On the defocusing semilinear wave equations in three space dimension with small power

In this paper, we continue our study on the global pointwise behaviors for solutions to the energy sub-critical defocusing semilinear wave equations

in $\mathbb{R}^{1+3}$ with small power $1<p\leq 2$.

The existence of global solutions in energy space is well known since the work [5] of Ginibre-Velo for the energy sub-critical case $1<p<5$. The long time dynamics of these global solutions concerns mainly two types of questions: The first type is the problem of scattering, namely comparing the nonlinear solutions with linear solutions as time goes to infinity. A natural choice for the linear solution is the associated linear wave. Since linear wave decays in time in space dimension $d\geq 2$, the nonlinear solution approaches to linear wave in certain sense for sufficiently large power $p$. This was shown by Strauss in [14] for the super-conformal case $3\leq p<5$ in space dimension three. Extensions could be found for example in [1], [6], [9], [13], [17]. The latest work [18] of the second author shows that the solution scatters to linear wave in energy space for $2.3542<p<5$ in $\mathbb{R}^{1+3}$. However the precise asymptotics of the solutions remains unclear for small power $p$. One of the difficulties is that the equation degenerates to linear Klein-Gordon equation when $p$ approaches to the end point $1$.

Another question is to investigate the asymptotic behaviors of the solutions in the pointwise sense, which is plausible in lower dimensions $d\leq 3$. The obstruction in higher dimension is that taking sufficiently many derivatives for energy sub-critical equations is not possible for general $p$. Even for energy critical equations, the global regularity result only holds in space dimension $d\leq 9$ (see for example [10]). The energy super-critical case remains completely open (see recent breakthrough in [4] for the blowing up result for the defocusing energy super-critical nonlinear Schrödinger equations). The scattering result of Strauss is based on the time decay $t^{\epsilon-1}$ of the solution, which has been improved in [2] by using conformal compactification method. However this method only works for super-conformal case when $p\geq 3$. To study the asymptotic behaviors of the solutions with sub-conformal power $p<3$, Pecher in [13], [7] observed that the potential energy decays in time with a weaker decay rate (comparing to $t^{-2}$ for the super-conformal case). This allows him to obtain polynomial decay in time of the solutions when $p>\frac{1+\sqrt{d^{2}+4}}{2}$ in space dimension $d=2$ and $3$.

However, Pecher’s observation was based on the conformal symmetry of Minkowski spacetime, that is, the time decay of the potential energy is derived by using the conformal Killing vector field as multiplier. As we have seen, the smaller power $p$ leads to the slower decay of the nonlinearity, hence making the analysis more difficult. More precisely, the power $p$ is closely related to the weights in the multipliers. For the super-conformal case $p\geq\frac{d+3}{d-1}$, one can use the conformal Killing vector field with weights $t^{2}$, which leads to the time decay $t^{-2}$ of the potential energy. For the end point case $p=1$, so far as we know, there is no similar weighted energy estimates for linear Klein-Gordon equations, without appealing to higher order derivatives. This suggests that it is important to use multipliers with proper weights depending on the power $p$ in order to reveal the asymptotic behaviors of the solutions. As the power $p$ varies continuously, it in particular calls for a family of weighted vector fields which are consistent with the structure of the equations.

The robust new vector field method originally introduced by Dafermos and Rodnianski in [3] provides such family of multipliers $r^{\gamma}(\partial_{t}+\partial_{r})$ with $0\leq\gamma\leq 2$. Combined with the well known integrated local energy estimates (see for example [12]), a pigeon-hole argument leads to the improved time decay of the potential energy. This enables the second author in [18] to show that the solution decays at least $t^{-\frac{1}{3}}$ for $2<p<5$ in three space dimension. The lower bound $p>2$ arises due to the fact that the pigeon-hole argument works only for $\gamma>1$. However the multipliers can be used for all $0\leq\gamma\leq 2$ and hence uniform boundedness (or spatial decay) of the solution holds for $\frac{3}{2}<p\leq 2$. We see that there is a gap regarding the time decay of the solutions between the cases $p>2$ and $p\leq 2$. Moreover this method fails in lower dimensions $d\leq 2$.

The philosophy that suitable weighted multiplier yields the time decay of the potential energy inspires us to introduce new non-spherically symmetric weighted vector fields as multipliers in [16],[15] to show the polynomial decay in time of the solution for all $p>1$ in space dimension one and two. This in particular extends the result of Lindblad and Tao in [11], where an averaged decay of the solution was shown for the defocusing semilinear wave equation in $\mathbb{R}^{1+1}$.

The aim of this paper is to investigate the asymptotic behaviors of the solutions in three space dimension with small power $p\leq 2$. Again, the essential idea is to introduce some new weighted vector fields as multipliers, which are partially inspired by our previous work [15] in space dimension two. This allows us to obtain potential energy decay for all $1<p<5$ and time decay of the solutions for all $p>\sqrt{2}$, and hence filling the gap left in [18].

To state our main theorem, for some constant $\gamma$ and integer $k$, define the weighted energy norm of the initial data

We prove in this paper that

We give several remarks.

Now let’s review the main ideas for studying the asymptotic behaviors for defocusing semilinear wave equations. The early pioneering works (for example [6], [13],[7]) relied on the following time decay of the potential energy

obtained by using the conformal Killing vector field $t^{2}\partial_{t}+r^{2}\partial_{r}$ ($r=|x|$) as multiplier. The new vector field method of Dafermos and Rodnianski can improve the above time decay in the following way: First the $r$-weighted energy estimates derived by using the vector fields $r^{\gamma}(\partial_{t}+\partial_{r})$ with $0\leq\gamma\leq 2$ as multipliers show that

However in order to obtain time decay of the potential energy, one then needs to combine the $r$-weighted energy estimate with the integrated local energy estimates. A pigeon-hole argument then leads to the energy flux decay through the outgoing null hypersurface $\mathcal{H}_{u}$ (constant $u$ hypersurface with $u=\frac{t-r}{2}$)

Integrating in $u$, we end up with a weighted spacetime bound

by assuming $\gamma>1$, which requires $p>2$. In view of the above $r$-weighted energy estimate, one then derives the time weighted spacetime bound

This improves the above time decay (2) for the sub-conformal case $p<3$ and is sufficient to conclude the time decay estimates of the solutions for $p>2$ in [18].

However, the above new vector field method works only in space dimension $d\geq 3$, due to the lack of integrated local energy estimates in lower dimensions. To improve the asymptotic decay estimates of the solution in space dimension two, we in [15] introduced non-spherically symmetric vector fields

as multipliers applied to regions bounded by the null hyperplane $\{t=x_{1}\}$ and the initial hypersurface. The advantage of using such non-spherically symmetric vector fields is that we can make use of the reflection symmetry $x_{1}\rightarrow-x_{1}$ as well as rotation symmetries. This enables us to derive the time decay of the potential energy

in space dimension two. This decay rate is consistent with that in higher dimensions. However we emphasize here that this method works for all $p>1$ while a lower bound $p>1+\frac{2}{d-1}$ was required in higher dimensions (see [17]).

For the three dimensional case when $p\leq 2$, we observe that it is not likely to use multipliers with weights higher than $t$. Although the vector fields $r^{\gamma}(\partial_{t}+\partial_{r})$ can be used for all $0\leq\gamma\leq 2$, it does not contain weights in time. In particular these vector fields can only lead to the spatial decay of the solution instead of time decay. Inspired by our previous work in space dimension two, we introduce new non-spherical symmetric weighted vector fields

as multipliers. By applying this vector field to the region bounded by the null hyperplane $\{t=x_{1}-1\}$, the initial hypersurface and the constant $t$-hypersurface, we can derive that

Using the symmetry $x_{1}\rightarrow-x_{1}$, we then conclude the time decay of the potential energy

for all $1<p\leq 2$. In view of this, we believe that such method can also lead to the time decay of potential energy in higher dimensions for the full energy sub-critical case.

Regarding the pointwise decay estimates for the solution, we rely on the representation formula. To control the nonlinearity, we apply the above vector fields to the region bounded by the backward light cone $\mathcal{N}^{-}(q)$ emanating from the point $q\in\mathbb{R}^{1+3}$. To simplify the analysis, we can assume that $q=(t_{0},x_{0})$, $x_{0}=(r_{0}=|x_{0}|,0,0)$. This gives the weighted energy estimate

which is sufficient to conclude the pointwise estimate for the solution in the interior region $|x|\leq t$. The better decay estimates in the exterior region are based on the weighted energy estimate

obtained by using the Lorentz rotation vector field $x_{1}\partial_{t}+t\partial_{1}$ as multiplier.

Acknowledgments. S. Yang is partially supported by NSFC-11701017.

Additional to the Cartesian coordinates $(t,x)=(t,x_{1},x_{2},x_{3})$ for the Minkowski spacetime $\mathbb{R}^{1+3}$, we will also use the null frame

with $\partial_{i}$ be shorthand for $\partial_{x_{i}}$. For fixed point $q=(t_{0},x_{0})\in\mathbb{R}^{1+3}$, let $(\tilde{t},\tilde{x})$ be the new Cartesian coordinates centered at $q$. More precisely, define

By translation invariance, note that

Here $\nabla$ is the spatial gradient while $\tilde{\nabla}$ is the associated one centered at $q$.

For vector fields $X,Y$ in $\mathbb{R}^{1+3}$, we use the geometric notation $\langle X,Y\rangle$ meaning the inner product of these two vector fields under the flat Minkowski metric $m_{\mu\nu}$ with non-vanishing components

Raising and lowering indices are carried out with respect to this metric in the sequel.

As the wave equation is time reversible, without loss of generality, we only consider the case in the future $t\geq 0$. For $q=(t_{0},x_{0})\in\mathbb{R}^{1+3}$, let $\mathcal{J}^{-}(q)$ be the causal past

The boundary contains the past null cone $\mathcal{N}^{-}(q)$ emanating from $q$, that is,

For $r>0$, we use $\mathcal{B}_{q}(r)$ to denote the spatial ball centered at $q=(t_{0},x_{0})$ with radius $r$. More precisely

The boundary of $\mathcal{B}_{q}(r)$ is the 2-sphere $\mathcal{S}_{q}(r)$.

Finally to avoid too many constants, we make a convention that $A\lesssim B$ means there exists a constant $C$, depending only on $p$ and the small constant $0<\epsilon<\frac{1}{2}$ such that $A\leq CB$.

Following the framework established early in [14] and developed in [13], [18], potential energy decay is of crucial importance to deduce the asymptotic long time behavior for the solution. We begin with the following weighted potential energy estimate through backward light cones.