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題目(和文)

半線形熱方程式における動的特異点

Title(English)

Time-dependent singularities in semilinear heat equations

著者(和文)

高橋仁

Author(English)

Jin Takahashi

出典(和文)

学位:博士（理学）,学位授与機関:東京工業大学,
報告番号:甲第10286号,
授与年月日:2016年9月20日,
学位の種別:課程博士,
審査員:柳田 英二,米田 剛,志賀 啓成,利根川 吉廣,磯部 健志,桒田 和正,二宮 広和

Citation(English)

Degree:,Conferring organization: Tokyo Institute of Technology,
Report number:甲第10286号,
Conferred date:2016/9/20,
Degree Type:Course doctor,
Examiner:,,,,,,

学位種別(和文)

博士論文

Category(English)

Doctoral Thesis

種別(和文)

要約

Type(English)Outline

Powered by T2R2 (Tokyo Institute Research Repository)

Time-dependent singularities in semilinear heat equations

Jin Takahashi

SummaryAcknowledgements
I am greatly indebted to my advisor Professor Eiji Yanagida for introducing me to thesubject of singular solutions and for giving me many suggestions and support. His
approach to a study of mathematics has given me strongly positive influence. I wish
to thank Professor Tsuyoshi Yoneda for his encouragement. I have gotten a lot of help
from him. I am also grateful to Dr. Toru Kan for fruitful discussions. I have learned
a lot of mathematics from him.

SummaryThis is a summary of the author’s Ph.D thesis [6] at Tokyo Institute of Technology.
In the thesis, we mainly study solutions with time-dependent singularities for heat
equations with a superlinear term. Here, by time-dependent singularity, we meana singularity with respect to the space variable whose position depends on the timevariable. The thesis consists of five chapters. In Chapter 1, we recall known results andsummarize our main results. Chapters 2–4 are devoted to the study of solutions withtime-dependent singularities in semilinear heat equations. In Chapter 5, we consider aCauchy problem by taking an approach similar to that of the previous chapters. Eachchapter of the thesis is based on a paper or a preprint, as listed in the last part of thissummary.
The study of time-dependent singularities dates back to the paper of Sato andYanagida [7]. They constructed a positive solution with a time-dependent singularity
for the following semilinear heat equation∂t u − ∆u = up(1)√√under the condition that N ≥ 3 and psg < p < p∗ := (N +2 N − 1)/(N −4+2 N − 1),
where the exponent psg is defined bypsg :=N.
N −2

The exponent psg is important for the existence of singular steady states. Indeed, if
p > psg , the equation −∆u = up in RN \ {0} has an explicit singular steady state2− p−1u∞ (x) := L|x|{,L :=1} p−12(N − 2).
(p − psg )(p − 1)2

By the perturbation of u∞ , Sato and Yanagida [7] constructed appropriate comparison
functions and proved the existence of a solution with a time-dependent singularity. The
exponent p∗ has a relation of the stability of the singular steady state in some sense.
For subsequent studies of time-dependent singularities of (1), we refer the reader tothe papers of Sato and Yanagida [8, 9, 10]. We note that, in the previous research1

on time-dependent singularities of solutions of (1), at least the condition p > psg isimposed.
In the case where 1 < p ≤ psg , we cannot apply the method based on the singularsteady state u∞ , because it was shown by Gidas and Spruck [4, Theorem A.1] that theequation −∆u = up in RN \ {0} has no nonnegative singular solutions if 1 < p ≤ psg .
For this reason, we need a method which is not based on the existence of singularsteady states.
In Chapters 2 and 3, we study time-dependent singularities of solutions of (1) under1 < p < psg by the method of external force. This method is motivated by Zhang
and Zhao [11]. They considered semilinear equations involving (1) with 1 < p < psg
and constructed solutions with a “time-independent” singularity. By their remark[11, Remark 4.1], one can see that they essentially solved the equation ∂t u − ∆u =up + cδ0 ⊗ µL . Here δ0 is the Dirac measure in RN concentrated at the origin, µL is theLebesgue measure on the real line and c > 0 is a small constant. We can observe thatthey considered semilinear equations involving a measure as an external force term,
where the support of the measure cδ0 ⊗ µL is contained in a straight line in RN +1 .
We sketch out the method of external force for time-dependent singularities by thefollowing Steps 1–3. Let us consider the following problem∂t u − ∆u = up ,x ∈ RN \ {ξ(t)}, t ∈ I,

(2)

where N ≥ 3, 1 < p < psg , I ⊂ R is an open interval and ξ(t) : I → RN is smoothenough.
Step 1. We prove that every nonnegative solution u of (2) belongs to Lploc (RN × I)
and satisfies the following equation involving a measure as an external force term∂t u − ∆u = up + (δ0 ⊗ µ) ◦ Tin D′ (RN × I).

(3)

Here T = Tξ is a translation operator defined by T (φ)(x, t) := φ(x + ξ(t), t) for afunction φ on RN × I and µ is a Radon measure on I determined by the solution u,so that the support of the measure (δ0 ⊗ µ) ◦ T is contained in the curve {(ξ(t), t) ∈RN +1 ; t ∈ I}. Note that the above result is a consequence of Theorem 2.1.1. We
remark that, by Theorem 2.1.2 (ii-b), we have µ = 0 if p ≥ psg .
Step 2. We show relations between the exponent p and the local growth rate of themeasure µ and specify the behavior of solutions of (3). Namely, µ must satisfy that,for any open interval I ′ ⊂ I with I ′ ⊂ I, there exists a constant C > 0 such thatC
if 1 < p < pF ,N)(−2N +21(4)µ((a, b)) ≤ C log
if p = pF :=,
b−aN

N1

C(b − a) 2 − p−1
if pF < p < psg
for all a, b ∈ I ′ with 0 < b − a ≤ 1/2. Moreover, every nonnegative solution u of (2)satisfies, for almost all t ∈ I,}{1Dµ(t) + o(1) |x − ξ(t)|2−N as x → ξ(t).
u(x, t) =N (N − 2)ωN2

1Here ωN is the volume of the unit ball in RN and Dµ ∈ Lloc(I) is the Radon-Nikodym
derivative of the absolutely continuous part of µ with respect to the Lebesgue measure.
By the above result, we see that our study of time-dependent singularities for theproblem (2) with 1 < p < psg will be done if we solve (3) under the condition (4). Notethat the above result is a consequence of Theorems 2.1.2 and 2.1.3.
Step 3. We construct a solution u of (3) satisfying that u is smooth on x ̸= ξ(t) for allt ∈ I. For the existence of a solution of (3) with a bounded and open interval I ̸= ∅,we impose the following growth condition on µ. For small constants C > 0 and ε > 0,
the measure µ satisfies

if 1 < p < pF ,
C1(())− p−1 −ε(5)µ((a, b)) ≤N11

if pF ≤ p < psg
 C(b − a) 2 − p−1 log e +b−afor any a, b ∈ I with a < b. This condition is stronger than (4). However, this conditionis sharp even if pF ≤ p < psg . Indeed, for any constant C > 0, there exists a Radonmeasure µ∗ on R satisfyingµ∗ ((a, b)) ≤ C(b − a)1
N− p−12( (log e +1b−a1))− p−1

if pF ≤ p < psg(6)

for any a, b ∈ R with a < b such that the problem (3) with µ∗ does not admit any local
in time and nonnegative solutions. Note that the results for existence and nonexistence
are consequences of Theorems 3.1.1 and 3.1.4, respectively. The existence for I = R
under pF < p < psg is also considered in Theorem 3.1.2.
In Chapter 4, we consider the following opposite-sign equation∂t u − ∆u = −up ,where N ≥ 3 and p > 1. For this equation, the structure of singular steady state is
diﬀerent from the equation (1). More precisely, for p < psg , the equation −∆u = −upin RN \ {0} has an explicit singular steady state2− p−1ũ∞ (x) := L̃|x|{,L̃ :=1} p−12(N − 2).
(psg − p)(p − 1)2

For p ≥ psg , it was proved by Brézis and Véron [3] that there are no singular steadystates. Therefore it seems that the structure of solutions of the following problem isalso diﬀerent from the problem (3).
∂t u − ∆u = −up ,x ∈ RN \ {ξ(t)}, t ∈ (0, ∞),(7)

where ξ(t) is smooth enough.
We first show that, for the problem (7) with p ≥ psg , there are no nonnegative
solutions with time-dependent singularities. This is stated in Theorem 4.1.1. In order3

to prove this result, we derive a universal pointwise estimate for nonnegative solutionsof (7) under p > 1 by using the method of Poláčik, Quittner and Souplet [5] which isbased on a Liouville type theorem and a scaling argument. Note that the universal
estimate are given by Corollary 4.2.3 (iii). We remark that Theorem 4.1.1 can be
regarded as an indirect consequence of the general removability result by Baras andPierre [1, THEOREME 3.1]. Their method is based on properties of some parabolic
capacity.
Next, we consider the case p < psg and construct two types of solution with atime-dependent singularity under the condition that ξ is smooth and ξ ′ is bounded.
Namely, we prove that there exist solutions uF and uS of (7) such that, uniformly fort ∈ (0, ∞),uF (x, t) = (w(t) + o(1))|x − ξ(t)|2−N as x → ξ(t),uS (x, t) = (L̃ + o(1))|x − ξ(t)|− p−12

as x → ξ(t),where w(t) is Hölder continuous uniformly for t ∈ R and satisfies 1/C ≤ w(t) ≤ C
for all t ∈ R and a constant C > 1. We also prove that, under some additional
assumption, every nonnegative and singular solution of (7) behaves like either uF oruS . These results are stated as Theorems 4.1.3 and 4.1.4.
In Chapter 5, we consider the solvability of the following Cauchy problem{∂t u − ∆u = up in RN × (0, T ),(8)u(·, 0) = λin RN ,where N ≥ 1, T > 0, p > 1 and λ is a Radon measure on RN . Baras and Pierre [2,PROPOSITION 3.2] gave an explicit necessary condition for existence. They provedthat if u ≥ 0 satisfies ∂t u − ∆u = up in RN × (0, T ), then there exists a unique Radon
measure λ on RN such that u(·, t) → λ as t ↓ 0 weakly as measures on each fixed openball. Furthermore, λ must satisfy
sup λ(B(x; 1)) < +∞if p < pF ,x∈RN

and for any compact subset K of RN there exists a constant C > 0 such that
 ()− N2

 C log 1
if p = pF ,ρλ(B(x; ρ)) ≤2
 N − p−1
Cρ
if p > pF ,(9)for any x ∈ K and ρ > 0 small. Here B(x0 ; r) stands for the N -dimensional open ballof radius r > 0 centered at x0 ∈ RN . It is also known [2, COROLLAIRE 3.4.i] thatthe above result under the case p < pF is also a suﬃcient condition for the existenceof local solutions.
One can observe that Steps 1 and 2 in the context of time-dependent singularities
correspond to the results of Baras and Pierre [2], especially, the exponent N/2−1/(p−1)
4

in (4) under the case pF < p < psg is a half of the exponent N − 2/(p − 1) in (9)under the case p > pF . The correspondence between those two exponents is natural,
because µ and λ are measures with respect to the time variable and the space variable,respectively, and the equation (1) is parabolic. In view of the above observation, weseem to be able to show the solvability of the problem (8) under conditions similar to(5) and (6).
We prove that (8) admits a local solution if there exist a constant C > 0 and asmall constant ε > 0 such that1))− p−1( (−ε21N − p−1λ(B(x; ρ)) ≤ Cρlog e +if p ≥ pF
ρfor any x ∈ RN and ρ > 0. On the other hand, the problem (8) does not admit any
local solutions for some initial data λ∗ satisfying
1( ())− p−121
N − p−1
log e +λ∗ (B(x; ρ)) ≤ Cρif p ≥ pF
ρfor any x ∈ RN and ρ > 0 with a constant C > 0. Note that these conditions on λimprove the known results listed in the first part of Chapter 5 and that these conditions
are stated in Theorems 5.1.2 (i) and 5.1.1.

References[1] P. Baras, M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures,
Applicable Anal. 18 (1984) 111–149.
[2] P. Baras, M. Pierre, Critère d’existence de solutions positives pour des équations
semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (3)(1985) 185–212.
[3] H. Brézis, L. Véron, Removable singularities for some nonlinear elliptic equations,Arch. Rational Mech. Anal. 75 (1980/81) 1–6.
[4] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear
elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525–598.
[5] P. Poláčik, P. Quittner, Ph. Souplet, Singularity and decay estimates in superlinear
problems via Liouville-type theorems. II. Parabolic equations, Indiana Univ. Math.
J. 56 (2007) 879–908.
[6] J. Takahashi, Time-dependent singularities in semilinear heat equations, Thesis,
Tokyo Institute of Technology, 2016.
[7] S. Sato, E. Yanagida, Solutions with moving singularities for a semilinear parabolicequation, J. Diﬀerential Equations 246 (2009) 724–748.
5[8] S. Sato, E. Yanagida, Forward self-similar solution with a moving singularity fora semilinear parabolic equation, Discrete Contin. Dyn. Syst. 26 (2010) 313–331.
[9] S. Sato, E. Yanagida, Singular backward self-similar solutions of a semilinear
parabolic equation, Discrete Contin. Dyn. Syst. Ser. S 4 (2011) 897–906.
[10] S. Sato, E. Yanagida, Appearance of anomalous singularities in a semilinear
parabolic equation, Commun. Pure Appl. Anal. 11 (2012) 387–405.
[11] Q. S. Zhang, Z. Zhao, Singular solutions of semilinear elliptic and parabolic equations, Math. Ann. 310 (1998) 777–794.

Original papers
1. T. Kan, J. Takahashi, Time-dependent singularities in semilinear parabolic equations: behavior at the singularities, J. Diﬀerential Equations 260 (2016) 7278–7319.
2. T. Kan, J. Takahashi, Time-dependent singularities in semilinear parabolic equations: existence of solutions, submitted.
3. J. Takahashi, E. Yanagida, Time-dependent singularities in a semilinear parabolic
equation with absorption, Commun. Contemp. Math. 18 (2016) 1550057, 27
pp.
4. J. Takahashi, Solvability of a semilinear parabolic equation with measures as
initial data, Springer Proc. Math. Stat., Springer, to appear.
Chapters 2–5 in [6] are based on the above 1–4, respectively.

Related papers
1. T. Kan, J. Takahashi, On the profile of solutions with time-dependent singularities for the heat equation, Kodai Math. J. 37 (2014) 568–585.
2. J. Takahashi, E. Yanagida, Time-dependent singularities in the heat equation,Commun. Pure Appl. Anal. 14 (2015) 969–979.

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