Recent developments on the lifespan estimate for classical solutions of nonlinear wave equations
in one space dimension

When $A_{0}=0$ and $B_{0}>0$, Zhou [38] obtained the estimates of $T({\varepsilon})$;

Here we denote the fact that there are positive constants, $C_{1}$ and $C_{2}$, independent of ${\varepsilon}$ satisfying $E({\varepsilon},C_{1})\leq T({\varepsilon})\leq E({\varepsilon},C_{2})$ by $T({\varepsilon})\sim E({\varepsilon},C)$. The classification by total integral of the initial speed $g$ is caused by strong Huygens’ principle such as (17). On the other hand, when $A_{0}>0$ and $B_{0}=0$, we have

For $q=0$, the upper bound in this estimate was obtained by Zhou [39], and the lower bound is due to Kitamura, Morisawa and Takamura [15]. For $q>1$, (5) was verified by Zhou [40] for the upper bound with integer $p,q$ satisfying $p\geq 1,q\geq 0,p+q\geq 2$, and by Li,Yu and Zhou [19, 20] for the lower bound with integer $p,q$ satisfying $p+q\geq 2$ including more general but smooth terms. Note that [40] is a preprint version of Zhou [39] in which only the case of $q=0$ is considered. But it is easy to apply its argument to the case of $q>1$. The lower bound in this case is due to Kido, Sasaki, Takamatsu and Takamura [12].

Therefore the natural expectation is that

in the case where $A_{0}>0$ and $B_{0}>0$. But, surprisingly, we have the following fact.

The brief proof of (8) will appear at the end of this section. We shall call this special phenomenon by “generalized combined effect” of two nonlinearities. The original “combined effect”, which means the case of $q=0$, was first observed by Han and Zhou [5] which targeted to show the optimality of the result by Katayama [10] on the lower bound of the lifespan of classical solutions of nonlinear wave equations with a nonlinear term $u_{t}^{3}+u^{4}$ in two space dimensions including more general nonlinear terms. It is known that $T({\varepsilon})\sim\exp\left(C{\varepsilon}^{-2}\right)$ for the nonlinear term $u_{t}^{3}$ and $T({\varepsilon})=\infty$ for the nonlinear term $u^{4}$, but Katayama [10] obtained only a much worse estimate than their minimum as $T({\varepsilon})\geq c{\varepsilon}^{-18}$. Surprisingly, more than ten years later, Han and Zhou [5] showed that this result is optimal as $T({\varepsilon})\leq C{\varepsilon}^{-18}$. They also considered (3) with $q=0$ for all space dimensions $n$ bigger than 1 and obtain the upper bound of the lifespan. Its counter part, the lower bound of the lifespan, was obtained by Hidano, Wang and Yokoyama [6] for $n=2,3$. See the introduction of [6] for the precise results and references. We note that the estimate (8) with $q=0$ coincides with the lifespan estimate for the combined effect in [5, 6] if one sets $n=1$ formally. Indeed, [5] and [6] showed that

holds for $n=2,3$ provided

Later, Dai, Fang and Wang [4] improved the lower bound of lifespan for the critical case in [6]. They also show that $T({\varepsilon})<\infty$ for all $p,r>1$ in case of $n=1$, i.e. (3) with $q=0$. For the non-Euclidean setting of the results above, see Liu and Wang [21] for example, in which the application to semilinear damped wave equations is included.

Here we strongly remark that our estimate in (8) is better than that of the general theory by Li, Yu and Zhou [19, 20] in the case of (7) with integer $p,q,r\geq 2$. Because our result on the lower bound of the lifespan can be established also for the smooth terms as $u_{tt}-u_{xx}=u_{t}^{p}u^{q}+u^{r}$. We note that there are infinitely many examples of $(p,q,r)=(m,m,2m+1)$ as the inequality

holds for $m=2,3,4,\ldots$. This fact shows a possibility to improve the general theory. We also note that, even for the original combined effect of $q=0$, the integer points satisfying (10) are $(p,r)=(2,3),(3,3),(3,4)$ for $n=2$ and $(p,r)=(2,2)$ for $n=3$, but (9) with $p=r$ agrees with the case of $A_{0}=0$ and $B_{0}>0$. See Introduction of Imai, Kato, Takamura and Wakasa [7] for references on the case of $A_{0}=0$ and $B_{0}>0$. Hence one can say that only the lifespan estimates with $(p,r)=(2,3),(3,4)$ for $n=2$ are essentially in the combined effect case. If $q\neq 0$, $p$ is replaced with $p+q$ in the results above. Therefore it has less meaningful to consider (3) in higher space dimensions, $n\geq 2$, if we discuss the optimality of the general theory. In spite of this situation, Han and Zhou [5] studied

with $1<p\leq 4,q>1$ to show the blow-up part of the generalized combined effect.

Of course, some special structure of the nonlinear terms such as “null condition” guarantees the global-in-time existence. See Tartar [32], Bianchini and G. Staffilani [3], Nakamura [23], Luli, Yang and Yu [22], Zha [36, 37] for this direction. But we are interested in the optimality of the general theory.

From now on, let us make sure the fact above. If one applies the result of the general theory (2) to our problem (3) with

one has the following estimates in each cases.

• •

  When $p+q<r$,

  then, we have to set $\alpha=p+q-1$ and $\beta_{0}=r-1$ which yield that

  $$\widetilde{T}({\varepsilon})\geq\left\{\begin{array}[]{ll}C{\varepsilon}^{-(p+q-1)/2}&\mbox{in general},\\ C{\varepsilon}^{-(p+q)(p+q-1)/(p+q+1)}&\mbox{if}\ \displaystyle\int_{{\bf R}}g(x)dx=0,\\ C{\varepsilon}^{-\min\{(r-1)/2,p+q-1\}}&\begin{array}[]{l}\mbox{if $\partial_{u}^{\beta}H(0)=0$}\\ \mbox{for $p+q\leq\forall\beta\leq r-1$}.\end{array}\end{array}\right.$$

  We note that the third case is available for (11). Therefore, for $p+q\leq(r+1)/2$, we obtain that

  $$\widetilde{T}({\varepsilon})\geq c{\varepsilon}^{-(p+q-1)}$$

  whatever the value of $\displaystyle\int_{{\bf R}}g(x)dx$ is. On the other hand, for $(r+1)/2<p+q$, i.e.

  $$\frac{r-1}{2}<p+q-1,$$

  we obtain

  $$\widetilde{T}({\varepsilon})\geq\left\{\begin{array}[]{ll}C{\varepsilon}^{-(r-1)/2}&\mbox{if}\ \displaystyle\int_{{\bf R}}g(x)dx\neq 0,\\ C{\varepsilon}^{-\max\{(r-1)/2,(p+q)(p+q-1)/(p+q+1)\}}&\mbox{if}\ \displaystyle\int_{{\bf R}}g(x)dx=0.\\ \end{array}\right.$$

• •

  When $p+q\geq r$,

  then, similarly to the case above, we have to set $\alpha=r-1$, which yields that

  $$\widetilde{T}({\varepsilon})\geq\left\{\begin{array}[]{ll}C{\varepsilon}^{-(r-1)/2}&\mbox{in general},\\ C{\varepsilon}^{-r(r-1)/(r+1)}&\mbox{if}\ \displaystyle\int_{{\bf R}}g(x)dx=0,\\ C{\varepsilon}^{-\min\{\beta_{0}/2,(r-1)\}}&\mbox{if $\partial_{u}^{\beta}H(0)=0$ for $r\leq\forall\beta\leq\beta_{0}$}.\end{array}\right.$$

  We note that the third case does not hold for (11) by $\partial_{u}^{r}H(0)\neq 0$.

As a conclusion, for the special nonlinear term in (11), the result of the general theory is

and

Therefore a part of our results in (8) is larger than the lower bound of $\widetilde{T}({\varepsilon})$ in that case. If one follows the proof of Theorem 1, one can find that it is easy to see that our results on the lower bounds also hold for a special term (11) by estimating the difference of nonlinear terms from above after employing the mean value theorem. This fact indicates that we still have a possibility to improve the general theory in the sense that the optimal results in (8) should be included at least.

Inspired by Theorem 1, Takamatsu [30] has recently proved the following theorem which is an improvement of the general theory related to the generalized combined effect.

We note that (12) is stronger than both the second and the third cases in (2) because we have

and

The sharpness of (12) can be verified by setting

or

in Theorem 1.

We shall omit the proof of Theorem 2, but point out that it is initiated by that of Thereom 1, especially the key is how to split $L^{\infty}$ norm of the solution itself according to the appropriate domains. Let us see it in the next section.

The key of our success to prove the generalized combined effect case, especially the lower bound of the lifespan estimate in (8), in Theorem 1 is to handle the system of integral equations of $(u-{\varepsilon}u^{0},u_{t}-{\varepsilon}u^{0}_{t})$, where $u^{0}$ is a solution of the free wave equation with the same initial data $(u,u_{t})|_{t=0}=(f,g)$;

The basic argument is due to John [8] in which classical solutions of semilinear wave equations in three space dimensions are constructed in a weighted $L^{\infty}$ space.

Assume that

which yields that the finiteness of the propagation seed of the wave,

by standard argument on small solutions of nonlinear wave equations. For example, see Appendix in John [9]. Then, set an “interior” domain $D$ by

and define a sequence $\{(U_{j},V_{j})\}$ by

where $L(v)$ for a function $v$ of space-time variables is from Duhamel’s term defined by

This sequence $\{(U_{j},V_{j})\}$ will converge to $(u-{\varepsilon}u^{0},u_{t}-{\varepsilon}u_{t}^{0})$ in a function space

where

and $\chi_{D}$ is a characteristic function of the interior domain $D$. In view of the expression of $u^{0}$ in (13), it follows from (14) and the assumption on $g$ yield the strong Huygens’ principle,

Then we have main a priori estimates;

with some positive constant $C$ independent of ${\varepsilon}$ and $T$ which give us the key estimates for $p+q<r$;

where $\lesssim$ stands for mod constant independent of ${\varepsilon}$ and $T$. Therefore the boundedness of the sequence, $\|U_{j}\|_{1},\|V_{j}\|_{2}\lesssim{\varepsilon}^{p+q}$ for all $j\in{\bf N}$, follows from the competition between the second and third terms in the right-hand side. When $(r+1)/2<p+q$, the third term is major, so that

is the required condition. The difference to convergence of the sequence as well as those for the spatial derivative is almost the same as the boundedness above. Therefore we obtain the desired lower bound;

For the upper bound in this generalized combined effect, its proof is easy if we follow the argument by higher dimensional case by Han and Zhou [5]. In fact, assume that $f(x)\geq 0(\not\equiv 0),\ g(x)\equiv 0$. Set

Then, the equation and (15) imply that

The second term and Hölder’s inequality yield that

This is the main inequality. Next we shall make use of the first term to obtain the better estimate than the single term. For this purpose, assume further that

Then, we have that

Plugging these estimates into the first term in (18), we obtain that

which implies that

These estimates (18) and (19) will give us the desired result according to the improved Kato’s lemma by Takamura [31]. But here we show a brief feeling that the result is correct as follows. Plugging (19) into (18), we obtain that

which implies that

The difference of the lower bound of $F$ between (19) and (20) is

This quantity is larger than some constant, we will reach to the desired lifespan estimate

by standard iteration argument. Because we don’t have to cut the time interval in each step, which means that this procedure can be repeated infinitely many times in the fixed time interval. In this way, we obtain the sharp estimate of the lifespan in the generalized combined effect case.

Other cases except for the generalized combined effect are similar to those above. However, we point out that the existence part under the assumption of $\displaystyle\int_{{\bf R}}g(x)dx\neq 0$ is most difficult practically among all the proofs in choosing appropriate weight functions.

Our motivation to consider such $A$ and $B$ above comes from an initial value problem for the semilinear damped wave equations;

with the same setting on the initial data. This problem is a very important model as it has a critical decay in damping term, namely scaling invariant damping. The time-decay $(1+t)^{-1}$ is a threshold between heat-like with weaker decay and wave-like with stronger decay in the sense that the each critical exponent is the same as Fujita one and Strauss one respectively. In the critical case, the size of the constant in front of the damping term is important. There is also a threshold on the constant and this case “2” is in the domain of heat-like in one dimension. See the introduction of Kato, Takamura and Wakasa [11] for precise results and references. In fact, D’Abbicco [1] showed for $p>3$ that the energy solution of (21) exists globally-in-time, while Wakasugi [35] obtained its counter part of finite-time blow-up of the energy solution for $1<p\leq 3$. This critical exponent $3$ is Fujita one in one space dimension, so one may expect that the solution behaves like one of semilinear heat equations for which $u_{tt}$ is neglected from (21). But, this is not true.

In fact, Liouville transform $u(x,t)=(1+t)v(x,t)$ shows that (21) is equivalent to

so that all the technique for semilinear wave equations are applicable to this problem, and we have the following results on the lifepsan estimates for (21). Wakasa [33] obtained that

This is the heat-like estimate in the sense that it coincides with the corresponding semilinear heat equations for which $u_{xx}$ is neglected in (21). In the original paper [33], the condition on the initial data is missing, but it was expected to be natural as the critical exponent is Fujita one. Later, Kato, Takamura and Wakasa [11] proved that

where $b=b({\varepsilon})$ is a positive number satisfying ${\varepsilon}^{2}b\log(1+b)=1$. This is the wave-like estimate. In deed, the cases $1<p<2$ and $p=3$ are the same forms as those of 3-dimensional semillinear wave equations.

In this way, it is very important to study the weighted nonlinear terms, and it is natural to extend the results to more general weighted terms than (22). The first simple question may go to the case of $x$-decay.

First we consider the following case of spatial weights;

where $c\in{\bf R}$, in (3). This setting was first introduced by Suzuki [29] under the supervision by Prof. M. Ohta (Tokyo Univ. of Sci. Japan), but with the assumption that the initial data is of non-compact support. Later, Kubo, Osaka and Yazici [17] improved the result and Wakasa [34] finalized it. For the compactly supported case, we have the following result.

The strategy of the proof of Theorem 3 for the existence part is similar to the one of Theorem 1 based on weighted $L^{\infty}$ estimates of the solution, so we don’t comment about it here. For the blow-up part, the iteration argument of the point-wise estimate is employed. The functional method like the proof of Theorem 1 cannot be applied to this case due to the effect of the weight.

After the work [14], one may study the counter case of

where $c\in{\bf R}$, in (3) with $q=0$. For this equation, one may expect that the result is not so interesting as $|x|\sim t$ in dealing with the nonlinear term $|u_{t}|^{p}$. But the result is

We note that Zhou [39] established the blow-up part with $c=-1$, namely non-weighted case. The proof of Theorem 4 is similar to the one of Theorem 3, so we don’t comment about it here also. The main difference between (25) and (28) is a possibility to obtain the global-in-time existence for (28) while there is no such a situation due to the effect of the origin in space for (25).

According to the result on (21) and Theorems 3 and 4, one may expect that it is sufficient to study the weights in the nonlinear terms by powers of $(1+t)$ or $\langle x\rangle$ for the purpose to extend the general theory, but we have to take into account of “characteristic weights” in handling nonlinear wave equations. Let’s see it in the next subsection.

It is well-known that the wave propagates along with the characteristic directions, so that it is not sufficient to study the weights of powers by $(1+t)$ or $\langle x\rangle$ only. Therefore, as a breakthrough to this direction, we set

where $a,b\in{\bf R}$, in (3). Then, we obtained the following result.