The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

1.   Introduction

In this paper, we consider the initial value problems;

where $p > 1$, and $T > 0$. We assume that $f$ and $g$ are given smooth functions of compact support and a parameter $\varepsilon > 0$ is "small enough". We are interested in the lifespan $T(\varepsilon)$, the maximal existence time, of classical solutions of (1.1). Our result is that there exists positive constants $C_1, \; C_2$ independent of $\varepsilon$ such that $T(\varepsilon)$ satisfies

We note that, even if $|u_x|^p$ is replaced with $|u_t|^p$, (1.2) still holds. Such a result is due to Zhou ^[18] for the upper bound of $T(\varepsilon)$, and Kitamura, Morisawa and Takamura ^[7] for the lower bound of $T(\varepsilon)$.

As model equations to ensure the optimality of the general theory for nonlinear wave equations by Li, Yu and Zhou ^[8,9], the nonlinear term $|u_t|^p$ is sufficient to be studied except for the "combined effect" case. See Morisawa, Sasaki and Takamura ^[10,11] and Kido, Sasaki, Takamatsu and Takamura ^[6] for this direction with a possibility to improve the general theory. See also Takamatsu ^[17] for such an improvement. But it is quite meaningful to deal with also $|u_x|^p$ because their proofs are technically different from each others. Moreover, there is no result on its blow-up boundary due to lack of the monotonicity of the solution, while the one for $|u_t|^p$ is well-studied by Sasaki ^[14,15], and Ishiwata and Sasaki ^[2,3]. See Remark 2.1 below. It is also remarkable that it can be studied if the nonlinear term has a special form of both $u_t$ and $u_x$. See Sasaki ^[16] for this direction. Our research may help us to study the blow-up boundary for the equation in (1.1) near future.

This paper is organized as follows. In the next section, the preliminaries are introduced. Moreover, (1.2) is divided into two theorems. Section 3 is devoted to the proof of the existence part, the lower bound of $T(\varepsilon)$, of (1.2). The main strategy is the iteration method for the system of integral equations for $(u, u_x)$, which is essentially due to Kitamura, Morisawa and Takamura ^[7]. They employed it for the system of integral equations for $(u, u_t)$ to construct a classical solution of the wave equation with nonlinear term $|u_t|^p$, which is originally introduced by John ^[4]. In the Section 4, following Rammaha ^[12,13], we prove the blow-up part, the upper bound of $T(\varepsilon)$, of (1.2). We note that the method to be reduced to $u$-closed integral inequality by Zhou ^[18] for the nonlinear term $|u_t|^p$ cannot be applicable to (1.1) because a time delay appears in the reduced ordinary differential inequality. Rammaha ^[12,13] overcomes this difficulty by employing weighted functionals along with the characteristic direction in studying two or three dimensional wave equations with nonlinear terms of spatial derivatives.

2.   Preliminaries and main results

Throughout this paper, we assume that the initial data $(f, g)\in C_0^2(\mathbb R)\times C^1_0(\mathbb R)$ satisfies

Let $u$ be a classical solution of (1.1) in the time interval $[0, T]$. Then the support condition of the initial data, (2.1), implies that

For example, see Appendix of John ^[5] for this fact.

It is well-known that $u$ satisfies the integral equation

where $u^0$ is a solution of the free wave equation with the same initial data

and a linear integral operator $L$ for a function $v = v(x, t)$ in Duhamel's term is defined by

Then, one can apply the time-derivative to (2.3) to obtain

and

where $L'$ for a function $v = v(x, t)$ is defined by

Therefore, $u_t$ is expressed by $u_x$. On the other hand, applying the space-derivative to (2.3), we have

and

where $\overline{L'}$ for a function $v = v(x, t)$ is defined by

Remark 2.1. In view of (2.9), it is almost impossible to obtain a point-wise positivity of $u_x$. This fact prevents us from studying its blow-up boundary as stated in Introduction.

Moreover, applying one more time-derivative to (2.9) yields that

and

Similarly, we have that

Therefore, $u_{tt}$ is expressed by $u_x, u_{xt}$ and so is $u_{xx}$.

First, we note the following fact.

Proposition 2.1. Assume that $(f, g)\in C_0^2(\mathbb R)\times C_0^1(\mathbb R)$. Let $w$ be a $C^1$ solution of (2.9) in which $u_x$ is replaced with $w$. Then,

is a classical solution of (1.1) in $\mathbb R\times[0, T]$.

Proof. This is easy along with the computations above in this section. □

Our results are divided into the following two theorems.

Theorem 2.1. Assume (2.1). Then, there exists a positive constant $\varepsilon_1 = \varepsilon_1(f, g, p, R) > 0$ such that a classical solution $u\in C^2(\mathbb R\times[0, T])$ of (1.1) exists as far as $T$ satisfies

where $0 < \varepsilon\leq \varepsilon_1$, and $C_1$ is a positive constant independent of $\varepsilon$.

Theorem 2.2. Assume (2.1) and

Then, there exists a positive constant $\varepsilon_2 = \varepsilon_2(f, p, R) > 0$ such that any classical solution of (1.1) in the time interval $[0, T]$ cannot exist as far as $T$ satisfies

where $0 < \varepsilon\leq \varepsilon_2$, and $C_2$ is a positive constant independent of $\varepsilon$.

The proofs of above theorems are given in following sections.

3.   Proof of Theorem 2.1

According to Proposition 2.1, we shall construct a $C^1$ solution of (2.9) in which $u_x = w$ is the unknown function. Let $\{w_j\}_{j\in \mathbb N}$ be a sequence of $C^1(\mathbb R\times[0, T])$ defined by

Then, in view of (2.12), $(w_j)_t$ has to satisfy

so that the functional space in which $\{w_j\}$ converges is

which is equipped with a norm

where

We note that (2.9) implies that

The following lemma provides us a priori estimate.

Proposition 3.1. Let $w\in C(\mathbb R\times[0, T])$ and supp\ $w\subset\{(x, t)\in \mathbb R\times[0, T]:|x|\leq t+R\}$. Then, the following a priori estimate holds:

where $C$ is a positive constant independent of $T$ and $\varepsilon$.

Proof. The proof of Proposition 3.1 is completely same as the one of Proposition 3.1 in Morisawa, Sasaki and Takamura ^[10]. □

Let us continue to prove Theorem 2.1. Set

The convergence of the sequence {w_j}

First we note that $\|w_1\| \leq M \varepsilon$ by (2.10). (3.1) and (3.3) yield that

because it is trivial that

Therefore, the boundedness of $\{w_j\}$, i.e.,

follows from

Assuming (3.5), one can estimate $\|w_{j+1}-w_j\|$ as follows:

Therefore, the convergence of $\{w_j\}$ follows from

provided (3.5) and

are fulfilled.

The convergence of the sequence {(w_j)_t}

First we note that $\|(w_1)_t\| \leq M \varepsilon$ by (2.13). Assume that (3.5) and (3.6) are fulfilled. Since (3.2) and (3.3) yield that

the boundedness of $\{(w_j)_t\}$, i.e.,

follows as long as it is fulfilled that

Assuming (3.7), one can estimate $\{(w_{j+1})_t-(w_j)_t\}$ as follows. Noting that

we have

Therefore, we obtain the convergence of $\{(w_j)_t\}$ provided

Continuation of the proof

The convergence of the sequence $\{w_j\}$ to $w$ in the closed subspace of $X$ satisfying $\|w\|, \|w_t\| \leq 2M \varepsilon$ is established by (3.5)–(3.8), which follow from

Therefore, the statement of Theorem 2.1 is established with

because $R\leq (2C_1)^{-1} \varepsilon^{-(p-1)}$ holds for $0 < \varepsilon\leq \varepsilon_1$. □

4.   Proof of Theorem 2.2

Following Rammaha ^[12], set

where $R_0$ is some fixed point with $0 < R_0 < R$. We may assume that there exists a point $x_0\in(R_0, R)$ such that $f(x_0) > 0$ because of the assumption (2.15) and of a possible shift of $x$-variable.

Then it follows that

where

By virtue of (2.15) and (4.1), we have that

where

Integrating this inequality in [$0, t$] twice and noting that $H'(0) = H(0) = 0$, we have

On the other hand, $F(t)$ can be rewritten as

From now on, we assume that

Then, inverting the order on $(y, x)$-integral, for $0 \leq s \leq t-R_1$, we have that

Similarly, for $t-R_1\leq s \leq t$, we also have that

Hence, we obtain that

Therefore, it follows from (4.3) and

that

In this way, (2.15), (4.1) and the estimate of $F(t)$ above yield that

Moreover, it follows from (2.2), integration by parts and Hölder's inequality that

where

Hence, we obtain that

Therefore, the argument in Rammaha ^[12] can be applied to (4.2) and (4.4) to ensure that there exist positive constants $\varepsilon_2 = \varepsilon_2(f, p, R)$ and $C_2$ independent of $\varepsilon$ such that a contradiction appears provided

holds for $0 < \varepsilon \leq \varepsilon_2$. The proof is now completed. □

5.   Conclusions

Our theorems could be extended to higher dimensional case basically along with our method, but we have to assume that the solution is radially symmetric at least, which is closely related to "Glassey's conjecture" for nonlinear term of $|u_t|^p$. See Hidano, Wang and Yokoyama ^[1] for this direction.

Use of AI tools declaration

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

Acknowledgments

The first author was partially supported by the Grant-in-Aid for Young Scientists (No. 18K13447), Japan Society for the Promotion of Science. The third author was partially supported by the Grant-in-Aid for Scientific Research (A) (No. 22H00097), Japan Society for the Promotion of Science.

Conflict of interest

The authors declare no competing interests in this paper.