Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping

1.   Introduction

In this paper, we consider the following initial boundary value problem

where $s, \alpha\in (0, 1)$, $1<p<N/s$, $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $0<T<\infty$ is a given constant and $Q_T = \Omega\times(0, T)$, $M:\mathbb{R}_0^+\rightarrow\mathbb{R}^+$ is a continuous function and $\Phi$ is a discontinuous and nonlinear set valued mapping by filling in jumps of a function $a(x, t, s): Q_T\times\mathbb{R}\rightarrow\mathbb{R}$, $[u]_{s, p}$ is the Gagliardo seminorm defined by

Here $(-\Delta )^s_{p}$ is the fractional $p$-Laplace operator which, up to a normalization constant, is defined as

along functions $\varphi\in C_0^\infty(\mathbb{R}^N)$. Henceforward $B_\varepsilon(x)$ denotes the ball of $\mathbb{R}^N$ centered at $x\in\mathbb{R}^N$ and radius $\varepsilon>0$. In particular, if $p = 2$ the fractional $p$-Laplacian $(-\Delta)_p^s$ reduces to the fractional Laplacian $(-\Delta)^s$.

Furthermore, we assume $a$ and $\Phi$ satisfy the following assumptions.

(H1) For each $\xi\in\mathbb{R}$, $a$ is a continuous function with respect to $(x, t)\in Q_T$ and for each $(x, t)\in Q_T$, $a\in L_{{\rm loc}}^\infty(\mathbb{R})$. Moreover, there exist positive constants $a_0, a_1, a_2$ such that

where $q\in(1, p_s^*)$ and $p_s^* = Np/(N-sp)$.

(H2) The set valued function $\Phi: Q_T\times\mathbb{R}\rightarrow2^{\mathbb{R}}$ is obtained by filling in jumps of the function $a$ in (H1) by means of the functions $\underline{a}_\varepsilon, \overline{a}_\varepsilon, \underline{a}, \overline{a}:\mathbb{R}\rightarrow\mathbb{R}$ as follows

(H3) $u_0\in W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$, $u_1\in L^2(\Omega)$, $f\in L^{q^\prime}(Q_T)$, where $q^\prime = \frac{q}{q-1}$.

Throughout the paper, without explicit mention, we assume that $M:[0, \infty)\rightarrow \mathbb{R}^+$ is continuous and verifies

$(M_1)$ there exists $m_0>0$ such that $M(\tau)\geq m_0$ for all $\tau\geq 0$.

A typical example of $M$ is given by $M(t) = m_0+m_1\, t$ for $t\geq0$, where $m_0>0, m_1\geq0$. When $M$ is of this type, problem (1) is said to be degenerate if $a = 0$, while it is called non–degenerate if $a>0$. As for some recent existence results on Kirchhoff-type problems, we refer the interested readers to [5,11]. Recently, the fractional Kirchhoff problems have received more and more attention. Some new existence results for fractional Kirchhoff problems were given, for example, in [18,19,20,24,25,26,27,34].

In recent years, fractional Laplacian operator and related equations have an increasingly wide utilization in many important fields, as explained by Caffarelli in [4], Laskin in [15] and Vázquez in [36]. In [8], a stationary Kirchhoff variational equation was first proposed by Fiscella and Valdinoci as a model to study the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. Indeed, the stationary problem of (1) is a fractional version of a model, the so-called stationary Kirchhoff equation, which was introduced by Kirchhoff in [12] as a model to study elastic string vibrations. The literature on elliptic type problems involving fractional Laplacian and its variant is rich and very vast, see for example [8,25,26,27,35] and the references cited there.

Recently, the fractional hyperbolic problems with continuous nonlinearities have been studied by many researchers. For example, Pan, Pucci and Zhang [29] studied the initial-boundary value problem of degenerate Kirchhoff-type

where $\Omega\subset\mathbb{R}^N$ is a bounded domian with Lipshcitz boundary, $\theta\in[1, 2_s^*/2)$, $p\in(2\theta-1, 2_s^*-1)$ and $[u]_{s}$ is the Gagliardo seminorm of $u$ defined by

Under some appropriate conditions, the authors obtained the global existence, vacuum isolating and blow up of solutions for (2) by using the Galerkin method combined with the potential wells theory. Moreover, the authors also investigated the global existence of solutions under the critical initial conditions. Furthermore, Pan et al. in [28] considered the following degenerate Kirchhoff equation with nonlinear damping term

$2<a<2\theta<b<2_s^*$. Under some natural assumptions, the authors obtained the global existence, vacuum isolating, asymptotic behavior and blow up of solutions for (3) by combining the Galerkin method with potential wells theory. In [20], Lin et al. studied the initial-boundary value problem of Kirchhoff wave equation

The authors established some sufficient conditions on initial data such that the solutions blow up in finite time for arbitrary positive initial energy by using an modified concavity method. Moreover, when $f(u) = |u|^{p-2}u$, the authors obtained the upper and lower bounds for blow up time. Concerning the related diffusion problems, for instance, we refer to [30,31,33,39] for more results and methods.

It is worth mentioning that problem (1) can be regard as a fractional version of the initial-boundary value problem of the following equation

In [32], Park and Kim studied the existence of solutions for problem (4) without the Kirchhoff function $M$. The research on differential inclusions is an interesting topic in recent years. These problems arise mainly from physics and optimization, especially continuum mechanics, where non-monotone, multi-valued constitutive laws lead to a class of differential inclusions (variational inequalities). For a brief account of works on such variational inequalities we refer to [21] for the details. For the analysis of nonlinear second order or fourth order or six order hyperbolic partial differential equations with damping, we refer to the seminal work of Lions and Strauss [22], see also [9,16,17,40,41] for more recent results. In recent years, partial differential equations of hyperbolic type with variable exponent growth conditions were studied by Antontsev [1], see also Autuori and Pucci [2,3] for Kirchhoff systems with $p(x)$-growth.

However, to the best of our knowledge, there are no papers that deal with the global existence and blow-up results for problems like (1). Inspired by above papers, we study the existence of global solutions that vanish at infinity or solutions that blow up in finite time for problem (1) involving the fractional $p$-Laplacian and discontinuous nonlinearity. Since our problem is nonlocal and the diffusion coefficient $M([u]_{s, p}^{p})$ is a function, our discussion is more elaborate than the papers in the literature.

The rest of this paper is organized as follows. In section 2, we will give some necessary definitions and properties of fractional Sobolev spaces. In section 3, we obtain the existence of weak solutions for problem (1.1) by Galerkin's approximation method. In section 4, we give an example to illustrate our result.

2.   Preliminaries

To study the existence of solutions for equation (1), let us first recall some results related to the fractional Sobolev space $W_0^{s, p}(\Omega)$. For convenience, we shortly denote by $\|\cdot\|_2$ the norm of $L^2(\Omega)$. Let $\Omega\subset \mathbb{R}^N$ be a bounded domain, $1<p<\infty$ and set

where the Gagliardo seminorm $[u]_{s, p}$ is defined as

Equipped with the norm

$W_0^{s, p}(\Omega)$ is a uniformly convex Banach space, and hence reflexive. The fractional critical exponent is defined by

Moreover, the fractional Sobolev embedding states that $W_0^{s, p}(\Omega)\hookrightarrow L^{q}(\Omega)$ is continuous if $sp<N$ and $1\leq q<p_s^*$. For more detailed account on the properties of $W_0^{s, p}(\Omega)$, we refer to [7].

Definition 2.1. Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with smooth boundary. We define

with the norm

Remark 1. Following the standard proof of Sobolev spaces, we can prove that $W(Q_{T})$ is a reflexive Banach space if $1<p, q<\infty$.

In the following, we give a useful result which will be used to get the existence of solutions for problem (1).

Proposition 1. Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ and let $\{\omega_i\}_{i = 1}^\infty$ be an orthogonal basis in $L^2(\Omega)$. Then for any $\varepsilon>0$, there exists a constant $N_{\varepsilon}>0$ such that

for all $u\in W_0^{s, p}(\Omega)$ where $2N/(N+2s)\leq p<N/s$.

Proof. Following the idea of [13], we first show that for any $\delta>0$ and $\varepsilon>0$ there exists positive integer number $N_{\varepsilon, \delta}$ such that

Arguing by contradiction, we assume that there exists an $\epsilon_0>0$ and a sequence $\{u_n\}_{n = 1}^\infty\subset W_0^{s, p}(\Omega)$ such that

for any $n$. Let $v_n = u_n/\|u_n\|_2$. Then

which means that

Since $2N/(N+2s)\leq p<N/s$ and the embedding of $W_0^{s, p}(\Omega)$ into $L^2(\Omega)$ is compact, there exists a subsequence $\{v_{n_k}\}_{k = 1}^\infty$ such that $\{v_{n_k}\}_{k = 1}^\infty$ strongly converges to some function $v$ in $L^2(\Omega)$. Hence $\|v\|_2 = \|v_n\|_2 = 1$. Since $\{\omega_i\}_{i = 1}^\infty$ is an orthogonal basis of $L^2(\Omega)$, we also have

converges strongly to $v$. Indeed, we have

as $k\rightarrow\infty.$ Using (6) and letting $k\rightarrow\infty$, we deduce $1\geq 1+\delta$, which is absurd. Thus, (5) holds true.

By (5), we obtain

where $C>0$ is the embedding constant of $W_0^{s, p}(\Omega)$ into $L^2(\Omega)$. Therefore, the proof is complete.

3.   Existence of weak solutions

In this section, we prove the main result of this paper.

Definition 3.1. A pair of functions $u, \mu:Q_T\rightarrow\mathbb{R}$ is called a weak solution of (1.1), if

and

hold for all $\varphi\in C^1(0, T;C_0^\infty(\Omega))$. Here $\langle u, \varphi\rangle_{s, p}$ and $\langle u_t, \varphi\rangle_{\alpha, 2}$ are defined as

and

We need a regularization of $a$ defined by

where $\rho\in C_0^\infty(-1, 1), \ \rho\geq0$ and $\int_{-1}^1\rho(\tau)d\tau = 1$.

Lemma 3.2. The function $a_n$ is continuous and satisfies the following inequalities

and

for each $(x, t, \xi)\in Q_T\times\mathbb{R}$, where $C_0 = \frac{a_0}{2}+a_1+a_2+a_2(\frac{2a_2}{a_0})^{q-1}.$

Proof. Since for each $(x, t)\in Q_T$, $a\in L^\infty_{\rm loc}(\mathbb{R})$, it's easy to show that $a_n\in C(Q_T\times\mathbb{R})$ for each $n\in\mathbb{N}$. From assumption (H2) and Young's inequality, for each $(x, t, s)\in Q_T\times\mathbb{R}$, we have

where $\tau_0\in [-1, 1]$. From the inequality

we obtain

where $C_0 = \frac{a_0}{2}+a_1+a_2+a_2(\frac{2a_2}{a_0})^{q-1}.$ Similarly, by (H2), we get

Thus, the proof is finished.

We choose a sequence $\{\omega_j\}_{j = 1}^\infty\subset C_0^\infty(\Omega)$ such that $C_0^\infty(\Omega)\subset \overline{\bigcup_{n = 1}^\infty V_n}^{C^{1}(\bar{\Omega})}$ and $\{\omega_j\}_{j = 1}^\infty$ is a complete orthonormal basis in $L^2(\Omega)$, where $V_n = \mathrm{span}\{\omega_1, \omega_2,$ $\ldots, \omega_n\}$, see [10,24].

Since $C_0^\infty(\Omega)\subset \overline{\bigcup_{n = 1}^\infty V_n}^{C^{1}(\bar{\Omega})}$, we have the following lemma.

Lemma 3.3. For the function $u_0\in W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$, there exists a sequence $\{\psi_n\}$ with $\psi_n\in V_n$ such that $\psi_n\rightarrow u_0$ in $W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$ as $n\rightarrow\infty$.

Proof. For $u_0\in W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$, there exists a sequence $\{v_n\}$ in $C_0^{\infty}(\Omega)$ such that $v_n\rightarrow u_0$ in $W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$. Since $\{v_n\}_{n = 1}^\infty\subset C_0^{\infty}(\Omega)\subset \overline{\bigcup\limits_{m = 1}^\infty V_m}^{C^{1}(\overline{\Omega})}$, we can find a sequence $\{v_n^k\}\subset\bigcup\limits_{m = 1}^{\infty}V_m$ such that for each $n\in\mathbb{N}$, there holds $v_n^k\rightarrow u_n$ in $C^1(\overline{\Omega})$ as $k\rightarrow\infty$. For $\frac{1}{2^n}$, there exists $k_n\geq1$ such that $\|v_n^{k_n}-u_n\|_{C^1(\overline{\Omega})}\leq \frac{1}{2^n}.$ Thus

That is $v_n^{k_n}\rightarrow u_0$ in $W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$ as $n\rightarrow\infty$. Denote $u_n = v_n^{k_n}$. Since $u_n\in \bigcup\limits _{m = 1}^{\infty}V_{m}$, there exists $V_{m_n}$such that $u_n\in V_{m_n}$, without lost of generality, we assume that $V_{m_1}\subset V_{m_2}$ as $m_1\leq m_2$. We suppose that $m_1>1$ and define $\psi_n$ as follows: $\psi_{n}(x) = 0$, $n = 1, \cdot\cdot\cdot, m_1-1$; $\psi_n = u_1, n = m_1, \cdot\cdot\cdot, m_2-1$; $\psi_n = u_2, n = m_2, \cdot\cdot\cdot, m_3-1;\cdot\cdot\cdot$, then we obtain the sequence $\{\psi_n\}$ and $\psi_n\rightarrow u_0$ in $W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$ as $n\rightarrow\infty$.

The existence of weak solutions for problem (1.1) is proved by Galerkin's approximation method. We shall find the sequence of approximate solutions with the form

The unknown functions $(\eta_n(t))_j$ are determined by an ordinary differential system as follows:

where $(U_{0n})_i = \int_\Omega \psi_n\omega_idx$, $(U_{1n})_i = \int_\Omega \phi_n\omega_idx$, $(F_n)_i = \int_\Omega f_n\omega_idx$, $\psi_n\in V_n, \phi_n\in V_n, f_n\in C_0^\infty(Q_T)$, and $\psi_n\rightarrow u_0$ strongly in $W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$, $\phi_n\rightarrow u_1$ strongly in $L^2(\Omega)$, $f_n\rightarrow f$ strongly in $L^{q^\prime(x, t)}(Q_T)$. Here the vector-valued function $P_n(t, \mu, \nu): [0, T]\times\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}^n$ is defined as:

where $\mu = (\mu_1, \cdot\cdot\cdot, \mu_n)$ and $\nu = (\nu_1, \cdot\cdot\cdot, \nu_n)$.

Let $\eta^\prime(t) = X(t), Y(t) = \big(\eta(t), X(t)\big)$ and $H_n(t, Y) = \big(X, F_n -P_n(t, \eta, X)\big)$. Then the problem (9) is transformed into the following problem

The inequality (7) implies

From (10) and Young's inequality, we obtain

where $\widetilde{M}([u_n]_{s, p}^p) = \int_0^{[u_n]_{s, p}^p}M(\tau)d\tau$. Denote $E_n(t) = \frac{1}{2}|Y|^2+\frac{1}{p}\widetilde{M}([u_n]_{s, p}^p)$. Then,

This together with Gronwall's inequality yields that

Thus, $|Y(t)-Y(0)|\leq 2 \sqrt{2C_n(T)}$. Denote

where $B\big(Y(0), 2 \sqrt{2C_n(T)}\big)$ is the ball of radius $2 \sqrt{2C_n(T)}$ with center at the point $Y(0)$ in $\mathbb{R}^{2n}$. From the definition of $H(t, Y)$, $H(t, Y)$ is continuous with respect to $(t, Y)$. By Peano's Theorem, we know that (10) admits a $C^1$ solution on $[0, \tau_n]$, that is, (9) has a $C^2$ solution on $[0, \tau_n]$ denoted by $\eta_n^1(t)$. Let $\eta(\tau_n), \frac{\partial\eta(\tau_n)}{\partial t}$ be the initial value of problem (9), then we can repeat the above process and get a $C^2$ solution $\eta_n^2(t)$ on $[\tau_n, 2\tau_n]$. Without lost of generality, we assume that $T = [\frac{T}{\tau_n}]\tau_n+(\frac{T}{\tau_n})\tau_n, 0<(\frac{T}{\tau_n})<1,$ where $[\frac{T}{\tau_n}]$ is the integer part of $\frac{T}{\tau_n}$, $(\frac{T}{\tau_n})$ is the decimal part of $\frac{T}{\tau_n}$. We can divide $[0, T]$ into $[(i-1)\tau_n, i\tau_n], i = 1, ..., L$ and $[L\tau_n, T]$ where $L = [\frac{T}{\tau_n}]$, then there exist $C^2$ solution $\eta^{i}_{n}(t)$ in $[(i-1)\tau_n, i\tau_n], i = 1, ..., L$ and $\eta^{L+1}_n(t)$ in $[L\tau_n, T]$. Therefore, we get a solution $\eta_n(t)$ $\in C^2([0, T])$ defined by

Lemma 3.4. (A priori estimate) There exists $C(T)>0$ independent of $n$ such that the following estimates

hold.

Proof. By (9), for each $1\leq i\leq n$, we have

Multiplying (11) by $\frac{d}{dt}(\eta_n(t))_i$, then summing up $i$ from $1$ to $n$, we obtain

The inequality (7), (11) and Young's inequality imply

Further,

Thus,

Since $u_n(x, 0) = \psi_n\rightarrow u_0$ strongly in $W^{s, p}(\Omega)\bigcap W_0^{\alpha, 2}(\Omega)$, $\frac{\partial u_n(x, 0)}{\partial t} = \phi_n\rightarrow u_1$ strongly in $L^2(\Omega)$ and $f_n\rightarrow f$ strongly in $L^{q^\prime}(Q_T)$, we deduce from (13) that

which together with assumption $(M_1)$ yields that

Moreover, integrating (12) with respect to $t$ over $(0, T)$, we have

Furthermore, for each $t\in [0, T]$, by Hölder's inequality, we get

Thus, we obtain that $[u_n(x, t)]_{\alpha, 2}^2\leq C(T)$ for each $t\in [0, T]$.

By Lemma 3.4, we have

Lemma 3.5. The estimate

holds uniformly with respect to $n$.

Proof. By (8), we have

From the fractional Sobolev inequality, there holds

Furthermore, $\int_{Q_T}|u_n|^{q}dxdt\leq C(T)$.

Theorem 3.6. Under the conditions $\rm(H1)$–$\rm(H3)$, problem (1.1) has a weak solution.

Proof. By Lemma 3.4 and Lemma 3.5, there exist a subsequence of $\{u_n\}_{n = 1}^\infty$ still denoted by $\{u_n\}_{n = 1}^\infty$ and $u, \mu: Q_T\rightarrow\mathbb{R}$ such that

First, we prove that there exists a subsequence of $\{u_n\}_{n = 1}^\infty$ (still denoted by $\{u_n\}_{n = 1}^\infty$) such that $\frac{\partial u_n}{\partial t}\rightarrow \frac{\partial u}{\partial t}$ strongly in $L^2(Q_T)$ and $u_n\rightarrow u$ strongly in $L^{q}(Q_T)$.

Since $(\eta^\prime_n(t))_j = \int_\Omega \frac{\partial u_n}{\partial t}\omega_jdx$, by Lemma 3.4, $(\eta^\prime_n(t))_j$ is uniformly bounded on $[0, T]$. For all $0\leq t_1<t_2\leq T$, integrating (11) with respect to $t$ from $t_1$ to $t_2$, we have

Hölder's inequality, Lemmas 3.4–3.5 yield

where $Q^{t_2}_{t_1} = \Omega\times(t_1, t_2)$. Thus, the sequence $\{(\eta_n^\prime(t))_j\}_{n = 1}^\infty$ is uniformly bounded and equi-continuous for fixed $j$. By Ascoli-Arzela Theorem, for $j = 1$, there exists a subsequence of $\{n\}$ denoted by $\{n_{1, k}\}$ such that $\{(\eta^\prime_{n_{1, k}}(t))_1\}$ converges uniformly on $[0, T]$ to some continuous function $\zeta_1(t)$; for $j = 2$, there exists a subsequence of $\{n_{1, k}\}$ denoted by $\{n_{2, k}\}$ such that $\{(\eta^\prime_{n_{2, k}}(t))_2\}$ converges uniformly on $[0, T]$ to $\zeta_2(t)$; generally, for $j$, there exists a subsequence of $\{n_{j-1, k}\}$ denoted by $\{n_{j, k}\}$ such that $\{(\eta^\prime_{n_{j, k}}(t))_j\}$ converges uniformly on $[0, T]$ to $\zeta_j(t)$; $\ldots$. The diagonal procedure imply that there exists a sequence of $\{(\eta_{n_{k, k}}^\prime(t))_j\}_{k = 1}^\infty$ still denoted by $\{(\eta_n^\prime(t))_j\}_{n = 1}^\infty$ such that $\{(\eta_n^\prime(t))_j\}$ converges uniformly on $[0, T]$ to $\zeta_j(t)$ for each $j = 1, 2, \cdot\cdot\cdot$.

For $r\leq n$ with $r\in\mathbb{N}$, by Lemma 3.4, we have

Letting $n\rightarrow\infty$, we get

Letting $r\rightarrow\infty$, we obtain

Denote $\overline{u}(x, t) = \sum\limits_{j = 1}^\infty \zeta_j(t)\omega_j(x)$, then $\sup\limits_{0\leq t\leq T}\|\overline{u}(x, t)\|_{L^2(\Omega)}\leq C(T)$ and for each $j\in\mathbb{N}$, there holds

uniformly on $[0, T]$. For each $\varepsilon_1>0$ and $\phi\in L^2(\Omega)$, by the completeness of $\{\omega_j\}$, there exists a $m_0>0$ such that $\|\phi-\sum\limits_{i = 1}^{m_0}(\int_\Omega\phi\omega_idx)\omega_i\|_{L^2(\Omega)}\leq\varepsilon_1$. Thus,

For the $\varepsilon_1>0$, by (14), there exists a $n_{\varepsilon_1}>0$ such that

From (15) and Hölder's inequality, we have

It follows from (16) that

uniformly on $[0, T]$ as $n\rightarrow\infty$. For each $\varphi\in C_0^\infty(Q_T)$, by Lebesgue's Dominated Convergence Theorem, we obtain

By integration by parts, we get

Letting $n\rightarrow\infty$, we have

Thus, we obtain that $\overline{u} = \frac{\partial u}{\partial t}$. Moreover, for each $j\in\mathbb{N}$, Lemma 3.3 and Lebesgue's dominated convergence theorem yield

Thus, for $\varepsilon>0$, by Proposition 1, there exists a positive number $N_\varepsilon$ independent of $n$ such that

From a discussion similar to that of (17), there is a $\widetilde{n}(\varepsilon)>0$ such that

Thus, $\frac{\partial u_n}{\partial t}\rightarrow \frac{\partial u}{\partial t}$ strongly in $L^2(Q_T)$. Further, there exists a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$ such that $\frac{\partial u_n}{\partial t}\rightarrow\frac{\partial u}{\partial t}$ a.e. on $Q_T$.

As $u_n\in L^{\infty}(0, T;W_0^{\alpha, 2}(\Omega))$ and $\frac{\partial u_n}{\partial t}\in L^2(Q_T)$, by the Lions-Aubin Lemma (see Lions [21]), there exists a subsequence of $\{u_n\}$ still labelled by $\{u_n\}$ such that $u_n\rightarrow u$ strongly in $L^2(Q_T)$ and a.e. on $Q_T$. Since $1\leq q<p_s^*: = Np/(N-sp)$, by Lemma 3.4 we have

Furthermore, $\int_0^T\int_\Omega|u_n|^{p^*_s}dxdt\leq C(T)$. For each measurable subset $U\subset Q_T$ with $|U|\leq1$, Hölder's inequality yields

Thus, the sequence $\{|u_n|^{q}\}_{n = 1}^\infty$ is equi-integrable on $L^1(Q_T)$. Vitali Theorem implies that $\lim_{n\rightarrow\infty}\int_{Q_T}|u_n-u|^{q}dxdt = 0,$ that is to say, $u_n\rightarrow u$ strongly in $L^{q}(Q_T)$.

For each $u\in L^p(0, T;W_0^{s, p}(\Omega))$, we define a linear functional $\mathcal L: L^p(0, T;W_0^{s, p} (\Omega))\rightarrow \mathbb{R}$ as:

for all $v\in L^p(0, T;W_0^{s, p}(\Omega))$. By Hölder's inequality, we have

This means that $\mathcal L(u)$ is a bounded linear functional on $L^p(0, T;W_0^{s, p}(\Omega)).$ Thus,

where $C>0$ independent of $n$. Further, up to a subsequence we assume that there exists $\chi\in (L^p(0, T;W_0^{s, p}(\Omega)))^*$ such that

Here $(L^p(0, T;W_0^{s, p}(\Omega)))^*$ denotes the dual space of $L^p(0, T;W_0^{s, p}(\Omega)).$ Then,

for all $v\in L^p(0, T;W_0^{s, p}(\Omega)).$ From (11), for $\varphi\in C^1(0, T, V_k)$ ($k\leq n$), we have

where $0<\tau\leq T$. Letting $n\rightarrow\infty$ in (18), we obtain

where $\varphi\in C^1(0, T;V_k)$ ($k\in\mathbb{N}$). Since $C_0^\infty(\Omega)\subset\overline{\bigcup_{n = 1}^\infty V_n}^{C^1(\overline{\Omega})}$, for each $\varphi\in C_0^\infty(\Omega)$, there exists a sequence $\{\varphi_{n_k}\}_{k = 1}^\infty$ with $\varphi_{n_k}\in V_{n_k}$ such that $\varphi_{n_k}\rightarrow\varphi$ in $C^1(\overline{\Omega})$. Taking $\varphi_{n_k}$ in (19) and letting $k\rightarrow\infty$, we get

Letting $\tau\rightarrow0$, then we have

Similarly, for $t_0\in [0, T]$, there holds

Furthermore, we obtain that $\frac{\partial u(x, 0)}{\partial t} = u_1(x)$ for $x\in \Omega$.

Since $u\in L^\infty(0, T;W_0^{\alpha, 2}(\Omega))$ and $\frac{\partial u}{\partial t}\in L^2(0, T;W_0^{\alpha, 2}(\Omega))$, we can assume that $u\in C(0, T;W_0^{\alpha, 2}(\Omega))$ (see Lions [21]). By Lemma 3.4 and the embedding $W_0^{\alpha, 2}(\Omega) \hookrightarrow L^2(\Omega)$ is continuous, we deduce that $\int_\Omega u_n^2(x, T)dx\leq C(T)$. Thus, there exist a subsequence of $\{u_n\}$ still denoted by $\{u_n\}$ and a function $\widehat{u}$ such that $u_n(x, T)\rightharpoonup \widehat{u}\ \ {\rm weakly\ in}\ L^2(\Omega).$ For each $\varphi\in C_0^\infty(\Omega)$ and $\eta\in C^1([0, T])$, we have

Letting $n\rightarrow\infty$, we get

Integration by parts yields

Choosing $\eta(T) = 1, \eta(0) = 0$ or $\eta(T) = 0, \eta(0) = 1$, we obtain that $\widehat{u} = u(x, T)$ and $u(x, 0) = u_0(x)$ for $x\in\Omega$. Similarly, we can prove that $u_n(x, T)\rightharpoonup u(x, T)$ weakly in $W_0^{\alpha, 2}(\Omega)$ and

Further, by the compactness of embedding $W_0^{\alpha, 2}(\Omega)$ to $L^2(\Omega)$, we get $u_n(x, T)\rightarrow u(x, T)$ strongly in $L^2(\Omega)$. Taking $\varphi = u_k$ in (19), then letting $k\rightarrow\infty$, we obtain

Finally, we prove that $\pi\in \Phi(x, t, \frac{\partial u}{\partial t})$ a.e. on $Q_T$ and $u_n\rightarrow u$ strongly in $L^p(0, T; W_0^{s, p}(\Omega))$. Since $\frac{\partial u_n}{\partial t}\rightarrow\frac{\partial u}{\partial t}$ a.e. on $Q_T$, for each $\delta>0$, by Lusin's theorem and Egoroff's theorem, we can choose a subset $E_\delta\subset Q_T$ such that meas($E_\delta$)$<\delta$, $\frac{\partial u}{\partial t}\in L^\infty(Q_T\setminus E_\delta)$ and $\frac{\partial u_n}{\partial t}\rightarrow\frac{\partial u}{\partial t}$ uniformly on $Q_T\setminus E_\delta$. Then, for each $0<\varepsilon<1$, there exists a $K>0$ such that

If $|\frac{\partial u_n(x, t)}{\partial t}-\tau|<\frac{1}{n}$, then we have $|\frac{\partial u(x, t)}{\partial t}-\tau|<\varepsilon$, for all $n>\max\{K, \frac{2}{\varepsilon}\}$ and $(x, t)\in Q_T\setminus E_\delta$. From the definition of $a_n$, there holds

Thus, $|\frac{\partial u_n(x, t)}{\partial t}-\tau|<\frac{1}{n}$, for each $(x, t)\in Q_T\setminus E_\delta$. Furthermore,

for all $n>\max\{K, \frac{2}{\varepsilon}\}$ and $(x, t)\in Q_T\setminus E_\delta$. Let $\varphi\in L^\infty(Q_T)$ with $\varphi\geq0$. Then,

Letting $n\rightarrow\infty$ and using the weak convergence of $a_n(x, t, \frac{\partial u_n}{\partial t})$, we get

By (H1), the definition of $\underline{a}_\varepsilon$ and $\overline{a}_\varepsilon$, we have

and

for each ${x, t, \xi}\in Q_T\times\mathbb{R}$. Thus, $\underline{a}_\varepsilon(x, t, \frac{\partial u_n}{\partial t})$ and $\overline{a}_\varepsilon(x, t, \frac{\partial u_n}{\partial t})$ are bounded on $Q_T\setminus E_\delta$. By $\underline{a} = \lim\limits_{\varepsilon\rightarrow0}\underline{a}_\varepsilon(x, t, \frac{\partial u}{\partial t})$, $\overline{a} = \lim\limits_{\varepsilon\rightarrow0}\overline{a}_\varepsilon(x, t, \frac{\partial u}{\partial t})$ and Lebesgue's dominated convergence theorem, we obtain

It's easy to check that $\underline{a}$ and $\overline{a}$ still satisfy the same conditions imposed on $a$ in (H2). Thus, $\underline{a}(x, t, \frac{\partial u}{\partial t})$ and $\underline{a}(x, t, \frac{\partial u}{\partial t})$ belong to the space $L^{q^\prime}(Q_T)$. Letting $\delta\rightarrow0$ in (23), we have

Without loss of generality, we assume that $\big|\{(x, t)\in Q_T: \pi(x, t)<\underline{a}(x, t, \frac{\partial u}{\partial t})\}\big|>0$. Taking

in the first inequality of (24), we have

where $(\underline{a}(x, t, \frac{\partial u}{\partial t})-\mu(x, t))^+ = \max\{\underline{a}(x, t, \frac{\partial u}{\partial t})-\mu(x, t), 0\}$. Thus, from the contradiction above, $\underline{a}(x, t, \frac{\partial u}{\partial t})\leq\mu(x, t)$ a.e. on $Q_T$. Similarly, we can deduce that $\mu(x, t)\leq\overline{a}(x, t, \frac{\partial u}{\partial t})$ a.e. on $Q_T$.

Multiplying (11) by $(\eta_n(t))_j$ and summing up $j$ from 1 to $n$, and integrating with respect to $t$ from $0$ to $T$, we have

Thus,

By (21), (22), the weak convergence of $\frac{\partial u_n(x, T)}{\partial t}\rightharpoonup \frac{\partial u(x, T)}{\partial t}$ in $L^2(\Omega)$ and the strong convergence of $u_n(x, T)\rightarrow u(x, T)$ in $L^2(\Omega)$, we get

This together with $(M_1)$ implies that

Similar to the discussion as in [37,26], we get $u_n\rightarrow u$ strongly in $L^p(0, T;W_0^{s, p} (\Omega))$ and $\langle \chi, u\rangle = M([ u]_{s, p}^{p})[u]_{s, p}^p$. It follows from (19) that the theorem is proved.

Remark 2. Obviously, if the function $a:Q_T\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous with respect to $(x, t, s)\in Q_T\times\mathbb{R}$, then $\mu(x, t) = a(x, t, \frac{\partial u}{\partial t})$.

4.   An example

We consider the following problem

where $m_0, m_1>0$ and the set valued function $\Phi$ is obtained by filling in jumps of the function $a$ defined by

After a simple calculation, we have

and

Then $\Phi(x, t, u_t) = [\underline{a}(x, t, u_t), \overline{a}(x, t, u_t)]$ a.e. on $Q_T$. Suppose that (H1) and (H4) are satisfied and let $q$ be defined as in (H3). Then by Theorem 3.6, there exists a weak solution for problem (25).

Acknowledgments

The authors would like to express their gratitude to the referees for valuable comments and suggestions. M. Xiang was supported by the Natural Science Foundation of China (No. 11601515) and the Tianjin Youth Talent Special Support Program. B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199), the Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q18109), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.