Improved decay of solution for strongly damped nonlinear wave equations

1.   Introduction

In this paper, we study the initial boundary value problem of solutions for the following strongly damped nonlinear wave equations:

where $\alpha > 0$, $\Omega\subset \mathbb{R}^n$ is a smooth bounded domain and $f$ is a given nonlinear function satisfying the following hypothesis.

In the one-dimensional case, Eq (1.1) models the longitudinal vibration of a uniform, homogeneous bar with the nonlinear stress law given by the function in ^[1]. In the two-dimensional and three-dimensional cases, Eq (1.1) describes antiplane shear motions of viscoelastic solids ^[2]. For $f(u) = \sin u$, Eq (1.1) can be used to describe the propagation of fluxons in the Josephson junction between two superconductors ^[3,4]. The concept of a strongly damped nonlinear wave equation was introduced to describe many nonlinear phenomena described by Eqs ^[5,6,7,8,9], and it has attracted a lot of attention from the mathematical perspective ^{[10,11,12,13]}. In ^[14] the problem described by Eqs (1.1)–(1.3) for $n\le 3$ was considered, and the global existence and asymptotic behavior of the strong solution were obtained for the positive-definite initial energy by properly adjusting the nonlinearity $f(u)$. Then, the authors of ^[15] further considered this problem and proved that the local existence of the solution is Lipschitz-continuous on a bounded domain, and that the solution is global and decays exponentially to zero as $t\to\infty$ under the assumption that $f(u)$ ensures the positive energy. In ^[16], the authors further showed the asymptotic behavior of the solution for the problem described by Eqs (1.1)–(1.3) by assuming that the nonlinearity takes the form to ensure the positive-definite initial energy.

Here, we would like to mention that the above results are based on the assumptions ensuring that the corresponding problem including Eq (1.1) has a positive-definite initial energy, or that the nonlinearity takes the linear form. In ^[17], the authors considered the nonlinear version of the model with non-positive initial energy. They showed the global existence of both weak and strong solutions to problem described by Eqs (1.1)–(1.3) by using the potential well method. It is natural to consider how the global solution behaves after we get the existence of the global solution, which involves describing the long time behavior of the solution as the time $t\to +\infty$.

Pata and Squassina ^[18] investigated the initial boundary value problem of the three-dimensional wave equation $u_{tt} - \Delta u - \alpha \Delta u_t + \phi(u) = f(u),$ and carefully described the long time behavior of the solution. The techniques and conclusions there depend on the the damping coefficient $\alpha$, which means that the weakly damping term plays a crucial role in the arguments. For the initial boundary value problem of the strongly damped nonlinear wave equation given by

Gazzola and Squassina in ^[19] conducted a kind of comprehensive study that involved proving the global well-posedness, finite time blowup and long time behavior of the solution; here, we focus on the conclusion concerning the decay as follows.

Theorem 1.1. (Polynomial decay obtained in ^[19]) Let $u$ be the unique local solution to (1.4). Assume that

$\lambda_1$ being the first eigenvalue of the operator $-\Delta$ under homogeneous Dirichlet boundary conditions, and assume that

In addition, assume that there exists $t_0\in[0, T_{max})$ such that $u(t_0)\in W'$ and $E\left(u(t_0), u_t(t_0)\right)\le d$. Then, for $T_{max} = \infty$ and every $t > t_0$, we have

where

and $C$ is independent of $\mu$, whereas $C_\mu$ only depends on $\mu$.

The above theorem tells that the global solution decays polynomially in the form of $\frac{1}{t}$ for the initial data trapped within the potential well and $E(0) < d$, where $d$ is the depth of the potential well, or as it is also called, the mountain pass level.

Later, Gerbi and Houari ^[20] again investigated the asymptotic behavior of solutions of Eq (1.4); they obtained the following theorem to improve the decay rate obtained in ^[19] by showing the exponential decay compared with the polynomial decay.

Theorem 1.2. (Exponential decay obtained in ^[20]) Assume that $u_{0}\in W'$, $E(0) < d$ and (1.5) hold. Then, there exist two positive constants $\bar{C}$ and $\xi$ independent of $t$ such that

The key tool in the proof of Theorem 1.2 is the construction of a suitable Lyapunov function to make a small perturbation of the energy. The method introduced by Gerbi and Houari ^[20] strongly depends on the weakly damping term $u_t$. In other words, an interesting question is if such a method may work for the case without the weakly damping term, i.e., $\mu = 0$ in Theorem 1.2, and a similar case can be found in ^[21].

The interesting point of the present paper is to consider the asymptotic behavior of the solution to the strongly damped nonlinear wave equation without the weakly damping term in the framework of the potential well theory. It is well known that, in order to prove the global existence of solution for the nonlinear hyperbolic equation, Sattinger ^[22] introduced the potential well method to treat the problem without positive-definite energy. Later, in ^[23], the potential well method was applied to prove the global nonexistence of the solution for semilinear hyperbolic and parabolic equations. In the recent decades, the theory relying on the single potential well was improved by proposing the theory of a family of potential wells ^[24]. Both of these two theories have been improved and applied in studies of many important mathematical models (see the theory of the single potential well and the family of potential wells ^{[25,26,27,28,29]}).

In the present paper, first, we define a family of potential wells, and then, by using them, we give the existence of the global weak solution of the problem described by Eqs (1.1)–(1.3). Although, for the problem described by Eqs (1.1)–(1.3), there have been some conclusions about the global well-posedness of the solution, most of them were not established in the framework of the family of potential wells; hence, it is necessary to rebuild it again to give a strict argument. Further we prove the asymptotic behavior of the solution, i.e., the solution of the problem described by Eqs (1.1)– (1.3) decays exponentially to zero as $t \to \infty$.

2.   Preliminary conclusions of setting up the potential wells theory

We shall introduce some necessary functionals and manifolds in order to setup the variational structure. Throughout the paper, we use the following denotations: $\|\cdot\|_p: = \|\cdot\|_{L^p(\Omega)}: = \left(\int_\Omega |u|^{p} {\rm d}x\right)^{\frac{1}{p}}$, $\|\cdot\|: = \|\cdot\|_2: = \left(\int_\Omega u^{2} {\rm d}x\right)^{\frac{1}{2}}$ and $(u, v): = \int_\Omega uv {\rm d}x$. First, for the problem described by Eqs (1.1)–(1.3), we define the potential energy:

where

Here, we also need the Nehari functional:

as well as its family version:

Next, in aid of the Nehari functional and the family of Nehari functionals, we can define the depth of a single potential well and the depths of a family of potential wells, respectively, as follows:

(i) for a single potential well,

(ii) for family of potential wells,

Lemma 2.1. Let $f(u)$ satisfy $({\bf H})$, $u\in H_0^1(\Omega)$ and $w(u): = \frac{f(u)}{u}$, $u\neq 0$. Then, it holds that

(i) $\lim_{u\to 0} w(u) = 0$;

(ii) $w(u)$ is an increasing function on $(0, \infty)$ and a decreasing function on $(-\infty, 0)$;

(iii) $f(u)u > 0$ for $u\neq 0$;

(iv) $f(u)$ is strictly increasing on $(-\infty, \infty)$;

(v) $0\le F(u)\le A_1 |u|^{q+1}$ for $u\in\mathbb{R}$ and some positive constant $A_{1}$.

Proof. (i) The assumption (ii) in $({\bf H})$ can directly give this conclusion.

(ii) In view of the assumption (i) in $({\bf H})$, we can prove this point by considering the following two cases. If $u > 0$, we see

and if $u < 0$, we see

which is the conclusion we desired here.

(iii) By the already established conclusions (i) and (ii) proved above, we can easily find that $w(u) > 0$ for any $u\neq 0$, which proves that $f(u)u > 0$ for any $u\neq 0$.

(iv) By the assumption (i) in $({\bf H})$ and the conclusion (iii) proved just now, we obtain

which proves the conclusion claimed in this item.

(v) By the fact that $f(0) = 0$, with the item (ii) in this lemma, we get $F(u)\ge 0$. Then, $F(u)\le A_1 |u|^{q+1 }$ follows from (ii) in $({\bf H})$.

Lemma 2.2 (Calculus in an abstract space ^[30]). Let $u\in W^{1, p}(0, T;X)$ for some $1\leq p < \infty$. Then, $u\in C\left([0, T]; X\right)$.

Now, let us introduce the following four sets:

3.   Global-in-time existence of solution

In this section, we prove the existence of a global weak solution for the problem described by Eqs (1.1)–(1.3) under the assumption $({\bf H})$.

Definition 3.1 The function $u = u(x, t)$ is called a weak solution of the problem described by Eqs (1.1)–(1.3) on $\Omega\times[0, T)$, where $T$ is the maximum existence time of the solution if $u\in C((0, T);H_0^1(\Omega))$ and $u_t\in L^\infty((0, T);L^2(\Omega))$ satisfy

(i)

(ii)

Theorem 3.1. Let $f(u)$ satisfy $({\bf H})$, $u_0(x)\in H_0^1(\Omega)$ and $u_1(x)\in L^2(\Omega)$. Assume that $E(0) < d$ and $u_0(x)\in W'$. Then, the problem described by Eqs (1.1)–(1.3) admits a global weak solution $u\in C((0, \infty); H_0^1(\Omega))$ with $u_t\in L^\infty\left((0, \infty); L^2 (\Omega)\right)\cap L^2 \left((0, \infty); H_0^1(\Omega)\right)$ and $u\in W$ for $0\le t < \infty$.

Proof. Let $\{w_j(x)\}$ be a system of base functions in $H_0^1(\Omega)$. Construct the approximate solutions of the problem described by Eqs (1.1)–(1.3)

satisfying

Testing the both sides of (3.1) by $g'_{sm}(t)$ and taking the sum for $s$, we obtain

and

where

Now, we need a conclusion about the approximate solution that $E_m(0) < d$ and $u_m(0)\in W'$ by a limit, which can be derived from the assumptions $E(0) < d$ and $u_0(x)\in W'$ with the settings of (3.2) and (3.3). Hence, from (3.5), we have

From (3.6) and an argument similar to that in ^[25,26,27], for $0\le t < \infty$ and a sufficiently large $m$, we can obtain

From this and

we can get

Hence, there exists a $u$ and a subsequence $\{u_\nu\}$ of $\{u_m\}$ such that, as $\nu\to\infty$,

$u_\nu\to u$ in $L^\infty((0, \infty); H_0^1(\Omega))$ weakly star and a.e. in $Q = \Omega\times[0, \infty)$,

$u_{\nu t}\to u_t$ in $L^\infty((0, \infty); L^2(\Omega))$ weakly star,

$f(u_\nu)\to f(u)$ in $L^\infty ((0, \infty); L^r(\Omega))$ weakly star.

Further, for an $s$ that is fixed, letting $m\rightarrow \infty$ in (3.1), we obtain

Since ${\omega_{j}(x)}$ is a system of base functions in $H_0^1(\Omega),$ for any $v\in H_0^1(\Omega)$, we deduce

Moreover, (3.2) indicates that $u_{m}(0)\rightarrow u_{0}\in H_0^1(\Omega)$ and (3.3) gives $u_{mt}(0)\rightarrow u_{1}\in L^2(\Omega)$. Hence, from the above disscussion, we know that the problem described by Eqs (1.1)–(1.3) admits a weak solution

with

Regarding the continuity in time $t$, from (3.8) and (3.9), we get

and then we obtain

by Lemma 2.2. By (3.7) and the compactness method, for $t\in[0, \infty)$, we derive $u(t)\in W$.

4.   Improved decay of solution

In this section, we discuss the asymptotic behavior of a solution for the problem described by Eqs (1.1)–(1.3). We prove that the global weak solution given in Theorem 3.1 decays exponentially to zero as $t\to \infty$.

First, we give the following lemma, which will be used to prove the decay of the solution. And, this lemma will be proved in aid of the family of potential wells.

Lemma 4.1. Let $f(u)$ satisfy $({\bf H})$, $u_0(x)\in H_0^1(\Omega)$, $u_1(x)\in L^2(\Omega)$ and $u_m$ be the approximate solutions. Then, the following holds:

(i)

(ii) For $0 < E(0) < d$, $u_0(x)\in W'$ and $0\le t < \infty$, it holds that

where $(\delta_1, \delta_2)$ is the maximal interval such that $d(\delta) > E(0)$.

Proof. (i) Multiplying (3.1) by $g_{sm}(t)$ and summing for $s$, we get (4.1).

(ii) First, from $E(0) < d(\delta)$ for $\delta\in (\delta_1, \delta_2)$, (3.2) and (3.3), it follows that $E_m(0) < d(\delta)$ for $\delta\in (\delta_1, \delta_2)$. Next, from (3.5), for $\delta\in (\delta_1, \delta_2)$ and $0\le t < \infty$, we get

From (4.2) and an argument similar to that in ^[25,26,27], we can prove that $u_m (t)\in W_\delta^\prime$ for $\delta\in (\delta_1, \delta_2)$, $0\le t < \infty$ and a sufficiently large $m$. Hence, we have that $I_\delta (u_m)\ge 0$ for $\delta\in (\delta_1, \delta_2)$ and $I_{\delta_1} (u_m)\ge 0$ for $0\le t < \infty$ and a sufficiently large $m$. Thereby, for $0\le t < \infty$, we get

Theorem 4.1. Let $f(u)$ satisfy $({\bf H})$, $u_0(x)\in H_0^1(\Omega)$ and $u_1(x)\in L^2(\Omega)$. Assume that $0 < E(0) < d$ and $u_0(x)\in W'$. Then, for the global weak solution, it holds that

for some positive constants $C_{2}$, $C_{3}$, $\gamma$ and $\lambda$.

Proof. We aim to prove the approximate solution

for $C_{2} > 0$, $C_{3} > 0$ and $\lambda > 0$. To do this, multiplying (3.4) by $e^{\gamma t} (\gamma > 0)$, we get

and

By Lemmas 2.1 and 4.1, we get

and

From

we have

Hence, there exits a $C > 0$ such that

From (4.5)–(4.7) and (4.9), it follows that there exist some positive constants $C_0$ and $C_1$ such that

where

Take $\gamma$ such that

Then, (4.10) gives

And, by the Gronwall inequality, we can obtain

i.e.,

and (4.4), where $C_{2}: = 2C_0d$, $C_{3}: = 4C_{0}C_{1}d\gamma^{2}$ and $\lambda: = \gamma (1-2C_1\gamma) > 0$. Finally, let $\{u_\nu\}$ be the subsequence of $\{u_m\}$ given in the proof of Theorem 3.1. Then, from (4.8) and (4.4), we get

which gives (4.3).

Acknowledgments

This work was supported by the Talents Launch in Scientific Research Development Fund for Zhejiang Agriculture and Forestry University (2022LFR014).

Conflict of interest

The authors declare that there is no conflict of interest.