A family of potential wells for a wave equation

1.   Introduction

In this paper, we study the following initial-boundary value problem for a wave equation with single source term

where $\Omega$ is a bounded domain of $\mathbb{R}^N$ ($N\geq1$) with a smooth boundary $\partial \Omega$, and the power index $p$ of the source term satisfies

Eq. (1)$_1$ is the classical wave equation that has been widely investigated. We omit further comments. We focus on a family of new potential wells and their applications to Eq. (1)$_1$. The potential well is proposed by Sattinger [15] (see also Payne and Sattinger [14]). In general, making use of the energy functional $J(u)$ and the Nehari functional $I(u)$, we are able to define the potential well

and its outside set

where the depth of the potential well

the Nehari manifold

When the initial energy is controlled by the depth of the potential well, the well-posedness of solutions for problem under consideration can be investigated by the potential well theory (see e.g. [2,3,4,6,13,16,18,22] and the references therein).

Moreover, Liu [8] introduced a family of potential wells and their outside sets

where $J_\delta(u)$ is an auxiliary functional originated from $J(u)$, and $d(\delta)$ (namely the depths of the potential wells) needs to be estimated in advance. Subsequently, Liu and Zhao [12] introduced

where

and $I_\delta(u)$ is an auxiliary functional originated from $I(u)$. Through the study on the properties of $d(\delta)$, $W_\delta$ and $V_\delta$, many nonlinear evolution equations can be handled (see e.g. [11,19,20,21]). Thus the potential well theory has been enriched and developed greatly.

The main purpose of this paper is to construct a family of new potential wells and their outside sets by modifying the depths of the potential wells inspired by [17]. Thus it is unnecessary to introduce the Nehari functional and the Nehari manifold. The innovation of this paper is that the depths of the potential wells can be computed exactly by an effective approach instead of being estimated so that it is easy to understand the spatial structure of the potential wells. Then the applications of this family of potential wells to the problems concerned can simplify the proofs of the corresponding results. We take problem (1) as an object to illustrate our ideas, which can be applied to a lot of other similar models [1,5,9,10].

This paper is organized as follows. In Section 2, we handle problem (1) by modifying the depths of the potential wells and constructing a family of potential wells. Global existence and finite time blow-up of weak solutions for problem (1) with the subcritical initial energy are obtained. In Section 3, global existence and finite time blow-up of weak solutions for problem (1) with the critical initial energy are obtained. In Section 4, through numerical simulations, we show some intuitive relations between the depths of the potential wells and several parameters.

2.   Problem (1) with the subcritical initial energy

2.1.   Introduction of a family of potential wells and computation of their depths

Throughout the paper, in order to simplify the notations, we denote

We define the total energy function related to problem (1)

which satisfies the energy identity $E(t) = E(0)$ for all $t\geq 0$. Moreover, we introduce the auxiliary functional

Now we are in a position to define the depths of the potential wells

where $y = \|\nabla u\|$,

and $C_1$ is the best Sobolev constant for the embedding from $H_0^1(\Omega)$ into $L^{p+1}(\Omega)$. Obviously, this definition is different from those in the works mentioned before.

Let $g^{\prime}_\delta(y) = 0$, then

Hence

By virtue of (2) and (3), we can get

Thus we can define a family of potential wells

and their outside sets

Obviously,

Lemma 2.1. Let $u\in H_0^1(\Omega)$.

(i) If $u\in W_\delta$ and $\|\nabla u\|\neq 0$, then $\delta\|\nabla u\|^2>\|u\|_{p+1}^{p+1}$.

(ii) If $u\in \partial W_\delta$, then $\delta\|\nabla u\|^2\ge\|u\|_{p+1}^{p+1}$.

(iii) If $\delta\|\nabla u\|^2<\|u\|_{p+1}^{p+1}$, then $u\in V_\delta$.

(iv) If $\delta\|\nabla u\|^2 = \|u\|_{p+1}^{p+1}$ and $\|\nabla u\|\neq 0$, then $u\in H_0^1(\Omega)\backslash W_\delta = V_\delta\cup\partial V_\delta$.

Proof. (ⅰ) Since $u\in W_\delta$, we have

which, together with (3), gives

We further get

Since $\|\nabla u\|\neq 0$, multiplying the above inequality by $\|\nabla u\|^2$, we obtain

and so

(ⅱ) From $u\in \partial W_\delta$ we have

By similar arguments in (i), we get $\delta\|\nabla u\|^2\ge\|u\|_{p+1}^{p+1}$.

(ⅲ) Taking into account $\|\nabla u\|\neq 0$, we obtain

i.e.,

We further get

Combining this with (3), we obtain

Hence $u\in V_\delta$.

(ⅳ) From the proof of (ⅲ) we know that $\delta\|\nabla u\|^2 = \|u\|_{p+1}^{p+1}$ and $\|\nabla u\|\neq 0$ imply

Hence $u\in H_0^1(\Omega)\backslash W_\delta = V_\delta\cup\partial V_\delta$.

2.2.   Invariance of the potential wells and their outside sets

In this subsection, we show that $W_\delta$ and $V_\delta$ are both invariant under the flow of problem (1) with the subcritical initial energy.

Definition 2.2. A function $u = u(x, t)$ is called a weak solution of problem (1) on $\Omega\times [0, T)$ if $u\in L^\infty(0, T;H_0^1(\Omega))$, $u_t\in L^\infty(0, T;L^2(\Omega))$, $u(0) = u_0$ in $H_0^1(\Omega)$, $u_t(0) = u_1$ in $L^2(\Omega)$, and

for all $v\in H_0^1(\Omega)$ and a.e. $t\in (0, T)$.

(4) implies that

where $\langle \cdot, \cdot\rangle$ denotes the duality pairing between $H_0^1(\Omega)$ and its dual space $H^{-1}(\Omega)$.

Theorem 2.3. Let $u$ be a solution of problem (1) on $\Omega\times [0, T)$. Assume that $u_0\in H_0^1(\Omega)$, $u_1\in L^2(\Omega)$ and $0<E(0)<d(\delta)$.

(i) If $u_0\in W_\delta$, then $u(t)\in W_\delta$ for all $t\in [0, T)$.

(ii) If $u_0\in V_\delta$, then $u(t)\in V_\delta$ for all $t\in [0, T)$.

Proof. $(i)$ Suppose that $u(t)\notin W_\delta$ for some $0<t<T$. Then we see from $u_0\in W_\delta$ that there exists the first time $0<t_0<T$ such that $u(t_0)\in\partial W_\delta$. Thus

Consequently, we deduce from (ⅱ) in Lemma 2.1 that

Clearly, this contradicts

Hence $u(t)\in W_\delta$ for all $t\in [0, T)$.

(ⅱ) Again arguing by contradiction, there exists the first time $0<t_0<T$ such that $u(t_0)\in\partial V_\delta$. The remainder of proof is the same as that in (i), and so it is omitted here.

2.3.   Global existence and blow-up

In this subsection, we address global existence and finite time blow-up of solutions for problem (1).

Theorem 2.4. Assume that $u_0\in W_\delta$, $u_1\in L^2(\Omega)$ and $0<E(0)<d(\delta)$. Then problem (1) admits a solution $u(t)\in W_\delta$ for all $t\in [0, \infty)$.

Proof. Let $\{w_j\}^{\infty}_{j = 1}$ be an orthogonal basis of $H^1_0(\Omega)$ and an orthonormal basis of $L^2(\Omega)$. We construct

which satisfies

Multiplying (5) by $\xi_{jn}^{\prime}(t)$ and summing for $j$, we get

Integrating this with respect to $t$ from $0$ to $t$, we get

where

It follows from (6) and (7) that $E_n(0)\to E(0)$, $0<E_n(0)<d(\delta)$ and $u_n(0)\in W_\delta$ for sufficiently large $n$. By similar arguments in (i) in Theorem 2.3, we have $u_n(t)\in W_\delta$ for all $t\in [0, \infty)$. Consequently,

and

When $\|\nabla u_n(t)\|\neq 0$, in terms of (ⅰ) in Lemma 2.1 and (8), we have

When $\|\nabla u_n(t)\| = 0$, by means of (8), the above inequality remains valid.

Therefore, there exist $u$, $\chi$ and a subsequence of $\{u_n\}$, always relabeled as the same and we shall not repeat, such that, as $n\to\infty$,

According to [7,Chapter 1,Lemma 1.3], we have $\chi = |u|^{p-1}u$.

For fixed $j$, integrating (5) with respect to $t$ and taking $n\rightarrow \infty$, we get

Moreover, it is easy to see from (6) and (7) that $u(0) = u_0$ in $H^1_0(\Omega)$, $u_t(0) = u_1$ in $L^2(\Omega)$. Therefore, $u$ is a solution of problem (1) in the sense of Definition 2.2. In addition, according to (ⅰ) in Theorem 2.3, we have $u(t)\in W_\delta$ for all $t\in [0, \infty)$.

Theorem 2.5. Assume that $u_0\in V_\delta$, $u_1\in L^2(\Omega)$ and $E(0)<d(\delta)$. Then solutions of problem (1) blow up in finite time.

Proof. Let $u$ be a solution of problem (1). Next, we prove $T<\infty$. If it is not true, then $T = \infty$. We consider the auxiliary function $M(t) = \|u(t)\|^2$, $t\in[0, \infty)$. A direct calculation yields $M^{\prime}(t) = 2(u(t), u_t(t))$, and

When $0<E(0)<d(\delta)$, by virtue of $u_0\in V_\delta$ and (ii) in Theorem 2.3, we have $u(t)\in V_\delta$ for all $t\in[0, \infty)$, and so

Hence

which, together with (9), gives

When $E(0)\le0$, on account of (9), the above inequality still holds. Therefore, there exists a $t_0>0$ such that $M(t_0)>0$ and $M^\prime(t)\geq M^\prime(t_0)>0$ for a.e. $t\in [t_0, \infty)$. Then $M(t)\geq M^\prime(t_0)(t-t_0)+M(t_0)>0$ for a.e. $t\in [t_0, \infty)$.

It follows from the Cauchy-Schwarz inequality that

Consequently,

and

for a.e. $t\in[t_0, \infty)$, where $\beta = \frac{p-1}{4}$. Then there exists a $T_0$ such that

which contradicts $T = \infty$. Thus the proof of Theorem 2.5 is complete.

3.   Problem (1) with the critical initial energy

Next, in the critical case $E(0) = d(\delta)$, we discuss the invariance of the outside sets of the potential wells as well as global existence and finite time blow-up of solutions for problem (1).

Lemma 3.1. Let $u\in H_0^1(\Omega)$ and $\|\nabla u\|\neq 0$. $J_\delta(\rho u)$ is strictly increasing for $\rho\in (0, \rho_{\ast, \delta})$, strictly decreasing for $\rho\in(\rho_{\ast, \delta}, \infty)$, and attains the maximum at $\rho = \rho_{\ast, \delta}$.

Proof. It follows from definition of $J_\delta(u)$ that

Hence

Clearly, there is a $\rho_{\ast, \delta} = \rho_{\ast, \delta}(u)>0$ such that $\delta\|\nabla u\|^2 = \rho_{\ast, \delta}^{p-1}\| u\|_{p+1}^{p+1}$, i.e., $\left.\frac{{\rm d}}{{\rm d}\rho}J_\delta(\rho u)\right|_{\rho = \rho_{\ast, \delta}} = 0$. Moreover, $\frac{{\rm d}}{{\rm d}\rho}J_\delta(\rho u)>0$ for $\rho\in (0, \rho_{\ast, \delta})$, $\frac{{\rm d}}{{\rm d}\rho}J_\delta(\rho u)<0$ for $\rho\in(\rho_{\ast, \delta}, \infty)$.

Theorem 3.2. Suppose that $u_0\in W_\delta$, $u_1\in L^2(\Omega)$ and $E(0) = d(\delta)$. Then problem (1) admits $u(t)\in \overline{W_\delta} = W_\delta\cup \partial W_\delta$ for all $t\in [0, \infty)$.

Proof. We may perform this proof by considering the following two cases.

(ⅰ) $\|\nabla u_0\|\neq 0$.

Let $u_{0m} = \lambda_mu_0$, where $\lambda_m = 1- \frac{1}{m}$, $m = 2, 3, \cdots$. Next, we consider the following problem

whose energy is

From $u_0\in W_\delta$ and Lemma 2.1 it follows that

Hence

and so

Consequently,

It follows from (11) and the proof of Lemma 3.1 that there exists a $\rho_{\ast, \delta} = \rho_{\ast, \delta}(u_0)>1$ such that $J_\delta(\rho u_0)$ attains its maximum. Thus, according to Lemma 3.1, $J_\delta(\rho u_0)$ is strictly increasing on $[\lambda_m, 1]$, and $J_1(\lambda_mu_0)<J_1(u_0).$ As a result,

In terms of Theorem 2.4, for each $m$, problem (10) admits a solution $u_m(t)\in W_\delta$ for all $t\in[0, \infty)$ satisfying

for all $v\in H_0^1(\Omega)$. Consequently,

When $\|\nabla u_m(t)\|\neq 0$, in terms of (i) in Lemma 2.1 and $E_m(t) = E_m(0)<d(\delta)$, we have

When $\|\nabla u_m(t)\| = 0$, the above inequality still holds. By the compactness arguments used by the proof of Theorem 2.4, there exists a $u$ such that, as $m\rightarrow\infty$ in (12),

Moreover, it follows from (10)$_3$ that $u(0) = u_0$ in $H^1_0(\Omega)$, $u_t(0) = u_1$ in $L^2(\Omega)$. Therefore, $u$ is a solution of problem (1). By virtue of (13), we have

Hence $u(t)\in \overline{W_\delta}$ for all $t\in [0, \infty)$.

(ⅱ) $\|\nabla u_0\| = 0$.

Let $u_{1m} = \lambda_mu_1$, where $\lambda_m = 1- \frac{1}{m}$, $m = 2, 3, \cdots$. Next, we consider

In this case, due to the fact that $J_1(u_0) = 0$ and $\frac{1}{2}\|u_1\|^2 = E(0)$, we get

Thus it follows from Theorem 2.4 that, for each $m$, problem (14) admits a solution $u_m(t)\in W_\delta$ for all $t\in[0, \infty)$. The remainder of proof is similar to that in (ⅰ).

The proof of Theorem 3.2 is complete.

Theorem 3.3. Let $u$ be a solution of problem (1) on $\Omega\times [0, T)$. Assume that $u_0\in V_\delta$, $u_1\in L^2(\Omega)$, $E(0) = d(\delta)$ and $(u_0, u_1)\ge 0$. Then $u(t)\in V_\delta$ for all $t\in[0, T)$.

Proof. Suppose that $u(t)\notin V_\delta$ for some $0<t<T$. Then we see from $u_0\in V_\delta$ that there exists the first time $0<t_0<T$ such that $u(t_0)\in\partial V_\delta$. Thus,

and

By (15) and (ⅱ) in Lemma 2.1, we have

Hence

Set $M(t) = \|u(t)\|^2$, $t\in[0, T)$. A direct calculation yields $M^{\prime}(t) = 2(u(t), u_t(t))$. From $E(0) = d(\delta)$, (16) and (9), it follows that $M^{\prime\prime}(t)>0$ for $t\in[0, t_0)$. Hence $M^{\prime}(t)$ is increasing on $[0, t_0]$. We further get $M^{\prime}(t_0)>M^{\prime}(0)$, which, together with $M^{\prime}(0) = (u_0, u_1)\ge 0$, gives $M^{\prime}(t_0) = (u(t_0), u_t(t_0))>0$. Thus $\|u(t_0)\|\|u_t(t_0)\|\ge (u(t_0), u_t(t_0))>0$. Again by

we get $J_1(u(t_0))<d(\delta)$, which contradicts (17). Therefore, $u(t)\in V_\delta$ for all $t\in[0, T)$.

Theorem 3.4. Assume that $u_0\in V_\delta$, $u_1\in L^2(\Omega)$, $E(0) = d(\delta)$ and $(u_0, u_1)\ge 0$. Then the solutions of problem (1) blow up in finite time.

Proof. Let $u$ be a solution of problem (1). Next, we prove $T<\infty$. If it is not true, then $T = \infty$. From $u_0\in V_\delta$, $(u_0, u_1)\ge 0$ and Theorem 3.3, it follows that $u(t)\in V_\delta$ for all $t\in[0, \infty)$. Hence

Combining this with $E(0) = d(\delta)$, we obtain

Hence, for $M(t)$ introduced in the proof of Theorem 3.3, it follows from (9) that

for a.e. $t\in [0, \infty)$. The remainder of proof is the same as that in Theorem 2.5, and so it is omitted here.

Taking $\delta = 1$, we see that $d = d(1)$ equals to that in [14].

4.   Numerical simulations

This section is devoted to some numerical simulations of $d(\delta)$. We see from the definition of $J_\delta(u)$ that the parameter $\delta$ is introduced in order to adjust the potential energy. We have Figure 1 that shows the relation between $d(\delta)$ and $\delta$.

Next, we explore the relations among $d(\delta)$, $C_1$, $y_{\delta}$ and $p$ under $N = 2$.

We first demonstrate the relations: $d(\delta)\thicksim C_1, p$ when $\delta = 0.5$ by Figure 2; $d(\delta)\thicksim p, \delta$ when $C_1 = 2$ by Figure 3.

By fixing $\delta = 1$, $C_1 = 2$, $p = 2, 3, 4, 5$, we obtain the relation: $g_\delta(y)\thicksim y_\delta$ by Figure 4.

Moreover, we would like to do some simulations about the relations: $y_\delta\thicksim d(\delta), p$ when $\delta = 1$ by Figure 5; $y_\delta\thicksim C_1, p$ when $\delta = 0.5$ by Figure 6.

Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 31920190090).