MOVEMENT OF TIME-DELAYED HOT SPOTS IN EUCLIDEAN SPACE FOR A DEGENERATE INITIAL STATE

. We consider the Cauchy problem for the damped wave equation under the initial state that the sum of an initial position and an initial velocity vanishes. When the initial position is non-zero, non-negative and compactly supported, we study the large time behavior of the spatial null, critical, max- imum and minimum sets of the solution. The behavior of each set is totally diﬀerent from that of the corresponding set under the initial state that the sum of an initial position and an initial velocity is non-zero and non-negative. The spatial null set includes a smooth hypersurface homeomorphic to a sphere after a large enough time. The spatial critical set has at least three points after a large enough time. The set of spatial maximum points escapes from the convex hull of the support of the initial position. The set of spatial minimum points consists of one point after a large enough time, and the unique spatial minimum point converges to the centroid of the initial position at time inﬁnity.


(Communicated by Irena Lasiecka)
Abstract. We consider the Cauchy problem for the damped wave equation under the initial state that the sum of an initial position and an initial velocity vanishes. When the initial position is non-zero, non-negative and compactly supported, we study the large time behavior of the spatial null, critical, maximum and minimum sets of the solution. The behavior of each set is totally different from that of the corresponding set under the initial state that the sum of an initial position and an initial velocity is non-zero and non-negative.
The spatial null set includes a smooth hypersurface homeomorphic to a sphere after a large enough time. The spatial critical set has at least three points after a large enough time. The set of spatial maximum points escapes from the convex hull of the support of the initial position. The set of spatial minimum points consists of one point after a large enough time, and the unique spatial minimum point converges to the centroid of the initial position at time infinity.

1.
Introduction. Let f and g be real-valued smooth functions defined on R n . We consider the damped wave equation with initial data (f, g), (1.1) The damped wave equation describes several phenomena. One is the propagation of the wave with friction. Another is the diffusion of the heat with finite propagation speed. In this paper, we study the equation (1.1) in the latter sense. We refer to [6] for the deriving process as the heat equation with finite propagation speed (see also [12,Introduction]).

SHIGEHIRO SAKATA AND YUTA WAKASUGI
It is one of the most fundamental properties of the damped wave equation that, as t goes to infinity, the unique classical solution of (1.1) approaches to that of the corresponding heat equation, P n (t)(f +g)(x) = 1 (4πt) n/2 R n exp − |x − y| 2 4t (f +g)(y)dy, x ∈ R n , t > 0, (1.2) if f + g does not vanish. Such a property is called the diffusion phenomenon and has been studied by several researchers [2,3,8,7,9,11,12,13]. We also refer to [4] for the diffusion phenomenon in an exterior domain. In order to compare the damped wave and heat equations more deeply, in [12], the authors investigated the shape of the solution of (1.1) when f and g are compactly supported, and f + g is non-zero and non-negative. In particular, the authors investigated spatial maximum points of the solution of (1.1). Spatial maximum points of the solution of the (usual) heat equation (with infinite propagation speed) are called hot spots. Since (1.1) is the heat equation with finite propagation speed, we call spatial maximum points of the solution of (1.1) time-delayed hot spots. Also, we call spatial minimum points of the solution of (1.1) time-delayed cold spots.
This paper is a continuation of [12]. We investigate the shape of the solution of (1.1) when f is non-zero, non-negative and compactly supported, and f + g entirely vanishes. In particular, we study the large time behavior of the spatial null, critical, maximum and minimum sets of the solution. The previous study [12] was under the case f + g ≥ 0 and not under the case f + g = 0. The behavior of each set is totally different from the previous one.
In order to state our main results concisely, let us prepare necessary notation (see also Notation 1): for a compactly supported function φ, we denote by CS(φ) and d φ the convex hull and diameter of the support of φ, respectively; the parallel body of a convex set K of radius ρ is defined by K + ρB n = {x + ρy |x ∈ K, y ∈ B n } = {x ∈ R n | dist(x, K) ≤ ρ}. (1.3) For the spatial null set of the solution, we study its location by the following procedure: Step I.1: After a large time, the solution is negative on the parallel body Step I.2: Let ϕ(t) be of small order of t. After a large time, the solution is positive on the region sandwiched between CS(f ) + √ 2nt and CS(f ) + ϕ(t) (Proposition 3).
Step I.3: After a large time, the solution is strictly increasing in each outer unit normal direction of CS(f ) on the region sandwiched between CS(f ) Proposition 4). Thus, after a large time, for each outer unit normal direction of CS(f ), the solution has a unique spatial zero in the region sandwiched between CS(f ) + ( √ 2nt − d f )B n and CS(f ) + √ 2ntB n . The implicit function theorem implies that the spatial null set contained in the sandwiched region is a smooth hypersurface homeomorphic to the (n − 1)-dimensional sphere (Theorem 3.2).
For the spatial critical set of the solution, we study its location by the following procedure: Step II.1: After a large time, the solution is strictly increasing in each outer unit normal direction of CS(f ) on the region sandwiched between CS(f ) and CS(f ) + ( (2n + 4)t − d f )B n (Proposition 5).
Step II.2: Let ϕ(t) be of small order of t. After a large time, the solution is strictly decreasing in each outer unit normal direction of CS(f ) on the region sandwiched between CS(f )+ (2n + 4)tB n and CS(f )+ϕ(t)B n (Proposition 6).
Step II.3: After a large time, the solution is strictly concave in each outer unit normal direction of CS(f ) on the region sandwiched between CS(f ) + ( (2n + 4)t − d f )B n and CS(f ) + (2n + 4)tB n (Proposition 7).
Thus, after a large time, the solution has spatial critical points in CS(f ) and the region sandwiched between CS(f )+( (2n + 4)t−d f )B n and CS(f )+ (2n + 4)tB n . The implicit function theorem implies that the solution has at least two spatial critical points in the sandwiched region (Theorem 4.2). Using the technique in [1, Theorem 1], we also show that spatial critical points of the solution contained in CS(f ) converge to the centroid of f as t goes to infinity (Theorem 4.3). For time-delayed hot spots, we show that all of them are contained in the region sandwiched between CS(f ) + ( (2n + 4)t − d f )B n and CS(f ) + (2n + 4)tB n after a large time (Theorem 5.1 (1)). Hence they escape from CS(f ).
For time-delayed cold spots, we show that all of them are contained in CS(f ) after a large time (Theorem 5.1 (2)), that the set of time-delayed cold spots consists of one point after a large time (Corollary 2), and that the unique time-delayed cold spot converges to the centroid of f as t goes to infinity (Corollary 1). Furthermore, we show that spatial maximum points of the absolute value of the solution are time-delayed cold spots after a large time.
Our argument is based on the so-called Nishihara decomposition. It decomposes the solution of (1.1) with f = 0 into the diffusive part and the wave part (see Proposition 1). It was introduced in [11] when n = 3, in [7] when n = 1, and in [3] when n = 2. For any dimensional case, Narazaki [9] gave a similar decomposition in terms of the Fourier transform. In the previous study [12], the authors gave the Nishihara decomposition for any space dimension without the Fourier transform. Combining the Nishihara decomposition and the asymptotic expansion of the diffusive part, we can minutely analyze the shape of the solution.
The presentation of the proofs in Sections 3, 4 and 5 is a bit lengthy, but the meaning of the phrase "a large (enough) time" is clear. The large time depends on n, d f , ϕ, f 1 , f W * ,∞ , the inradius (the radius of a maximal inscribed ball) of the support of f and the mass of f around an incenter (the center of a maximal inscribed ball) of the support of f .
At the end of this section, let us review the previous results. In [12], the authors showed the following properties when f and g are compactly supported, and f + g is non-zero and non-negative: (1) After a large time, any time-delayed hot spot belongs to CS(f + g).
(2) After a large time, we have a unique time-delayed hot spot.
(3) As t goes to infinity, the unique time-delayed hot spot converges to the centroid of f + g.
The first and last properties correspond to the results of hot spots shown in [1, Theorem 1]. The second property corresponds to the fact of hot spots indicated in [5,Introduction]. In other words, the large time behavior of time-delayed hot spots is the same as that of hot spots. However, the short time behavior of time-delayed hot spots is not similar to that of hot spots. Precisely, in [12], the authors gave an example of (f, g) such that any time-delayed hot spot is in the exterior of CS(f + g) for some small time.
The investigation for time-delayed hot spots in [12] includes the large time behavior of the (restricted) spatial critical set of the solution since time-delayed hot spots are spatial critical points of the solution. Precisely, for a given ϕ(t) of small order of t, the following statements hold: (1) After a large time, any spatial critical point in CS(f + g) + ϕ(t)B n belongs to CS(f + g).
(2) After a large time, the number of spatial critical points in CS(f + g) + ϕ(t)B n is just one, and the unique spatial critical point is the time-delayed hot spot. These results immediately imply the large time behavior of the spatial minimum set of the solution. Namely, for a given ϕ(t) of small order of t, there exists a large time T such that, for any t ≥ T , any spatial minimum point does not belong to CS(f + g) + ϕ(t)B n . Roughly speaking, the spatial minimum set of the solution escapes from CS(f + g).
The large time behavior of the spatial null set of the solution was not investigated in [12], but it can be done in the same manner as in [12]. Under the same assumption as in [12], for a given ϕ(t) of small order of t, there exists a large time T such that, for any t ≥ T , the solution is positive on CS(f + g) + ϕ(t)B n (see Proposition 13).
In order to summarize and compare results of this paper and [12], we give Tables 1 and 2 (see also Notations 1 and 3). Let f and g be compactly supported smooth functions, and u the unique classical solution of (1.1). In Table 1, we assume that f is non-zero and non-negative, and that f + g entirely vanishes. In Table 2, we assume that f + g is non-zero and non-negative. Let ϕ(t) be of small order of t as t goes to infinity. There exists a large enough T such that, for any t ≥ T , the properties in Tables 1 and 2 hold: The object Its property Table 1. Results of this paper (when f + g = 0). Table 2. Results of [12] (when f + g ≥ 0). Notation 1. Throughout this paper, we use the following notation:

The object Its property
• We denote the usual L p norm by · p , that is, • For a natural number , we denote the Sobolev norm by · W ,∞ , that is, • For a compactly supported function φ, we denote by CS(φ) and d φ the convex hull and diameter of the support of φ, respectively. • For a non-zero, non-negative and compactly supported function φ, we denote by m φ the centroid of φ, that is, • Let B n and S n−1 be the n-dimensional unit closed ball and the (n − 1)dimensional unit sphere, respectively. • For two real numbers a and b, and for two sets X and Y in R n , we use the notation (Minkowski sum) aX + bY = {ax + by |x ∈ X, y ∈ Y } . When Y is a singleton {y}, we write aX + Y = aX + y, for short. In particular, we write B n t (x) = tB n + x and S n−1 t (x) = tS n−1 + x, that is, the n-dimensional closed ball of radius t centered at x and the (n − 1)-dimensional sphere of radius t centered at x, respectively.
• For a set X in R n , we denote by X c , X • andX the complement, interior and closure of X, respectively. • Let σ n denote the n-dimensional Lebesgue surface measure.
• The letter r is always used for r = |x − y|.
Notation 2. Let I be the modified Bessel function of order defined in (6.4). We use the following notation from Section 2: (1) Let n = 1. Put (2) Let n be an odd number greater than one. Put (3) Let n be an even number. Put Notation 3. Let us list up our notation for Sections 3, 4 and 5.

SHIGEHIRO SAKATA AND YUTA WAKASUGI
(f ) Let f be a non-zero and non-negative smooth function with compact support.
We denote by ρ f and i f the inradius and an incenter of the support of f , respectively.

Expression of the solution.
Let f be a smooth function, and u the unique classical solution of (1.1) with f + g = 0. In this subsection, we list up the explicit form of u and its derivatives. We denote by S n (t)g(x) the solution of (1.1) with f = 0. Then, the solution of (1.1) is given by Thus, the solution of (1.1) with f + g = 0 is given by The explicit form of S n (t)g(x) was given in [12, Proposition 2.1].

Remark 1 ([12, Remark 2.5]).
(1) Let n be an odd number greater than one. The kernel k (s) has the following properties: (2) Let n be an even number. The kernel k (s) has the following properties: The explicit form of u directly follows from [12, Proposition 2.8].
Proposition 1. Let f be a smooth function, and u the unique classical solution of (1.1) with f + g = 0.
(1) Let n = 1. Put Then, we have (2) Let n be an odd number greater than one. Put Then, we have (3) Let n be an even number. Put Then, we have The explicit form of derivatives of u directly follows from [12, Lemma 3.5].
Lemma 2.1. Let f be a smooth function, and u the unique classical solution of (1.1) with f + g = 0.

SHIGEHIRO SAKATA AND YUTA WAKASUGI
(1) Let n = 1. Then, we have the following identities: (2) Let n be an odd number greater than one. We have the following identities: where ω ∈ S n−1 .

2.2.
Expansion of the principal terms. Let f be a smooth function, and u the unique classical solution of (1.1) with f + g = 0. We denote by P(u(x, t)), P(ω · ∇u(x, t)) and P((ω · ∇) 2 u(x, t)) the first terms of u(x, t), ω · ∇u(x, t) and (ω · ∇) 2 u(x, t), respectively, in Proposition 1 and Lemma 2.1. For example, they are given by when n is an even number. We call P(u(x, t)), P(ω · ∇u(x, t)) and P((ω · ∇) 2 u(x, t)) the principal terms of u(x, t), ω · ∇u(x, t) and (ω · ∇) 2 u(x, t), respectively, in what follows. They play important roles in the study of the large time behavior of u(·, t).
In this subsection, we give asymptotic expansions of the kernels of the principal terms.  (1) This is a direct consequence of the fact (6.6).
(2) Using the recursion in Remark 1, we can show the expansion by induction.
Remark 2 directly implies the asymptotic expansions of the kernels of the principal terms with tedious computation.
Lemma 2.2. Let ψ(t) be of order of √ t as t goes to infinity. Then, we have (1) Let n be an odd number. We have as t goes to infinity. (2) Let n be an even number. We have as t goes to infinity. Lemma 2.3. Let ϕ(t) be of small order of t as t goes to infinity. Then, we have (1) Let n be an odd number. We have as t goes to infinity. (2) Let n be an even number. We have as t goes to infinity. (1) If 0 ≤ r ≤ d, then we have as t goes to infinity. (2) The first property immediately follows from the first assertion. We give a proof of the second property for the even dimensional case. The argument works for the odd dimensional case.
By the recursion in Remark 1, we have Thus, the first assertion implies as t goes to infinity.

2.3.
Estimates of the error terms. Let f be a smooth function, and u the unique classical solution of (1.1) with f + g = 0. Put We call E(u(x, t)), E(ω · ∇u(x, t)) and E((ω · ∇) 2 u(x, t)) the error terms of u(x, t), ω·∇u(x, t) and (ω·∇) 2 u(x, t), respectively, in what follows. We should pay attention to them in the study of the large time behavior of u(·, t). In this subsection, we give estimates of the error terms.
Lemma 2.5. Let f be a smooth bounded function, and u the unique classical solution of (1.1) with f + g = 0.
By Lemma 2.1, there exists a positive constant C(n) such that, for any ω ∈ S n−1 , x ∈ R n and t > 0, we have From [12, Lemma 3.1] with f + g = 0 and α = 1, we obtain the conclusion.
The estimate of the error term of (ω · ∇) 2 u(x, t) is the same as above.
3. Spatial zeros. Let f be as in Notation 3, and u the unique classical solution of (1.1) with f + g = 0. In this section, we discuss the large time behavior of the (non-trivial) spatial null set of u, Proof. We give a proof for the even dimensional case. The argument works for the odd dimensional case. We take T ≥ 2n, which implies that if t ≥ T , then, for any x ∈ CS(f ) + ( √ 2nt − d f )B n , the ball B n t (x) contains CS(f ). By Proposition 1 and Lemma 2.2, if t is large enough, then we have P (u(x, t)) = γ n 2 n/2 4t n/2 for any x ∈ CS(f ) + √ 2ntB n . Here, for the first inequality, we used the fact that if t ≥ n 2 /4, then, for any 0 ≤ r ≤ √ 2nt, we have and equality holds if r = √ 2nt. For the second inequality, we took a large enough T such that if t ≥ T , then we have 1 + O(1/t) > 1/2. The last inequality follows from r ≤ √ 2nt. By Lemma 2.5, if t is large enough, then we have |E (u(x, t))| ≤ C(n)e −t/2 (1 + t) n f W n/2,∞ < − nγ n 2 n/2 16t n/2+2 e −n f 1 for any x ∈ CS(f ) + √ 2ntB n . Hence if t is large enough, then we obtain u(x, t) < − nγ n 2 n/2 16t n/2+2 e −n f 1 < 0 for any x ∈ CS(f ) + √ 2ntB n .
Lemma 3.1. Let f be as in Notation 3. There exists a constant T > 0 such that if t ≥ T , then, for any x ∈ A f ( √ 2nt, (2n + 1)t), we have u(x, t) > 0.
Proof. We give a proof for the even dimensional case. The argument works for the odd dimensional case.
We take a large enough T such that if t ≥ T , then we have t ≥ (2n + 1)t + d f , which implies that, for any x ∈ CS(f ) + (2n + 1)tB n , the ball B n t (x) contains We decompose the principal term of u as k n 2 (r, t)f (y)dy =: P 1 + P 2 .
By Lemma 2.2, if t is large enough, then we have Here, the first inequality follows from r ≥ √ 2nt + ρ f /2. For the second inequality, we took a large enough T such that if t ≥ T , then we have 1 + O(1/ √ t) > 1/2. For the last inequality, we took a large enough T such that if t ≥ T , then we have We also used Remark 3 in the last inequality.
Therefore, if t is large enough, then we obtain P (u(x, t)) > − nγ n 2 n/2 2t n/2+2 e −n/2 f 1 + for any x ∈ A f ( √ 2nt, (2n + 1)t). By Lemma 2.5, if t is large enough, then we have for any x ∈ A f ( √ 2nt, (2n + 1)t). Hence if t is large enough, then we obtain Proposition 3. Let f be as in Notation 3. Let ϕ(t) be of small order of t as t goes to infinity with (2n + 1)t ≤ ϕ(t) ≤ t for any t > 0. There exists a constant T > 0 such that if t ≥ T , then, for any x ∈ A f ( √ 2nt, ϕ(t)), we have u(x, t) > 0.
Proof. We give a proof for the even dimensional case. The argument works for the odd dimensional case.
We take a large enough T such that if t ≥ T , then we have t ≥ ϕ(t) + d f , which implies that, for any x ∈ CS(f ) + ϕ(t)B n , the ball B n t (x) contains CS(f ). By Lemma 3.1, we show the existence of a constant T > 0 such that if t ≥ T , then, for any x ∈ A f ( (2n + 1)t, ϕ(t)), we have u(x, t) > 0.
By Proposition 1 and Lemma 2.3, if t is large enough, then we have for any x ∈ A f ( (2n + 1)t, ϕ(t)). Here, the first and third inequalities follow from r ≥ (2n + 1)t. For the second inequality, we took a large enough T such that if t ≥ T , then we have By Lemma 2.5, if t is large enough, then we have Hence if t is large enough, then we obtain for any x ∈ A f ( (2n + 1)t, ϕ(t)).
Proposition 4. Let f be as in Notation 3. There exists a constant T ≥ d 2 f /(2n) such that if t ≥ T , then, for any (ξ, ν) ∈ NCS(f ) and Proof. We give a proof for the even dimensional case. The argument works for the odd dimensional case.
We take a large enough T such that if t ≥ T , then we have t ≥ √ 2nt + d f , which implies that, for any x ∈ CS(f ) + √ 2ntB n , the ball B n t (x) contains CS(f ). By Lemmas 2.1 and 2.2, if t is large enough, then we have Here, for the first and third inequalities, we took a large enough T such that if t ≥ T , then we have r ≤ √ 2nt + d f ≤ (2n + 1)t.
For the second inequality, we took a large enough T such that if t ≥ T , then we have 1 + O(1/t) > 1/2. For the last inequality, we took a large enough T such that if t ≥ T , then we have and y ∈ B n t (x). By Lemma 2.5, if t is large enough, then we have . Hence if t is large enough, then we obtain ν · ∇u(x, t) > 3 √ 2n − 1γ n 2 n/2+1 128t n/2+3/2 exp − 2n + 1 2 f 1 > 0 for any (ξ, ν) ∈ NCS(f ) and Theorem 3.2. Let f be as in Notation 3. Let ϕ(t) be of small order of t as t goes to infinity, and (2n + 1)t ≤ ϕ(t) ≤ t for any t > 0. There exists a constant and the following statements hold: (1) If n = 1, then the restriction of u(·, t) to CS(f ) + ϕ(t)B 1 has just two zeros.
Proof. We give a proof for the even dimensional case. The argument works for the odd dimensional case.
Proof. We give a proof for the even dimensional case. The argument works for the odd dimensional case. We take a large enough T such that if t ≥ T , then we have t ≥ (2n + 5)t + d f , which implies that, for any x ∈ CS(f ) + (2n + 5)tB n , the ball B n t (x) contains CS(f ). For every (ξ, ν) ∈ NCS(f ), put ξ = ξ − (ρ f /2)ν. We decompose the principal term of ν · ∇u as P (ν · u(x, t)) By Lemma 2.2, if t is large enough, then we have for any (ξ, ν) ∈ NCS(f ) and x ∈ A f ( (2n + 4)t, (2n + 5)t) ∩ (Pos{ν} + ξ). Here, for the first inequality, we used the fact that, for any r ≥ (2n + 4)t, we have and equality holds if r = (2n + 4)t. The second inequality is usual. The third inequality follows from r ≥ (2n + 4)t. For the last inequality, we took a large enough T such that if t ≥ T , then we have ν · (x − y) ≤ r ≤ (2n + 5)t + d f ≤ (2n + 6)t.
for any r ≥ ϕ(t). Here, the first inequality follows from the triangle inequality and the monotonicity of k . For the last inequality, we took a large enough T such that if t ≥ T , then we have Therefore, if t is large enough, then we have P (u(x, t)) < 3γ n 2 n/2 4t n/2 exp for any x ∈ (CS(f ) + ϕ(t)B n ) c . By Lemma 2.5, if t is large enough, then we have |E (u(x, t))| ≤ C(n)e −t/2 (1 + t) n f W n/2,∞ < 3γ n 2 n/2 8t n/2 exp − ϕ(t) 2 4t f 1 for any x ∈ (CS(f ) + ϕ(t)B n ) c . Hence if t is large enough, then we have u(x, t) < 3γ n 2 n/2 8t n/2 exp − ϕ(t) 2 4t f 1 .
Theorem 5.1. Let f be as in Notation 3. There exists a constant T ≥ d 2 f /(2n + 4) such that if t ≥ T , then we have the following inclusion relations: Proof. Let ϕ(t) be of small order of t as t goes to infinity such that 1/ϕ(t) is of small order of 1/ √ t as t goes to infinity, and (2n + 5)t ≤ ϕ(t) ≤ t for any t > 0. We take a large enough T such that if t ≥ T , then we have (2n + 4)t−d f ≥ √ 2nt. (1) By the behavior of strictly increasing and decreasing of u(·, t) in Propositions 5 and 6, respectively, if t is large enough, then there is no spatial maximum point of u in A f (0, (2n + 4)t − d f ) ∪ A f ( (2n + 4)t, ϕ(t)).
Propositions 2 and 3 guarantee the negativity and positivity of u(·, t) on CS(f ) and A f ( (2n + 4)t − d f , (2n + 4)t), respectively. Thus, if t is large enough, then there is no spatial maximum point of u in CS(f ).
By Propositions 10 and 11, if t is large enough, then there is no spatial maximum point of u in the complement of CS(f ) + ϕ(t)B n . Hence, we obtain the first assertion.
In order to complete the proof, we use the contradiction argument. For any natural number N ≥ T 0 (ϕ), we assume the existence of t N ≥ N such that the function u(·, t N ) vanishes at a point x N in CS(h) + ϕ(t N )B n . Let r N = |x N − y|. We remark that, for any y ∈ CS(h) ∪ CS(f ), By the above asymptotic expansion, we obtain 0 = 2 n/2 t n/2 N c n exp as N goes to infinity, which contradicts to the non-negativity of h.
Proposition 13. Under the same assumption as in Lemma 6.1, there exists a time T ≥ T 0 (ϕ) such that, for any t ≥ T , u(·, t) is positive on the parallel body CS(h) + ϕ(t)B n .