x}$ the following holds: If $S_{+},S_{-}\subset S$ are Borel sets such that

[TABLE]

then $S_{+}\cap S_{-}\not=\emptyset$. In particular, $\operatorname{supp}\mu^{pp+}_{x}\cap\operatorname{supp}\mu^{pp-}_{x}\not=\emptyset$.

Denote by $s_{\alpha,\beta}(u)=\max(\alpha,\min(\beta,u))$ the cut-off function, where $\alpha,\beta\in\mathbb{R}$, $\alpha<\beta$. Let $\psi(u)=(\psi_{1}(u),\ldots,\psi_{N}(u))\in C(\mathbb{R},\mathbb{R}^{N})$ be a continuous vector-function. Suppose that for every $\alpha,\beta\in\mathbb{R}$, $\alpha<\beta$ the sequences of distributions

[TABLE]

where the Sobolev space $H^{-1}_{loc}(\Omega)$ is a locally convex space, consisting of distributions $l\in\mathcal{D}^{\prime}(\Omega)$ such that for all $f\in C_{0}^{\infty}(\Omega)$ the distribution $fl\in H^{-1}=H^{-1}(\mathbb{R}^{N})$ (the latter space is dual to the Sobolev space $H^{1}=W_{2}^{1}(\mathbb{R}^{N})$), the topology in $H^{-1}_{loc}(\Omega)$ is generated by seminorms $\|fl\|_{H^{-1}}$. Extracting a subsequence if necessary, we can assume that H-measures $\mu^{pq\pm}_{x}$ are well-defined. Let $H_{+}(x)$, $H_{-}(x)$ be the minimal linear subspaces of $\mathbb{R}^{N}$ containing $\operatorname{supp}\mu^{p_{0}p_{0}+}_{x}$, $\operatorname{supp}\mu^{p_{0}p_{0}-}_{x}$, respectively, where $p_{0}\in\mathbb{R}$ is fixed.

Proposition 4.4**.**

For a.e. $x\in\Omega$ for all $\xi\in H_{+}(x)\cap H_{-}(x)$ the function $\xi\cdot\psi(u)$ is constant in some vicinity $|u-p_{0}|<\delta$ of $p_{0}$.

The statement of Proposition 4.5 follows from a general result of [18, Theorem 3.2].

In the case of the sequence $u_{k}=u(k^{2}t,k^{2}\tilde{x}+k\bar{x})$ it follows from Proposition 4.2 that $\mu^{pp\pm}_{t,x}(S\setminus S_{0})=0$. More precisely, we assume that $u(t,x)\in L^{\infty}(\Pi)$ is an $x$-periodic function with the standard lattice of periods $\mathbb{Z}^{n}$. We consider the sequence $u_{k}=u(k^{2}t,k^{2}\tilde{x}+k\bar{x})$. Passing to a subsequence (and keeping the notation $u_{k}$ for this subsequence), we may assume that $u_{k}\to\nu_{t,x}$ in the sense of relation (4.1), and that the H-measures $\mu^{pq\pm}_{t,x}$ are defined.

Proposition 4.5**.**

Assume that the family $u(t,\cdot)$, $t>0$, is pre-compact in $L^{2}(\mathbb{T}^{n})$ (possibly, after correction on a set of $t$ of full measure). Then there exists a set $\Pi^{\prime}\subset\Pi$ of full Lebesgue measure such that for all $(t,x)\in\Pi^{\prime}$ and all $p,q\in\mathbb{R}$ $\mu^{pq\pm}_{t,x}(S\setminus S_{0})=0$.

Proof.

Let $M=\|u\|_{\infty}$. Observe that by Lemma 4.1 for each $g(\lambda)\in C(\mathbb{R})$ the weak-$*$ limit of $g(u_{k})$ does not depend on $x$:

[TABLE]

Let $G\subset C(\mathbb{R})$ be a countable dense set. Then the set $A\subset\mathbb{R}_{+}$ of common Lebesgue points of the functions $v_{g}(t)$, $g\in G$ has full measure. Obviously, for each $t\in A$ the functional $g\to v_{g}(t)$ is uniquely extended to a linear continuous functional on the whole $C(\mathbb{R})$. Therefore, for some compactly supported measure $\nu_{t}$

[TABLE]

Since $v_{g}(t)=0$ whenever $g\equiv 0$ on $[-M,M]$, we see that $\operatorname{supp}\nu_{t}\subset[-M,M]$. By (4.32) we conclude that $\nu_{t}$ is a limit measure valued function of the sequence $u_{k}$. Hence, the measure valued function $\nu_{t,x}=\nu_{t}$ does not depend on $x$. By (4.29), (4.2) we find that the measure $\mu^{pp}=\mu^{pp}_{t,x}(\hat{\xi})dtdx$ coincides with the H-measure $\mu=\mu(t,x,\hat{\xi})$ corresponding to the sequence $U_{k}(t,x,p)=U(k^{2}t,k^{2}\tilde{x}+k\bar{x},p)-v(t,p)$, where $U(t,x,p)=\theta(u(t,x)-p)$ ($\theta(r)$ being the Heaviside function), $v(t,p)=\nu_{t}((p,+\infty))$ is a weak-$*$ limit of the sequence $U_{k}(t,x,p)$. By Proposition 4.2 we claim that $\mu(\Pi\times(S\setminus S_{0}))=0$. From (4.30) it follows that for all $f(t,x)\in C_{0}(\Pi)$, $g(\hat{\xi})\in C(S)$

[TABLE]

Obviously, (4.33) remains valid for Borel functions $g(\hat{\xi})$. Taking $g(\hat{\xi})$ being the indicator function of the set $S\setminus S_{0}$, we arrive at the relation

[TABLE]

Therefore, $\langle\mu^{pp}_{t,x}(\hat{\xi}),g(\hat{\xi})\rangle=0$ for almost all $(t,x)\in\Pi$. Now we fix some countable dense set $D\subset E$ and choose a set of full Lebesgue measure $\Pi_{0}\subset\Pi$ such that $\langle\mu^{pp}_{t,x}(\hat{\xi}),g(\hat{\xi})\rangle=0$ for all $(t,x)\in\Pi_{0}$ and $p\in D$. By the definition of measures $\mu^{pp+}_{t,x}$, $\mu^{pp-}_{t,x}$, they are, respectively, right- and left-continuous in $M(S)$ with respect to $p\in\mathbb{R}$, for some set $\Pi^{\prime}\subset\Pi_{0}$ of full measure of values $(t,x)$. Therefore, $\langle\mu^{pp\pm}_{t,x}(\hat{\xi}),g(\hat{\xi})\rangle=0$, that is, $\mu^{pp\pm}_{t,x}(S\setminus S_{0})=0$ for all $p\in\mathbb{R}$, $(t,x)\in\Pi^{\prime}$. To complete the proof, it only remains to apply relation (4.31). ∎

5 The proof of the decay property

We assume that $u(t,x)$ is an $x$-periodic e.s. of (1.1), (1.3). As was demonstrated in section 3, we may suppose that conditions (R1)–(R4) hold. In view of (1.10) for every $k\in\mathbb{N}$ the function $u_{k}=u(k^{2}t,k^{2}\tilde{x}+k\bar{x})$ is an e.s. of the equation

[TABLE]

where $\varphi^{k}(u)$ is the vector with components $\varphi^{k}_{i}(u)=\varphi_{i}(u)$, $i=1,\ldots,d$, $\varphi^{k}_{i}(u)=k\varphi_{i}(u)$, $i=d+1,\ldots,n$, while the symmetric matrix $a^{k}(u)$ has the entries $(a^{k}(u))_{ij}=\epsilon_{ki}\epsilon_{kj}a_{ij}(u)$, $1\leq i,j\leq n$, where $\epsilon_{ki}=k^{-1}$ for $1\leq i\leq d$, $\epsilon_{ki}=1$ for $d<i\leq n$.

If $a(u)=b(u)^{\top}b(u)$ is an admissible representation then $a^{k}(u)=b_{k}(u)^{\top}b_{k}(u)$, $(b_{k})_{rj}=\epsilon_{kj}b_{rj}$, is an admissible representation for the matrix $a^{k}(u)$. Let $\alpha,\beta\in\mathbb{R}$, $\alpha<\beta$, $s_{\alpha,\beta}(u)=\max(\alpha,\min(\beta,u))$ be the corresponding cut-off function, $A^{k}(u)$ be a primitive of the matrix $a^{k}(u)$. By Definition 1.1(i,ii) and Remark 1.1

[TABLE]

where the matrix $B_{k}$ is a primitive of $b_{k}$, $(B_{k})^{\prime}(u)=b_{k}(u)$, and $(B_{k})_{r}$ is the vector with components $(B_{k})_{rj}(u)$, $j=1,\ldots,n$. By $\chi_{\alpha,\beta}(u)$ we denote the indicator function of the interval $(\alpha,\beta)$. By Lemma 2.2 $\operatorname{div}_{x}(B_{k})_{r}(u_{k})\in L^{2}_{loc}(\Pi)$, and

[TABLE]

where $\delta>0$ is a common Lebesgue point of the functions $\displaystyle t\to\int_{\mathbb{T}^{n}}|u_{k}(t,x)|^{2}dx$, $k\in\mathbb{N}$. Since the maps $x\to y=k^{2}\tilde{x}+k\bar{x}$ keep the Lebesgue measure on the torus $\mathbb{T}^{n}$, then for each $t>0$

[TABLE]

and it follows from (5) that for almost each $\delta>0$

[TABLE]

which implies that for all $r=1,\ldots,n$

[TABLE]

as $k\to\infty$. It follows from (5) that for $1\leq i\leq n$ the distributions

[TABLE]

This implies that

[TABLE]

as $k\to\infty$ in the Sobolev space $H^{-1}_{loc}(\Pi)$.

Observe that for every $g(u)\in C(\mathbb{R})$

[TABLE]

where $\operatorname{sign}^{+}u=(\max(u,0))^{\prime}$ is the Heaviside function. Using this relation and Lemma 2.1, we obtain

[TABLE]

in $\mathcal{D}^{\prime}(\Pi)$, where $\mu^{k}_{r}$, $r\in\mathbb{R}$ are nonnegative measures defined by (2.1) with $\varphi(u)=\varphi^{k}(u)$, $A(u)=A^{k}(u)$, $u=u_{k}$, and $\eta(u)=(u-r)^{+}\doteq\max(u-r,0)$. By Corollary 2.1(iii)

[TABLE]

and the sequence $\mu^{k}_{\beta}-\mu^{k}_{\alpha}$ is bounded in the space of measures $M_{loc}(\Pi)$ equipped with standard locally convex topology generated by semi-norms $p_{K}(\mu)=|\mu|(K)$, where $K$ is an arbitrary compact subset of $\Pi$. By Murat’s interpolation lemma [11], it follows from (5.6) and (5.5) that the sequence of distributions

[TABLE]

is pre-compact in $H^{-1}_{loc}(\Pi)$.

By requirement (ii) of Definition 1.1

[TABLE]

as $k\to\infty$ in $L^{2}_{loc}(\Pi)$ and therefore in $H^{-1}_{loc}(\Pi)$ as well. Since

[TABLE]

we claim that the sequences of distributions

[TABLE]

in $H^{-1}_{loc}(\Pi)$ for all $r=1,\ldots,n$. In particular, these sequences are pre-compact in $H^{-1}_{loc}(\Pi)$.

Taking into account Remark 4.2 and Corollary 4.1, we conclude that the sequence $u_{k}$ converges as $k\to\infty$ to a constant measure valued function $\nu_{t,x}\equiv\nu$ (in the sense of relation (4.1) ). Let $[a_{0},b_{0}]$ be the smallest segment containing $\operatorname{supp}\nu$. We suppose that $a_{0}<b_{0}$ and are going to get a contradiction.

Notice that by Corollary 2.1(ii) for a.e. $t>0$ $\displaystyle\int_{\mathbb{T}^{n}}u(t,x)dx=\int_{\mathbb{T}^{n}}u_{0}(x)dx=I.$ By Lemma 4.1 we find that $u_{k}\rightharpoonup I$ as $k\to\infty$ weakly-$*$ in $L^{\infty}(\Pi)$. In view of relation (4.1) $I=\int\lambda d\nu(\lambda)\in(a_{0},b_{0})$. By our assumption (R3) the functions $\varphi_{i}(u)$, $i=d+1,\ldots,n$, are constant on some interval $(a_{1},b_{1})\ni I$. Without loss of generality we can assume that $(a_{1},b_{1})\subset(a_{0},b_{0})$.

Lemma 5.1**.**

There exists an interval $(a_{2},b_{2})\subset(a_{1},b_{1})$ such that

[TABLE]

Proof.

Denote $I_{1}=\int s_{a_{1},b_{1}}(\lambda)d\nu(\lambda)$. If $I_{1}=I$, there is nothing to prove, we set $(a_{2},b_{2})=(a_{1},b_{1})$. If $I_{1}<I$, then the continuous function $f(a)=\int s_{a,b_{1}}(\lambda)d\nu(\lambda)$ takes values $f(a_{1})=I_{1}<I$, $f(I)>I$. Therefore, there exists $a_{2}\in(a_{1},I)$ such that $f(a_{2})=I$. Taking $b_{2}=b_{1}$, we conclude that (5.9) holds. In the case $I_{1}>I$ we consider the continuous function $g(b)=\int s_{a_{1},b}(\lambda)d\nu(\lambda)$ and observe that $g(I)<I$, $g(b_{1})=I_{1}>I$. Therefore, there is a value $b_{2}\in(I,b_{1})$ such that $g(b_{2})=I$ and (5.9) follows with $a_{2}=a_{1}$. ∎

Let $(a_{2},b_{2})$ be an interval indicated in Lemma 5.1. We consider the sequence $v_{k}=s_{a_{2},b_{2}}(u_{k})$. In correspondence with (4.1)

[TABLE]

Observe that in the case when $(\alpha,\beta)\cap(a_{2},b_{2})=(\alpha_{1},\beta_{1})\not=\emptyset$

[TABLE]

Otherwise, $s_{\alpha,\beta}(v_{k})\equiv c=\mathrm{const}$. Since the flux functions $\varphi_{i}(u)$, $i=d+1,\ldots,n$, are constant on the segment $[a_{2},b_{2}]$, we conclude that $\varphi_{i}(s_{\alpha,\beta}(v_{k}))_{x_{i}}=0$ and in view of (5.7) for all $\alpha,\beta\in\mathbb{R}$, $\alpha<\beta$ the sequences of distributions

[TABLE]

are pre-compact in $H^{-1}_{loc}(\Pi)$. Similarly, relation (5.8) implies that the sequences

[TABLE]

are pre-compact in $H^{-1}_{loc}(\Pi)$ as well.

As follows from relation (4.1), the limit measure valued function for the sequence $v_{k}$ is a constant measure valued function $\nu_{t,x}\equiv\nu_{1}=s_{a_{2},b_{2}}^{*}\nu$ is push-forward measure of $\nu$ under the map $s_{a_{2},b_{2}}(\lambda)$. Since $[a_{2},b_{2}]\subset[a_{0},b_{0}]$, $[a_{2},b_{2}]$ is the minimal segment containing $\operatorname{supp}\nu_{1}$. Passing to a subsequence if necessary, we can consider H-measures $\mu^{pq\pm}_{t,x}$ corresponding to the sequence $v_{k}$. Let $p\in(a_{2},b_{2})$, $H_{+}(t,x)$, $H_{-}(t,x)$ be the linear spans of supports $\operatorname{supp}\mu^{pp+}_{x}$ and $\operatorname{supp}\mu^{pp-}_{x}$, respectively. As follows from Proposition 4.4 and pre-compactness of sequences (5.10), (5.11), for a.e. $(t,x)\in\Pi$ for all $(\tau,\xi)\in H_{+}(t,x)\cap H_{-}(t,x)$ the functions

[TABLE]

in some vicinity of $p$. For fixed $(t,x)$ we denote $H_{\pm}=H_{\pm}(t,x)$ and introduce the sets $S_{\pm}=H_{\pm}\cap S_{0}$. Since $S\setminus S_{\pm}\subset(S\setminus H_{\pm})\cap(S\setminus S_{0})$ then

[TABLE]

where we use Proposition 4.5. We remark that by Corollary 2.2 the family $u(t,\cdot)$, $t>0$ is pre-compact in $L^{2}(\mathbb{T}^{n})$, and the conditions of Proposition 4.5 are actually satisfied.

Applying Lemma 4.2, we obtain that for a.e. $(t,x)\in\Pi$ the set $H_{+}(t,x)\cap H_{-}(t,x)\cap S_{0}$ is not empty and for all $(\tau,\xi)\in H_{+}(t,x)\cap H_{-}(t,x)\cap S_{0}$ identities (5.12), (5.13) hold. In particular, we can take $p=I\in(a_{2},b_{2})$. Then it follows from (5.12) and assumption (R2) that $\tilde{\xi}=0$, $\tau=0$. Further, in view of (5.13) $b(u)\bar{\xi}=\frac{d}{du}B(u)\bar{\xi}=0$ in some vicinity of $I$. This implies that $a(u)\bar{\xi}=b(u)^{\perp}b(u)\bar{\xi}=0$ a.e. in this vicinity. By assumption (R4), we claim that $\bar{\xi}=0$. Hence, $\hat{\xi}=(\tau,\xi)=0$, which contradicts to the condition $\hat{\xi}\in S$. We conclude that $a_{0}=b_{0}=I$, that is, $\nu=\delta(\lambda-I)$. This means that the limit measure valued function $\nu_{t,x}(\lambda)\equiv\nu(\lambda)=\delta(\lambda-I)$ of the sequence $u_{k}$ is regular, and by Theorem 4.1, $u_{k}\to I$ as $k\to\infty$ in $L^{1}_{loc}(\Pi)$. This implies that for a.e. $t>0$ $u_{k}(t,\cdot)\to I$ as $k\to\infty$ in $L^{1}(\mathbb{T}^{n})$. We fix such $t=t_{0}$. Then

[TABLE]

For $t>k^{2}t_{0}$ we use Lemma 2.3 for e.s. $u_{1}=u$, $u_{2}\equiv I$ and find that for almost each such $t$

[TABLE]

which, together with (5), implies the desired decay relation (1.11).

Remark 5.1**.**

Using methods of paper [19], we can extend our results to the case of almost periodic (in the Besicovitch sense) initial function $u_{0}(x)$. Let

[TABLE]

[TABLE]

be the mean $L^{1}$-norm of a function $u(x)\in L^{1}_{loc}(\mathbb{R}^{n})$. Recall, ( see [2] ) that Besicovitch space ${\mathcal{B}}^{1}(\mathbb{R}^{n})$ is the closure of trigonometric polynomials, i.e., finite sums $\sum a_{\lambda}e^{2\pi i\lambda\cdot x}$, with ${i^{2}=-1}$, $\lambda\in\mathbb{R}^{n}$, in the quotient space $B^{1}(\mathbb{R}^{n})/B^{1}_{0}(\mathbb{R}^{n})$, where

[TABLE]

The space ${\mathcal{B}}^{1}(\mathbb{R}^{n})$ is a Banach space equipped with the norm $\|u\|_{1}=N_{1}(u)$.

It is known [2] that for each $u\in{\mathcal{B}}^{1}(\mathbb{R}^{n})$ there exist the mean value

[TABLE]

and, more generally, the Bohr-Fourier coefficients

[TABLE]

The set

[TABLE]

is at most countable and is called the spectrum of an almost periodic function $u$. We denote by $M(u)$ the smallest additive subgroup of $\mathbb{R}^{n}$ containing $Sp(u)$.

Suppose that $u_{0}\in{\mathcal{B}}^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$, \displaystyle I=\mathchoice{{\vbox{\hbox{\textstyle- }}\kern-7.44173pt}}{{\vbox{\hbox{\scriptstyle- }}\kern-5.60507pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.51254pt}}{{\vbox{\hbox{\scriptscriptstyle- }}\kern-4.03754pt}}\!\int_{\mathbb{R}^{n}}u_{0}(x)dx, $M_{0}=M(u_{0})$, and that $u=u(t,x)$ is an e.s. of (1.1), (1.3). Arguing as in [19], we may conclude that $u(t,\cdot)\in{\mathcal{B}}^{1}(\mathbb{R}^{n})$ and that $M(u(t,\cdot))\subset M_{0}$ for a.e. $t>0$. The decay property is modified as follows.

Theorem 5.1**.**

Assume that for all $\xi\in M_{0}$, $\xi\not=0$ there is no vicinity of $I$, where simultaneously the function $\xi\cdot\varphi(u)$ is affine and the function $a(u)\xi\cdot\xi\equiv 0$. Then

[TABLE]

6 Acknowledgements

This work was supported by the Ministry of Education and Science of the Russian Federation (project no. 1.445.2016/1.4) and by the Russian Foundation for Basic Research (grant 18-01-00258-a.)