Global existence of weak solutions to a class of higher-order nonlinear evolution equations

1.   Introduction

In this paper, we study the following initial boundary value problem for n-dimensional higher-order nonlinear wave equations with dispersive and dissipative terms:

where $U\subset R^{n}$ is a bounded domain with sufficiently smooth boundary $\partial U$, $K = 1, 2, 3, \cdots$, $D^{\alpha} = \frac{\partial^{|\alpha|}}{\partial x^{\alpha_{1}}_{1}\partial x^{\alpha_{2}}_{2}\cdots\partial x^{\alpha_{n}}_{n}}$ means multi-index derivative operator, $\alpha = (\alpha_1, \alpha_2, \cdots, \alpha_n)$ is multi-index of nonnegative integers $\alpha_i(i = 1, 2, \cdots, n)$, $\big|\alpha\big| = \alpha_1+\alpha_2+\cdots+\alpha_n, \; u_0(x)\in H_{0}^{K}(U)$ and $u_1(x)\in H_{0}^{K}(U)$. Moreover,

with $p > 1$ satisfying:

Problems (1.1)–(1.3) come from viscoelastic mechanics. As $K = 1$, the nonlinear evolution equation

describes the propagation of longitudinal strain waves in a slender elastic rod ^[1,2]. Similar equations containing a strong damping term $u_{xxt}$ appear in the framework of the Mooney–Rivlin viscoelastic solids of second grade (see ^[3]). Concerning the higher-dimensional equation

a unique existence result of a global strong solution for the initial boundary problem of Eq (1.6) was proved in ^[4] under some assumptions on $f(u)$ for the positive definite energy. Xu et al. ^[5] also proved that the global strong solution of Eq (1.6) decays to zero exponentially as the time approaches infinity by using the multiplier method for the positive definite energy.

In ^[6], Gazzola and Squassina considered the initial boundary value problem of the following equation with both strong and weak damping terms

where $T > 0, \omega\geq0$ and $\mu > -\omega \lambda_{1}$ ($\lambda_{1}$ is the first eigenvalue of the operator $-\Delta$ under homogeneous Dirichlet boundary condition). They got the global existence of solutions with initial data in the potential well, which was first introduced by Sattinger (see ^[7]). Moreover, they proved the finite time blow up for solutions starting in the unstable set and constructed the high energy initial data for which the solution blows up. In ^[8], Lian and Xu also obtained the global well-posedness of equation $u_{tt}-\Delta u-\omega\Delta u_{t}+\mu u_{t} = u\ln|u|$.

In ^[9], Xu and Yang studied the following nonlinear wave equation with dispersive–dissipative terms and weak damping

Using the technique of ^[6] and the concavity method, Xu and Yang derived a sufficient condition on the initial data with arbitrarily positive initial energy such that the corresponding local solution of Eq (1.8) blows up in a finite time. However, the global existence of weak solutions and strong solutions for Eq (1.8) is still open.

As $K = 2$, Eq (1.1) represents the elastic plate equation with dispersive and dissipative effects ^[10,11]. In ^[12], Xu et al. studied the global well-posedness of the initial boundary value problem for a class of fourth-order wave equations with a nonlinear damping term and a nonlinear source term, which was introduced to describe the dynamics of a suspension bridge. By the potential well method, in ^[13] Lin et al. derived global weak solutions and global strong solutions of the initial boundary value problem for a class of damped nonlinear evolutional equations

Up to now, there is no result on the existence of global solutions to the initial boundary value problem for the nonlinear wave equation, including dispersive term $\triangle u_{tt}$, dissipative term $\triangle u_{t}$ and $u_{t}.$

Fourth-order equation models with the main part $u_{tt}+\Delta^{2}u +\cdots$ containing weak and strong damping terms such as $u_{t}, f(u_{t}), \Delta u_{t}$ and nonlinear strain $\sum^{n}_{i = 1}\frac{\partial}{\partial x_{i}}\sigma_{i}(u_{x_{i}})$ also attract a lot of attention (see ^{[14,15,16,17,18]}). A recent work by Lian et al. (see ^[19]) considered the solutions of the following equation

The global existence, asymptotic behavior, and blow-up of solutions for subcritical initial energy and critical initial energy of Eq (1.9) were obtained, and the blow-up of solutions in finite time for the positive initial energy case was also proved.

As $K > 2$, Problems (1.1)–(1.3) also appear in physics. For example, when $n = 2$ and $K = 4$, Eq (1.1) can represent the model of two-dimensional quasicrystal elasticity with dispersive and dissipative effect; when $n = 3$ and $K = 6$, Eq (1.1) represents the model of three-dimensional quasicrystal elasticity with dispersive and dissipative effect (see ^[20]). As mentioned above, Eq (1.1) has an important physical background, while the mathematical achievements for the arbitrary higher order wave equation with both dispersive and dissipative terms (for any positive integer $K\geq1$) are scarce. So the aim of the present paper is to establish a global existence result of weak solutions to such an evolution problem.

Motivated by previous papers ^[6,8,13], we shall use the potential well theory to establish conditions under which the initial boundary value problems $(1.1)$–(1.3) have global weak solutions. This method proposed by Sattinger (see ^[7,21]) and its improvements (see ^{[22,23,24,25]}) allow us to consider the hyperbolic equations without positive definite energy. For example, concerning about Eq (1.7) in ^[6], in the framework of the potential well method, a Nehari manifold $N$, a stable set $\mathbf{W}$ (potential well) and an unstable set $\mathbf{V}$ (outside the potential well), should be introduced, and the mountain pass energy level $d$ (also known as potential well depth) can be characterized as

However, when dealing with the present models with higher-order dispersive term $\Delta^{K}u_{tt}$, higher-order energy of motion $\|\nabla^{K}u_{t}\|_{L^{2}(U)}$ should be contained in energy functionals. This work brings complicated construction of potential well $\mathbf{W}$, and consequently a detailed computational formula of the modified potential well depth $d$ is needed.

Therefore, in Section 2, we present some notations and definitions for the energy functionals, modified potential well $\mathbf{W}$, and modified potential well depth $d$. Then we concentrate on the detailed equivalent definition of $d$ and prove that $d > 0$.

In Section 3, using the potential well method and Galerkin method, we construct a global weak solution to the evolution Problems (1.1)–(1.3) when the initial data stars from stable set $\mathbf{W}$.

There are also interesting problems for further studies.

1) As $K = 1$, the blow-up property of corresponding local solution, has been derived in ^[12], whereas the uniqueness of solution, vacuum isolation of solutions and decay or blow-up properties of solutions are still open for dispersive–dissipative models with any arbitrary higher-order $K$.

2) The Cauchy problem of such kinds of higher-order evolution equations has not been of concerned so far.

3) It is well worth considering some important physical properties and physical structures in numerical analysis, such as positivity preservation, maximum principle ^[26], long-term behavior ^[27], and singular solutions.

2.   Notations and preliminaries

Let us give some explanations for constraint $(1.5)$ of exponent $p$. Note that $2^{*} = \frac{2n}{n-2K}$ is the critical Sobolev exponent for $q$ in the embedding $H^{K}_{0}(U)\hookrightarrow L^{q}(U)$ (see ^[28]); it follows from $(1.5)$ that $H^{K}_{0}(U)\hookrightarrow L^{p+1}(U)$, hence the following functionals $I(u), J(u)$, and $E(t)$ introduced in $(2.1), (2.2)$, and $(2.4)$ should be well defined. Furthermore, by assumption $(1.5)$, we can control the $L^{2}$ norm of the nonlinear term $(1.4)$ by using Sobolev embedding $H^{K}_{0}(U)\hookrightarrow L^{2p}(U)$. It will lead to global existence results for the nonlinear ordinary differential systems (3.7) and (3.8) associated to Problems (1.1)–(1.3) when the Galerkin method is applied.

We denote by $\big\|\cdot\big\|_{q}$ the $L^{q}(U)$ norm for $1\leq q\leq\infty$, by $\big\|\cdot\big\|$ the $L^{2}(U)$ norm, and by $\big\|\cdot\big\|_{k, p}$ the $W^{k, p}(U)$ norm. Let

For any $0 < t < T$, we define functionals $I, J:Z\rightarrow \mathbb{R}$ by

and

We define the potential well depth (also the mountain pass value of $J$) as

The energy functional $E:Z\rightarrow \mathbb{R}$ is defined by

where $F(u) = \int_{0}^u f(s)ds$.

We introduce the modified Nehari manifold (for Nehari manifold we refer to ^[29] and ^[30]) as

Finally, for any $0 < t < T$ the modified potential well is defined as

Theorem 2.1. The depth of the potential well (denoted by $d$ in $(2.3)$) can also be characterized as

In order to prove Theorem 2.1, we introduce the following two lemmas.

Lemma 2.1. If $0 < t < T, u\in Z \; and \; \|\nabla^{K}u\|^{2}+\|\nabla^{K}u_{t}\|^{2} \neq0$, we have

(i) $\underset{a\rightarrow +\infty}{\mathrm{Lim}}J(a u) = -\infty, \ \underset{a\rightarrow 0}{\mathrm{Lim}}J(a u) = 0.$

(ii) There exists a unique positive number $\tilde{a} = \tilde{a}(u)$ such that $\frac{dJ(a u)}{da}\big|_{a = \tilde{a}} = 0.$

(iii) When $a = \tilde{a}$, $\frac{d^{2}J(a u)}{da^{2}} < 0.$

(iv) $J(a u)$ increases with $a$ as $0\leq a\leq\tilde{a}$; $J(a u)$ decreases with $a$ as $\tilde{a}\leq a < +\infty$.

Proof. (i) is true because

Calculate

Solving $\frac{dJ(a u)}{da} = 0$, there is an unique solution

(ii) is true as $\|\nabla^{K}u_{t}\|^{2}+ \|\nabla^{K}u\|^{2} \neq0$.

In order to obtain (iii), substitute $(2.8)$ into the expression

gives

At last, from $(2.7)$ and $(2.8)$, we have

Hence $\frac{dJ(a u)}{da} > 0$ as $0 < a < \tilde{a}$ and $\frac{dJ(a u)}{da} < 0$ as $\tilde{a} < a < +\infty$. So (iv) is true.

Lemma 2.2. If $0 < t < T, u\in Z$ and $\|\nabla^{K}u_{t}\|^{2}+ \|\nabla^{K}u\|^{2} \neq0$, $J(\alpha u) = \sup \limits_{a\geq 0}J(a u)$ is equivalent to $I(\alpha u) = 0$.

Proof. Since

$\tilde{a}$ in $(2.8)$ coincides with the solution to equation $I(\alpha u) = 0$. From Lemma $2.1$ (iv), $J(\alpha u) = \sup \limits_{a\geq 0}J(a u).$

Conversely, if $J(\alpha u) = \sup \limits_{a\geq 0}J(a u)$, Lemma 2.1 gives $\alpha = \tilde{a}$ in $(2.8)$, then

Proof of Theorem $2.1$: By Lemma 2.1, the depth of potential well (see (2.3)) should be

Let $w = \tilde{a}u$ then from Lemma $2.2$ we have $d = \inf J(w),$ where the infimum is taken for all $t\in (0, T)$ and all functions $w\in Z$ satisfying that $I(u)$ attains $0$ on $(0, T)$ with $\| \nabla^{K}u \|^{2}+\|\nabla^{K}u_{t}\|^{2}\neq0$, which means $w\in N$.

The proof of Theorem 2.1 is completed.

Lemma 2.3. As $p > 1$ satisfies $(1.5)$, a computational formula for the potential well depth is

Here

and

Moreover,

where $S_{p+1}$ is the best Sobolev constant for the embedding $H^{K}_{0}(U)\hookrightarrow L^{p+1}(U)$, i.e.,

Proof. As the process of proof in Theorem 2.1, substituting $(2.8)$ into the computation of $J(a u)$ (see $(2.6)$) we obtain

Value of $\kappa$ and $\Lambda$ in $(2.11), (2.12)$ gives

Therefore

Furthermore,

It follows that

Lemma 2.4. If $J(u)\leq d$, then $I(u) > 0$ is equivalent to

Proof. From $(2.1)$ and $(2.2)$, the following equality holds:

Since $d = \frac{1}{\kappa \Lambda^{\kappa}}$ where $\kappa = \frac{2(p+1)}{p-1}$, for $I(u) > 0$ and $J(u)\leq d$ we have

Conversely, if

then

By value of $\Lambda$ in $(2.12)$,

Hence

Thus $I(u) = \big\|\nabla^{K}u\big\|^{2} +\big\|\nabla^{K}u_{t}\big\|^{2} -\big\|u\big\|_{p+1}^{p+1} > 0.$

3.   Global weak solution

We denote the inner product in $L^{2}(U)$ by

A continuous linear functional defined on the locally convex linear topological space $\mathfrak{D}(0, T)$ is called the "distribution" or the "generalized function" (see ^[31], Chapter 8). We denote the space of generalized functions on $(0, T)$ by $\mathfrak{D}'(0, T)$.

Definition 3.1. For $T > 0$, if the function $u(x, t)\in Z$ satisfies:

1) for any $v(x)\in H_{0}^{K}(U)$ and for almost $t\in [0, T)$,

2) $u(x, 0) = u_0(x)$ in $H_{0}^{K}(U)$ and $u_t(x, 0) = u_1(x)$ in $L^{2}(U).$

Then we call $u = u(x, t)$ a global weak solution to Problems (1.1)–(1.3).

For $u_{0}\in H_{0}^{K}(U)$ and $u_{1} \in H_{0}^{K}(U)$, we introduce the following initial functionals:

Theorem 3.1. If $T > 0$, $f(s) = |s|^{p-1}s$ where $p$ satisfies $(1.5)$ and $E(0) < d$, there exists a global weak solution to Problems (1.1)–(1.3) as long as $I(0) > 0$, $J(0) < d$ for $u_{0}\in H^{K}_{0}(\Omega)$ and $u_{1}\in H^{K}_{0}(\Omega)$. Moreover, for any $0\leq t < T$, $u\in\mathbf{ W}$.

3.1.   Step 1: Galerkin method.

Let $\{\omega_k(x)\} (k = 1, 2, 3, \cdots)$ be a complete orthogonal basis in $H^{2K}(\Omega)\cap H_{0}^{K}(\Omega)$, which solves the following eigenvalue system

It is also a complete orthonormal basis for $L^{2}(U)$ and a complete orthogonal basis for $H_{0}^{K}(U)$. (see ^[32,33]).

Based on the Galerkin method, an approximate solution to Problems (1.1)–(1.3) can be constructed by

where $u_{m}(x, t)$ satisfies a system of nonlinear ordinary differential equations

with initial values

for $k = 1, 2, \cdots, m$.

Since $u_{0}(x)\in H_{0}^{K}(U)$ and $u_{1}(x)\in H_{0}^{K}(U)$, when $m \rightarrow +\infty$ there exist $a_{km}$ and $b_{km}\ (k = 1, 2, \cdots, m)$ such that

Notice that $f(s) = |s|^{p-1}s \; (p > 1)$ is locally Lipschitz continuous with respect to $s$. According to classical existence theory for nonlinear ordinary differential equations (see ^[34], corollary 1.1.1), systems (3.7) and (3.8) with initial data satisfying (3.9) and (3.10) have a local solution $u_{m}(x, t)$ for each $m$. In order to extend it to a global solution on $[0, T)$, we will make priori estimates of $u_{m}(x, t) \; (m = 1, 2, \cdots)$.

Multiplying by $g_{km}^{'}(t)$ on both sides of (3.7) and summing up from $k = 1$ to $k = m$, we obtain

That is,

Integrating the above equality by parts with respect to $x$,

Let $F(u_{m}) = \int^{u_{m}}_{0}f(s)ds$, calculation

gives that

Let $E_{m}(t) = E(u_{m})$, from $(2.4)$ we have

and

Integrating $(3.11)$ with respect to $t$ on $(0, t)$ for $0\leq t < T$ gives

It concludes that

3.2.   Step 2: proving $u_m(x, t)\in \mathbf{W}$ for sufficiently large $m$ and $0 < t < T$.

First, we claim that there exists $N_{1} > 0$ such that

From (3.9), (3.10), and $(1.5)$, using Sobolev imbedding theorem we find $\big\|\nabla^{K}u_{m}(x, 0)\big\|$ converges to $\big\|\nabla^{K}u_{0}(x)\big\|$, $\big\|u_m(x, 0)\big\|_{p+1}^{p+1}$ converges to $\big\|u_0(x)\big\|_{p+1}^{p+1}$ and $\big\|\nabla^{K}u_{mt}(x, 0)\big\|$ converges to $\big\|\nabla^{K}u_{1}(x)\big\|$ as $m\rightarrow +\infty$. Therefore, when $m$ tends to $+\infty$,

converges to $I\big(0\big)$ and

converges to $J\big(0\big)$.

Since $I(0) > 0$ and $J(0) < d$, we conclude that $I(u_{m})(0) > 0$ and $J(u_{m})(0) < d$ for sufficiently large integer $m$, which implies $(3.16)$.

Next we prove that $\int_{U}F\big(u_m(x, 0)\big)dx$ converges to $\int_{U}F\big(u_0(x)\big)dx$ when $m$ increases to $+\infty$.

By mean value theorem of integral, there exists $\xi^{(m)}$ between $u_{0}(x)$ and $u_{m}(x, 0)$ such that

hence

Under condition $(1.5)$ of $p$, $H^{K}_{0}(U)$ is embedded in $L^{p+1}(U)$. Since $u_{m}(x, 0)$ converges to $u_{0}(x)$ in $H^{K}_{0}(U)$ as $m$ increases to $+\infty$ (see $(3.9)$), $\big\|u_m(x, 0)-u_0(x)\big\|_{p+1}\rightarrow 0$ as $m\rightarrow +\infty$ and $\big\| |\xi^{(m)}|^{p}\big\|_{\frac{p+1}{p}}$ is uniformly bounded for $m = 1, 2, \cdots$. So we arrive at

By $(3.9), (3.10)$, and $(3.17)$, $E_{m}(0)$ in $(3.13)$ converges to

that is,

Since $E(0) < d$, there exists $N_{2} > 0$ satisfying $E_m(0) < d$ for all $m > N_{2}$.

Recalling $\big|f(u)\big| = \big|u\big|^{p}$, a control of $F(u) = \int_{0}^{u}f(s)ds$ is

then

Combining with $(3.19)$, it follows from $(2.2), (3.12)$ that

Therefore, from $(3.14)$ we have

Hence

In what follows, we prove that $u_{m}(x, t)\in W$ for sufficiently large integer $m$. Set $m > \max\{N_{1}, N_{2}\}$ and $T > 0$, if there exists $t_0 = t_0(m)\in(0, T)$ such that $u_m(x, t)$ attains $\partial \mathbf{W}$ at $t = t_{0}$, then $I(u_m)(t_{0}) = 0$ with $\big\|\nabla^{K}u_m(x, t_0)\big\|+\big\|\nabla^{K}u_{mt}(x, t_0)\big\| \neq0$ or $J(u_m)(t_0) = d.$

Inequality $(3.21)$ means $J(u_m)(t_0) = d$ is impossible; on the other hand, by Theorem 2.1 we find $J(u_{m})(t_{0})\geq d,$ which also contradicts $(3.21)$. Therefore, when $m$ is large enough and $0 < t < T$, $u_m(x, t)$ always stays in $\mathbf{W}$. That is, $I(u_{m}) > 0$ and $J(u_m) < d$.

3.3.   Step 3: existence of a global solution for nonlinear ordinary differential systems (3.7) and (3.8).

Substituting

into $(3.20)$, we obtain

and

Since $I(u_m) > 0$, when sufficiently large $m$ we have the estimates

and

Inequality $(3.24)$ shows $\big\|u_{m}\big\|_{H_{0}^{K}(U)}$ and $\big\|u_{mt}\big\|_{H_{0}^{K}(U)}$ are uniformly bounded for $m = 1, 2, \cdots$. Consequently $\big\|u_{m}\big\|$ and $\big\|\nabla u_{m}\big\|$ are also uniformly bounded for $m = 1, 2, \cdots$.

From (1.5), the Sobolev space $H^{K}_{0}(U)$ is embedded in $L^{p+1}(U)$ and $L^{2p}(U)$. Thus $\big\|u_{m}\big\|_{p+1}$ and $\big\|f(u_m)\big\|^{2} = \|u_{m}\|^{2p}_{2p}$ are also uniformly bounded for $m = 1, 2, \cdots$.

When $0 < t < T$,

should be uniformly bounded for $m = 1, 2, \cdots$, where $\big\{\omega_k(x)\big\}^{+\infty}_{k = 1}$ is a complete orthonormal basis in $L^{2}(\Omega)$ with $\big\|\omega_i(x)\big\| = 1 \ (i = 1, 2, \cdots)$.

Now we conclude that there exist global solutions

to problems (3.7) and (3.8) on $[0, T)$, according to classical theory of nonlinear ordinary differential system (see ^[34]).

3.4.   Step 4: deriving global weak solutions that satisfy (3.2).

Let $q = \frac{p+1}{p}$ and $Q_{T} = U\times[0, T)$. It follows from

that

Furthermore,

By $(3.24)$ there exists a subsequence of $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty}$ (still denoted by $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty}$), a function $u(x, t)$ satisfying the following two:

By $(3.26)$, there exists another subsequence of $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty}$ (still denoted by $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty}$ again), a function $X(x, t)$ satisfy that

By $(3.25)$, $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty} \text{ is uniformly bounded in } H^{1}(Q_T).$ Since $H^{1}(Q_{T})$ is compactly imbedded into $L^{2}(Q_T)$, there exists a subsequence of $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty}$ (still denoted by $\big\{u_m(x, t)\big\}_{m = 1}^{+\infty}$) such that

and then

Moreover,

because $f(s) = |s|^{p-1}s$ is continuous.

On the other hand, $f(u_{m}(x, t))$ is bounded in $L^{q}(Q_{T})$ from $(3.27)$, according to J. L. Lions' Lemma (^[35], Lemma 1.3) we find

For $0\leq t < T,$ integrating by parts with respect to $x$ and integrating with respect to $t$ from 0 to t on both sides of (3.7), we obtain

Let $m\rightarrow +\infty,$ we obtain

Since $\big\{\omega_k(x)\big\}_{k = 1}^{+\infty}$ is a complete orthogonal basis in $H_{0}^{K}(U),$ the above equality still holds if we replace $\omega_{k}$ by arbitrary $v \in H_{0}^{K}(U).$

3.5.   Step 5: verifying $u(x, 0)=u_0(x)$ in $H^{K}_{0}(U)$ and $u_{t}(x, 0)=u_{1}(x)$ in $L^{2}(U)$.

According to Lemma 1.2 in ^[35], it can be deduced from (3.28) and (3.29) that $u_{m}(x, t)\in C\big(0, T;H_{0}^{M}(U)\big)$, $u(x, t)\in C\big(0, T;H_{0}^{K}(U)\big)$. Then

On the other hand, from $(3.9)$ we see that $u_{m}(x, 0)$ strongly converges to $u_{0}(x)$ in $H_{0}^{K}(U)$ as $m$ increases to $+\infty$, so

Next, we will verify $u_{t}(x, 0) = u_{1}(x)$ in $L^{2}(U)$. Integrating by parts with respect to $x$ on both sides of $(3.7)$, we get

By (3.28)–(3.30), when $m\rightarrow +\infty$, for $k = 1, 2, 3, \cdots$,

So the right side in (3.32) converges to $(X, \omega_{k})-(\nabla^{K}u, \nabla^{K}\omega_{k}) -(\nabla^{K}u_{t}, \nabla^{K}\omega_{k})-\big(u_{t}, \omega_{k}\big)$ in $L^{\infty}(0, T)$ weakly-star as $m\rightarrow +\infty$, which means that the left side of (3.32) is also convergent in $L^{\infty}(0, T)$ weakly-star.

Moreover, by $(3.36)$ and $(3.35)$ when $m\rightarrow +\infty$, for $k = 1, 2, 3, \cdots$

and

Furthermore, when $m\rightarrow +\infty$, for $k = 1, 2, 3, \cdots,$

and

Hence the left side in (3.32) converges to $\big(u_{tt}, \omega_{k}\big)+\big(\nabla^{K}u_{tt}, \nabla^{K}\omega_{k}\big)$ in $\mathfrak{D}'(0, T)$, and then it converges to the same limit in $L^{\infty}(0, T)$ weakly-star by the uniqueness of limit.

From the above discussions, for $k = 1, 2, \cdots$

and

Again, using Lemma 1.2 in ^[35], for $k = 1, 2, \cdots$ we have

and

Therefore, when $m\rightarrow +\infty$, for $k = 1, 2, \cdots$

On the other hand, from $(3.10)$ when $m\rightarrow +\infty$, for $k = 1, 2, \cdots$

By uniqueness of limit, for $k = 1, 2, \cdots$

Integrating by parts with respect to $x$,

It follows that

Equivalently,

Since all eigenvalues $\lambda_{k} > 0$ ($k = 1, 2, \cdots)$ (see Theorem 7.23 in ^[32]), there should be

Thus, we have $u_t(x, 0)-u_1(x) = 0$ in $L^{2}(U)$, that is, $u_t(x, 0) = u_1(x)$ in $L^{2}(U)$.

We finally have a global weak solution $u(x, t)\in L^{\infty}(0, T;H^{K}_{0}(U))$ with $u_{t}(x, t)\in L^{\infty}(0, T;H^{K}_{0}(U))$ to Problems (1.1)–(1.3).

3.6.   Step 6: proving $u(x, t)\in \mathbf{W}$ for $0\leq t < T$.

The process will be an analogue as in Step 2.

We denote the inner product in $L^{\infty}(0, T;L^{2}(U))$ by

Making an inner product in $L^{2}(U)$ by $u_{t}$ on both sides of $(1.1)$, we obtain

Integrating by parts with respect to $x$, we obtain

Integrating with respect to $t$ from 0 to $t$ ($0 < t < T$), we have

That is,

Hence

Similar to $(3.20)$, by some computations we obtain

Since $E(0) < d$,

If there exists $t_{0}\in (0, T)$ such that $u\in \mathbf{W}$ for $0\leq t < t_{0}$ and $u$ attains $\partial\mathbf{W}$ at $t = t_{0}$, then the nontrivial solution $u\in Z$ satisfies that $J\big(u\big)(t_{0}) = d$ or $I\big(u\big)(t_{0}) = 0$ with $\big\|\nabla^{K}u_{t}(x, t_0)\big\|+\big\|\nabla^{K}u(x, t_0)\big\| \neq 0$.

Inequality $(3.37)$ implies that $J\big(u\big)(t_{0}) = d$ is impossible. On the other hand, if $u(x, t)\in Z$ satisfies that $I\big(u\big)(t_{0}) = 0$ and $\big\|\nabla^{K}u_{t}(x, t_0)\big\|+\big\|\nabla^{K}u(x, t_0)\big\| \neq 0$, that is $u(x, t_{0})\in N$, from Theorem 2.1 we should have that $J\big(u\big)(t_{0})\geq d,$ which is also in contradiction with (3.37). So we conclude that $u(x, t)\in \mathbf{W}$ for $0\leq t < T$.

The proof of Theorem 3.1 is completed.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by Teaching Team Project, Guangdong Provincial Department of Education 2018 (No. 99161010120) and Guangdong Provincial NSF under Grants 2018A030313546.

Conflict of interest

The authors declare there are no conflicts of interest.