Movement of time-delayed hot spots in Euclidean space

Abstract

We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution. Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give $L^p$-$L^q$ estimates of the solution.

Appendix

1.1 Proofs of preliminary estimates

Proof of Lemma 3.1

Let us give a proof for even dimensional cases. The other cases go parallel.

Changing the variable as $y=x+tz$ with $z \in B^n$, we have

$$\begin{aligned} \int _{B_t^n(x)} \frac{1}{\sqrt{t^2-r^2}} g(y) dy = t^{n-1} \int _{B^n} \frac{1}{\sqrt{1-\left| z \right| ^2}} g(x+tz) dz. \end{aligned}$$

When we estimate the function

$$\begin{aligned} {\varvec{W}}_n(t)g(x)&=2c_n \sum _{j=0}^{(n-2)/2} \frac{1}{8^j j !}\left( \frac{1}{t} \frac{\partial }{\partial t} \right) ^{(n-2)/2-j} \left( t^{n-1} \int _{B^n} \frac{1}{\sqrt{1-\left| z \right| ^2}} g(x+tz) dz \right) , \end{aligned}$$

the worst term with respect to the growth order of t is given by $j= (n-2)/2$. We can bound it above by $C(1+t)^{n-1} \Vert g \Vert _{L^\infty }$. Furthermore, we can bound the term of $j =0$ above by $C (1+t)^{n/2} \Vert g \Vert _{W^{n/2-1 ,\infty }}$. The other terms are bounded above by these two quantities (up to a constant multiple). Hence we obtain the estimate for ${\varvec{W}}_n(t)g(x)$.

From Proposition 2.8, we have

$$\begin{aligned} \widehat{{\varvec{W}}}_n(t) f(x) = \frac{c_n t^n}{2^{\frac{3n-2}{2}}\left( \frac{n}{2} \right) !} \int _{B^n} \frac{1}{\sqrt{1-\left| z \right| ^2}} f(x+tz) dz, \end{aligned}$$

which implies the estimate for $\widehat{{\varvec{W}}}_n(t) f(x)$. Also, we have

$$\begin{aligned} \widetilde{{\varvec{W}}}_n(t;f,g) (x) = \frac{1}{2} {\varvec{W}}_n (t) f(x) + {\varvec{W}}_n (t) g(x) + \widehat{{\varvec{W}}}_n(t) f(x) + \frac{\partial }{\partial t} {\varvec{W}}_n(t)f(x). \end{aligned}$$

Combining the above estimates for ${\varvec{W}}_n(t)g(x)$ and $\widehat{{\varvec{W}}}_n(t) f(x)$, we obtain the conclusion.

$\square$

Proof of Lemma 3.2

Using Remark 2.5, integration by parts implies the identities. $\square$

Proof of Lemma 3.3

If ${{\mathrm{dist}}}(x, CS (h) ) \le t-d_h$ and $t\ge d_h$, then the intersection $S^{n-1}_t(x) \cap CS(h)$ is a null set with respect to the $(n-1)$-dimensional spherical Lebesgue measure. Hence Lemma 3.2 guarantees the conclusion.

Proof of Lemma 3.4

• 1. We give a proof for even dimensional cases. The other cases go parallel. We remark that, from the definition of $c_n$ (2.3), we have

  $$\begin{aligned} E_n (r,t) = \frac{e^{-t/2}}{2^{3n/2+1}\pi ^{n/2}} k_{\frac{n}{2}+1}\left( \frac{1}{2}\sqrt{t^2-r^2} \right) . \end{aligned}$$

  From Theorem 2.4, we have

  $$\begin{aligned}&k_{\frac{n}{2}+1} \left( \frac{1}{2} \sqrt{t^2 -\varphi (t)^2} \right) \\&= \frac{1}{2} \left( \frac{2}{\sqrt{t^2 -\varphi (t)^2}} \right) ^{n/2+1} \exp \left( \frac{\sqrt{t^2 -\varphi (t)^2}}{2} \right) \left( 1+ O \left( \frac{1}{\sqrt{t^2 -\varphi (t)^2}} \right) \right) \end{aligned}$$

  as t goes to infinity, which implies the conclusion.

• 2. Applying the fact (5.11) to the first assertion, we obtain the conclusion.

• 3. Applying the fact (5.12) to the second assertion, we obtain the conclusion.

Proof of Lemma 3.5

Using Remark 2.5, integration by parts implies the identities. $\square$

Proof of Lemma 3.6

• 1. This is a direct consequence of (5.8) in Lemma 5.2.

• 2. We give a proof for even dimensional cases. The other cases go parallel. From the asymptotic expansion in Theorem 2.4, we have

  $$\begin{aligned}&e^{-t/2} \left( t k_{\frac{n}{2}+2} \left( \frac{1}{2} \sqrt{t^2 -r^2} \right) -2 k_{\frac{n}{2}+1} \left( \frac{1}{2} \sqrt{t^2 -r^2} \right) \right) \\&=\frac{2^{n/2+1}}{\left( t^2 -r^2 \right) ^{n/4+1/2}} \exp \left( \frac{-t+\sqrt{t^2 -r^2}}{2} \right) \\&\quad \times \left[ \frac{t}{\sqrt{t^2 -r^2}} \left( 1+ O \left( \frac{1}{\sqrt{t^2-r^2}} \right) \right) - \left( 1+ O \left( \frac{1}{\sqrt{t^2-r^2}} \right) \right) \right] . \end{aligned}$$

  Using the facts (5.11) and (5.12), the above expansion coincides with

  $$\begin{aligned} 2^{n/2+1} t^{-n/2-1} \left( O \left( \frac{1}{t} \right) + O \left( \frac{\psi (t)^2}{t^2} \right) \right) , \end{aligned}$$

  which implies the conclusion.

• 3. Using the asymptotic expansion in Theorem 2.4, in the same manner as in the second assertion, we obtain the conclusion.

$\square$

1.2 Frequently used Taylor’s expansions

Let us list up frequently used Taylor’s expansions:

• As s tends to zero, we have

  $$\begin{aligned} (1-s^2)^\alpha = 1 - \alpha s^2 + \frac{\alpha (\alpha -1)}{2} s^4 + O \left( s^6 \right) . \end{aligned}$$

  (5.10)

• If a function $\varphi (t)$ is of small order of t as t goes to infinity, then we have

  $$\begin{aligned} \begin{aligned}&\left( t^2-\varphi (t)^2 \right) ^\alpha \\&\quad = t^{2\alpha } \left( 1 -\alpha \left( \frac{\varphi (t)}{t} \right) ^2 + \frac{\alpha (\alpha -1)}{2} \left( \frac{\varphi (t)}{t} \right) ^4 + O\left( \left( \frac{\varphi (t)}{t} \right) ^6 \right) \right) \end{aligned} \end{aligned}$$

  (5.11)

  as t goes to infinity.

• If a function $\varphi (t)$ is of small order of $\sqrt{t}$ as t goes to infinity, then, as t goes to infinity, we have the following expansions:

  $$\begin{aligned}&\exp \left( \frac{-t+\sqrt{t^2-\varphi (t)^2}}{2} \right) = 1 - \frac{\varphi (t)^2}{4t} + O\left( \frac{\varphi (t)^4}{t^2} \right) , \end{aligned}$$

  (5.12)
  $$\begin{aligned}&\exp \left( \frac{\varphi (t)^2}{4t}+\frac{-t+\sqrt{t^2-\varphi (t)^2}}{2} \right) = 1 + \frac{1}{t}O\left( \frac{\varphi (t)^4}{t^2} \right) . \end{aligned}$$

  (5.13)

1.3 Properties of modified Bessel functions

In this section, we collect some properties of the modified Bessel functions

$$\begin{aligned} I_{\nu }(s)=\sum _{j=0}^\infty \frac{1}{j ! \Gamma (j+\nu +1)} \left( \frac{s}{2}\right) ^{2j+\nu } \end{aligned}$$

(5.14)

used in this paper from [19]:

• For a positive constant a, we have

  $$\begin{aligned} \int _{-a}^a\frac{e^{s/2}}{\sqrt{a^2-s^2}}ds=\pi I_0\left( \frac{a}{2}\right) . \end{aligned}$$

  (5.15)

• Direct computation shows the following recursion:

  $$\begin{aligned} I_0^{\prime } (s)=I_1(s), \ I_1^{\prime }(s)=I_0(s)-\frac{1}{s}I_1(s), \ \frac{1}{s} \frac{d}{ds}\left( \frac{I_\ell (s)}{s^\ell } \right) =\frac{I_{\ell +1}(s)}{s^{\ell +1}}. \end{aligned}$$

  (5.16)

• The modified Bessel function $I_\nu (s)$ has the expansion

  $$\begin{aligned} \begin{aligned} I_\nu (s)&=\frac{e^s}{\sqrt{2\pi s}} \left( 1-\frac{(\nu -1/2)(\nu +1/2)}{2s} \right. \\&\quad \left. +\,\frac{(\nu -1/2)(\nu -3/2)(\nu +3/2)(\nu +1/2)}{2!2^2s^2} -\cdots \right. \\&\quad \left. +\,(-1)^\ell \frac{1}{\ell !2^\ell s^\ell } \prod _{j=1}^{\ell } \left( \nu -(j-1/2) \right) \left( \nu +(j-1/2)\right) +O\left( \frac{1}{s^{\ell +1}} \right) \right) \end{aligned} \end{aligned}$$

  (5.17)

  as s goes to infinity.