Normalized solutions for Choquard equations with general nonlinearities

1.   Introduction

This paper deals with periodic measures of the following reaction-diffusion lattice systems driven by superlinear noise defined on the integer set $\mathbb Z^k$ :

along with initial conditions:

where $i = (i_1, i_2, \ldots, i_k)\in \mathbb Z^k$, $\lambda(t), \nu(t)$ are continuous functions, $\lambda(t) > 0$, $(f_i)_{i\in\mathbb Z^k}$ and $(\hat{\sigma}_{i, j})_{i\in\mathbb Z^k, j\in\mathbb N}$ are two sequences of continuously differentiable nonlinearities with arbitrary and superlinear growth rate from $\mathbb R \times \mathbb R\to \mathbb R$, respectively, $g = (g_i)_{i\in\mathbb Z^k}$ and $h = (h_{i, j})_{i\in\mathbb Z^k, j\in\mathbb N}$ are two time-dependent random sequences, and $\delta = (\delta_{i, j})_{i\in\mathbb Z^k, j\in\mathbb N}$ is a sequence of real numbers. The sequence of independent two-sided real-valued Wiener processes $(W_j)_{j\in\mathbb N}$ is defined on a complete filtered probability space $(\Omega, \mathcal F, \{\mathcal F_t\}_{t\in\mathbb R}, \mathbb P)$. Furthermore, we assume that system (1.1) is a time periodic system; more precisely, there exists $T > 0$ such that the time-dependent functions $\lambda, \nu, f_i, g, h, \sigma_{i, j}(i\in\mathbb Z^k, j\in\mathbb N)$ in (1.1) are all $T$-periodic in time.

Lattice systems are gradually becoming a large and evolving interdisciplinary research field, due to wide range of applications in physics, biology and engineering such as pattern recognition, propagation of nerve pulses, electric circuits, and so on, see ^{[1,2,3,4,5,6]} and the references therein for more details. The well-posedness and the dynamics of these equations have been studied by many authors, ^[7,8,9,10] for deterministic systems and ^{[11,12,13,14,15,16,17,18,19]} for stochastic systems where the existence of random attractors and probability measures have been examined. Especially, the authors research the limiting behavior of periodic measures of lattice systems in ^[15].

Nonlinear noise was proposed and studied for the first time in ^[19], the authors researches the long-term behavior of lattice systems driven by nonlinear noise in terms of random attractors and invariant measures. Before that, the research on noise was limited to additive noise and linear multiplicative noise, which can be transformed into a deterministic system. However, if the diffusion coefficients are nonlinear, then one cannot convert the stochastic system into a pathwise deterministic one, and thereby this problem cannot be studied under the frameworks of deterministic systems aforementioned. As an extension of ^[19], a class of reaction-diffusion lattice systems driven by superlinear noise, where the noise has a superlinear growth order $q\in [2, p)$, is studied by taking advantage of the dissipativeness of the nonlinear drift function $f_i$ in (1.1) to control the superlinear noise in ^[20].

In the paper, we will study the existence of periodic measures of reaction-diffusion lattice systems drive by superlinear noise. One of the main tasks in our analysis is to solve the superlinear noise terms. We remark that if the noise grows linearly, then the estimates we need can be obtained by applying the standard methods available in the literature. We adopt the ideas that take advantage of the nonlinear drift terms' the polynomical growth rate $p\ (p\ge2)$ to control the noise polynomical rate $q\in[2, p)$. Furthermore, notice that $l^2$ is an infinite-dimensional phase space and problem (1.1)–(1.2) is defined on the unbounded set $\mathbb Z^k$. The unboundedness of $\mathbb{Z}^k$ as well as the infinite-dimensionalness of $l^2$ introduce a major difficulty, because of the non-compactness of usual Sobolev embeddings on unbounded domains. We will employ the dissipativeness of the drift function in (1.1) as well as a cutoff technique to prove that the tails of solutions are uniformly small in $L^2(\Omega, l^2)$. Based upon this fact we obtain the tightness of distribution laws of solutions, and then the existence of periodic measures.

In the next section, we discuss the well-poseness of solutions of (1.1) and (1.2). Section 3 is devoted to the uniform estimates of solutions including the uniform estimates on the tails of solutions. In Section 4, we show the existence of periodic measures of (1.1) and (1.2).

2.   Global well-posedness of reaction-diffusion lattice systems with superlinear noise

In this section, we prove the existence and uniqueness of solutions to system (1.1) and (1.2). We first discuss the assumptions on the nonlinear drift and diffusion terms in (1.1).

We begin with the following Banach space:

The norm and inner product of $l^2$ are denoted by $(\cdot, \cdot)$ and $\|\cdot\|$, respectively. For the nonlinear drift function $f_i\in C^1(\mathbb R\times\mathbb R, \mathbb R)$ in the equation we assume that for all $s\in\mathbb R$ and $i\in\mathbb Z^k$,

where $p > 2$ and $\gamma_1 > 0$ are constants. For the sequence of continuously differentiable diffusion functions $\hat{\sigma} = (\hat{\sigma}_{i, j})_{i\in\mathbb Z^k, j\in\mathbb N}$, we assume, for all $s\in\mathbb R$ and $j\in\mathbb N$,

where $q\in[2, p)$ is a constant. For processes $g(t) = (g_i(t))_{i\in\mathbb Z^k}$ and $h(t) = (h_{i, j})_{i\in\mathbb Z^k, j\in\mathbb N}$ are both continuous in $t\in\mathbb R$, which implies that for all $t\in\mathbb R$,

In addition, we assume $\delta = (\delta_{i, j})_{i\in\mathbb Z^k, j\in\mathbb N}$ satisfies

We will investigate the periodic measures of system (1.1)–(1.2) for which we assume that all given time-dependent functions are T-periodic in $t\in\mathbb R$ for some $T > 0$; that is, for all $t\in\mathbb R, i\in\mathbb Z^k$ and $k\in\mathbb N$.

If $m:\mathbb R\to \mathbb R$ is a continuous T-periodic function, we denote

We want to reformulate problem (1.1)–(1.2) as an abstract one in $l^2$. Given $1\le j\le k, u = (u_i)_{i\in\mathbb Z^k}\in l^2$ and $i = (i_1, i_2, \ldots, i_k)\in\mathbb Z^k$. Let us define the operators from $l^2$ to $l^2$ by

and

For all $1\le j\le k, u = (u_i)_{i\in\mathbb Z^k}\in l^2$ and $v = (v_i)_{i\in\mathbb Z^k}\in l^2$ we see

Again, define the operators $f, \sigma_j:\mathbb R\times l^2\to l^2$ by

It follows from (2.3) that there exists $\theta\in (0, 1)$ such that for $p > 2$ and $u, v\in l^2$,

This together with $f(t, 0)\in l^2$ by (2.2) yields $f(t, u)\in l^2$ for all $u\in l^2$, and thereby $f:\mathbb R\times l^2\to l^2$ is well-defined. In addition, we deduce from (2.9) that $f:\mathbb R\times l^2\to l^2$ is a locally Lipschitz continuous function, that is, for every $n\in\mathbb N$, we can find a constant $c_1(n) > 0$ satisfying, for all $u, v\in l^2$ with $\|u\|\le n$ and $\|v\|\le n$,

For $q\in[2, p)$ and $u\in l^2$, one can deduce from(2.4), (2.7) and Young's inequality that for all $\varpi > 0,$

where $\gamma_1$ is the same number as in (2.1). From (2.11) and $l^2\subseteq l^p$ for $p > 2$, we find that $\sigma_j(t, u)\in l^2$ for all $u\in l^2$. Then $\sigma_j:\mathbb R\times l^2\to l^2$ is also well-defined. In addition, it yields from (2.5) and (2.7) that there exists $\eta\in (0, 1)$ such that for $q\in [2, p)$ and $u, v\in l^2$,

This implies that $\sigma_j:\mathbb R\times l^2\to l^2$ is also locally Lipschitz continuous, more precisely, for every $n\in\mathbb N$, one can find a constant $c_2(n) > 0$ satisfying, for all $u, v\in l^2$ with $\|u\|\le n$ and $\|v\|\le n,$

and

By above notations one is able to rewrite (1.1)–(1.2) as the following system in $l^2$ for $t > 0$ :

with initial condition:

in the present article, the solutions of system (2.15)–(2.16) are interpreted in the following sense.

Definition 2.1. Suppose $u_0\in L^2(\Omega, l^2)$ is $\mathcal F_0$-measurable, a continuous $l^2$-valued $\mathcal F_t$-adapted stochastic process $u$ is called a solution of equations (2.15) and (2.16) if $u\in L^2(\Omega, C([0, T], l^2))\cap L^p(\Omega, L^p(0, T;l^p))$ for all $T > 0$, and the following equation holds for all $t\ge0$ and almost all $\omega\in\Omega$:

Similar to Ref.^[20], we can get (2.15) and (2.16) exist global solutions in the sense of Definition 2.1.

3.   Uniform estimates

In this section, we derive the uniform estimates of solutions of (2.15)–(2.16). These estimates will be used to establish the tightness of a set of probability distributions of $u$ in $l^2$.

We assume that

Lemma 3.1. Let (2.1)–(2.7) and (3.1) hold. Then the solutions $u(t, 0, u_0)$ of system (2.15) and (2.16) with initial data $u_0$ at time $0$ satisfy, for all $t\ge0$,

where $L_1 > 0$ is a positive constant which depends on $\underline{\alpha}, p, q, \gamma, c_\delta, t,$ but indepentent of $u_0$.

Proof. Applying Ito's formula to (2.15) we get

This implies

For the second term on the left-hand side of (3.3), we have

For the first term on the right-hand side of (3.3), we get from (2.1) that

For the second term on the right-hand side of (3.3), we have

For the last term on the right-hand side of (3.3), we infer from (2.11) with $\omega = 2$ that

By (3.3)–(3.7) we get

implies that

where $C_1 = \frac{p-q}{p}\Big(\frac{p\gamma_1}{2q}\Big)^{-\frac{q}{p-q}} (4c_\delta)^{\frac{p}{p-q}}\|\varphi_1\|^{\frac{2p}{p-q}} _{\frac{2p}{p-q}}+4c_\delta\|\varphi_2\|^2+2\|\phi_1\|_1$. Multiplying (3.9) by $e^{\underline{\alpha}t}$ and integrating over $(0, t)$ to obtain

where $C_2 = 2\sum_{j = 1}^{\infty}\|\overline{h_j}\|^2+\frac{1}{\underline{\lambda}}\|\overline{g}\|^2+C_1$. This completes the proof.

Lemma 3.2. Let (2.1)–(2.7), and (3.1) be satisfied. Then for compact subset $\mathcal K$ of $l^2$, one can find a number $N_0 = N_0(\mathcal K)\in\mathbb N$ such that the solutions $u(t, 0, u_0)$ of (2.15) and (2.16) satisfy, for all $n\ge N_0$ and $t\ge 0$,

where $u_0\in\mathcal K$ and $\|i\|: = \max_{i\le j\le k}|i_j|$.

Proof. Define a smooth function $\xi:\mathbb R\to[0, 1]$ such that

Denote by

Similar notations will also be used for other terms. It follows from (2.15) that

By Ito's formula and (3.14) we have

This yields

For the second term on the left-hand side of (3.16), we have

We first deal with the first term on the right-hand side of (3.17). Notice that

By the definition of function $\xi$, there exists a constant $C_3 > 0$ such that $|\xi^{'}(s)|\le C_3$ for all $s\in\mathbb R$. Then the first term on the right-hand side of (3.18) is bounded by

By the definition of $|B_ju|_i$, the last term on the right-hand side of (3.18) is bounded by

Then we find from (3.18) to (3.20) that the first term on the right-hand side of (3.17) is bounded by

In addition, we find that the last term on the right-hand side of (3.17) can be bounded by

By (3.21), (3.22) and (3.17), we infer that the second term on the left-hand side of (3.16) satisfied

where $C_4 = 4kC_3(2+C_3)$. For the first term on the right-hand side of (3.16), we find from (2.1) that

For the second term on the right-hand side of (3.16), we infer from Young's inequality that

For the last term on the right-hand side (3.16), we infer from (2.4) and Young's inequality that

Substituting (3.23)–(3.26) into (3.16) we get

where $C_5 = 2+\frac{1}{\underline{\lambda}}+\frac{p-q}{p}(\frac{p\gamma_1}{q})^{-\frac{q}{p-q}}(4c_\delta)^{\frac{p}{p-q}}+4c_\delta$. One can multiply (3.27) by $e^{\underline{\alpha}t}$ and integrate over $(0, t)$ in order to obtain

Since $\mathcal K$ is a compact subset of $l^2$ we infer from (3.1) that

By Lemma 3.1, we find that for all $u_0\in\mathcal K$ and $t\ge 0$, as $n\to\infty$,

where $L_1$ is the same number of (3.1) and $C_{6} > 0$ is a constant depending only on $u_0$.By $\varphi_1\in l^{\frac{2p}{p-q}}, \varphi_2\in l^2, \phi_1\in l^1$, (2.6) and (3.1), we infer that

It follows from (3.28) to (3.31) that as $n\to\infty$,

Then for every $\varepsilon > 0$ we can find a number $N_0 = N_0(\mathcal K)\in\mathbb N$ satisfying, for all $n\ge N_0$ and $t\ge 0$,

uniformly for $u_0\in\mathcal K$ and $t\ge 0$. This concludes the proof.

4.   Existence of periodic measures

In the sequel, we use $\mathcal L(u(t, 0, u_0))$ to denote the probability distribution of the solution $u(t, 0, u_0)$ of (2.15)–(2.16) which has initial condition $u_0$ at initial time $0$. Then we have the following tightness of a family of distributions of solutions.

Lemma 4.1. Suppose (2.1)–(2.7) and (3.1) hold. Then the family $\{\mathcal L(u(t, 0, u_0)):t\ge 0\}$ of the distributions ofthe solutions of (2.15)–(2.16) is tight on $l^2$.

Proof. For simplicity, we will write the solution $u (t, 0, u_0)$ as $u(t)$ from now on. It follows from Lemma 3.1 that there exists a constant $c_1 > 0$ such that

By Chebyshev's inequality, we get from (4.1) that for all $t\geq0$,

Hence for every $\epsilon > 0$, there exists $R_1 = R_1(\epsilon) > 0$ such that for all $t\geq0$,

By Lemma 3.2, we infer that for each $\epsilon > 0$ and $m\in \mathbb{N}$, there exists an integer $n_m = n_m(\epsilon, m)$ such that for all $t\geq 0$,

and hence for all $t\geq 0$ and $m\in \mathbb{N}$,

It follows from (4.3) for all $t\geq 0$,

which shows that for all $t\geq 0$,

Given $\epsilon > 0$, set

and

By (4.2) and (4.4) we get, for all $t\geq0$,

Now, we show the precompactness of $\left\{ {v:v \in Y_\epsilon } \right\}$ in $l^2$. Given $\kappa > 0$, choose an integer $m_0 = m_0 \left(\kappa \right) \in \mathbb{N}$ such that $2^{m_0 } > \frac{8}{{\kappa ^2 }}$. Then by (4.6) we obtain

On the other hand, by (4.5) we see that the set $\left\{ {(v_i)_{|i|\leq m_0}:v \in Y_\epsilon } \right\}$ is bounded in the finite-dimensional space $R^{2m_0+1}$ and hence precompact. Consequently, $\left\{ {v:v \in Y_\epsilon } \right\}$ has a finite open cover of balls with radius $\frac{\kappa }{2}$, which along with (4.9) implies that the set $\left\{ {v:v \in Y_\epsilon } \right\}$ has a finite open cover of balls with radius $\kappa$ in $l^2$. Since $\kappa > 0$ is arbitrary, we find that the set $\left\{ {v: v\in Y_\epsilon } \right\}$ is precompact in $l^2$. This completes the proof.

If $\phi:l^2\to\mathbb R$ is a bounded Borel function, then for $0\le r\le t$ and $u_0\in l^2$, we set

and

where $\Gamma\in\mathcal B(l^2)$ and $1_\Gamma$ is the characteristic function of $\Gamma$. The operators $p_{s, t}$ with $0\le s\le t$ are called the transition operators for the solutions of (2.15)–(2.16). Recall that a probability measure $\nu$ on $l^2$ is periodic for (2.15)–(2.16) if

Lemma 4.2. ^[21]Let $\varrho(\psi, \omega)$ be a scalar bounded measurable randomfunction of $\psi$, independent of $\mathcal F_s$. Let $\varsigma$ be an$\mathcal F_s$-measurable random variable. Then

The transition operators $\{p_{r, t}\}_{0\le r\le t}$ have the following properties.

Lemma 4.3. Assume that (2.1)–(2.7) and (3.1) hold. Then:

(i) $\{p_{r, t}\}_{0\le r\le t}$ is Feller; that is, for every bounded andcontinuous $\phi: l^2\to\mathbb R$, the function $p_{r, t}\phi: l^2\to\mathbb R$is also bounded and continuous for all $0\le r\le t.$

(ii) The family $\{p_{r, t}\}_{0\le r\le t}$ is T-periodic; that is, for all $0\le r\le t$,

(iii) $\{u(t, 0, u_0)\}_{t\ge 0}$ is a $l^2$-valued Markov process.

Finally, we present our main result on the existence of periodic measures for problem (2.15)–(2.16).

Theorem 4.4. Assume that (2.1)–(2.7) and (3.1) hold. Then problem (2.15)–(2.16) has a periodic measure on $l^2$.

Proof. We apply Krylov-Bogolyubov's method to prove the existence of periodic measures of (2.15)–(2.16), define a probability measure $\mu_n$ by

By Lemma 4.1 we see the sequence $\{\mu_n\}^\infty_{n = 1}$ is tight on $l^2$, and hence there exists a probability measure $\mu$ on $l^2$ such that, up to a subsequence,

By (4.10)–(4.11) and Lemma 4.3, we infer that for every $t\ge0$ and every bounded and continuous function $\phi:l^2\to\mathbb R,$

which shows that $\mu$ is a periodic measure of (2.15)–(2.16), as desired.

Conflict of interest

The author declares there is no conflict of interest.