Nonexistence of positive global solutions to the differential equation

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NONEXISTENCE OF POSITIVE GLOBAL SOLUTIONS TO THE DIFFERENTIAL EQUATION u00_{− t}−p−1_up_{= 0}

MENG-RONG LI, TSUNG-JUI CHIANG-LIN, YOUNG-SHIUAN LEE, DANIEL WEI-CHUNG MIAO

Abstract. In this article we consider the ordinary differential equation u00− t−p−1up= 0.

We show the blow-up for solutions of this equation, under certain on the initial data.

1. Introduction

In articles [1]–[6], [8]–[10] we studied the semi-linear wave equation u+f (u) = 0 under some conditions, and we found some interesting results on up, blow-up rate and estimates for the life-span of solutions, but no information on the singular set. So we want to study some particular cases for lower dimensional wave equations, therefrom we hope that we gain some experience for for studying particular lower dimension later.

It is clear that the functions t−p−1_up_{, with p > 1, u ≥ 0 and t ≥ 1 is locally} Lip-schitz. By standard theory, the existence and uniqueness of classical local solutions holds for the equation

u00− t−p−1up= 0, p ∈ (1, ∞), u(1) = u0, u0(1) = u1.

(1.1) Notation and fundamental Lemmas. First we make a substitution

u = tv, u0= v + tv0, u00= 2v0+ tv00, tp+1u00= 2tp+1v0+ tp+2v00= tpvp,

2tv0+ t2v00= vp. Set s = ln t, v(t) = w(s), then tv0_{= w}

s, t2v00= wss− ws.

For a given function w in this work we use the following abbreviations: Ew(0) = (u1− u0)2− 2 p + 1u p+1 0 , aw(s) = w2:= a(s), K(s) := Kw(s) := Z s 0 w2_s(r)dr, J (s) := Jw(s) = w(s)− p−1 2 _.

2010 Mathematics Subject Classification. 34A34, 34C05.

Key words and phrases. Blow-up; global solution; nonlinear differential equation. c

2016 Texas State University.

Submitted May 15, 2016. Published July 13, 2016.

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The equation (1.1) can be transformed into

wss+ ws= wp, (1.2)

w(0) = w0= u0, (1.3)

ws(0) = w1= u1− u0. (1.4)

Using some elementary calculations we obtain the following lemmas. Lemma 1.1. Suppose that w is the solution of (1.2), then

Ks(s) + 2K(s) − 2 p + 1w p+1_{(s) = E} w(0), (1.5) ass(s) + as(s) = 2(w(s)p+1+ Ks(s)). (1.6) Lemma 1.2. Suppose that w is the solution of (1.2), then

K(s) = Ew(0) 2 (1 − e −2s_{) +} 2 p + 1e −2sZ s 0 e2rwp+1(r)dr, (1.7) Ks(s) = Ew(0)e−2s+ 2 p + 1w(s) p+1₋ 4 p + 1e −2sZ s 0 e2rwp+1(r)dr, (1.8) as(s) = (as(0) + 2Ew(0))e−s− 2Ew(0)e−2s + 2 p + 1 Z s 0 (p − 1 + 4er−s)er−swp+1(r)dr, (1.9) as= −(p + 1)Ew(0) + ((p + 1)Ew(0) + as(0))e−s

+ (p + 3)K(s) + (p − 1) Z s 0 er−sK(r)dr, (1.10) Jss(s) = −(p + 1)a(s)− p+1 2 −1_{((−((p + 1)E}_w_{(0) + a}_s_(0))e−s₎₎ − (p2_{− 1)a(s)}−p+1 2 −1 Z s 0 er−sK(r)dr. (1.11)

Proof. By (1.5) and (1.6) we have

Ks(s) + 2K(s) = Ew(0) + 2 p + 1w(s) p+1_, (e2sK(s))s= e2s(Ew(0) + 2 p + 1w(s) p+1_), e2sK(s) = Ew(0) 2 (e 2s_{− 1) +} 2 p + 1 Z s 0 e2rwp+1(r)dr. Thus (1.7) and (1.8) are obtained.

By (1.6) and (1.7) we obtain es ass(s) + as(s) = 2es w(s)p+1+ Ks(s), esas(s) = as(0) + Z s 0 2er(wp+1+ Ks)(r)dr = as(0) + 2esK(s) + Z s 0 2er(wp+1− K)(r)dr,

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esas(s) = as(0) + 2esK(s) + Z s 0 2erw(r)p+1dr − Z s 0 2er(Ew(0) 2 (1 − e −2r_{) +} 2 p + 1e −2rZ r 0 e2ηwp+1(η)dη)dr = as(0) + 2esK(s) + Z s 0 2erwp+1(r)dr − Z s 0 Ew(0)(er− e−r)dr − 4 p + 1 Z s 0 e−r Z r 0 e2ηwp+1(η)dη dr, esas(s) = as(0) + 2esK − Ew(0)(es+ e−s− 2) + Z s 0 2erwp+1(r)dr − 4 p + 1 Z s 0 e−r Z r 0 e2ηwp+1(η)dηdr, esas(s) = as(0) + 2esK(s) − Ew(0)(es+ e−s− 2) + Z s 0 2erwp+1(r)dr + 4 p + 1 e−s Z s 0 e2ηwp+1(η)dη − Z s 0 e−re2rwp+1(r)dr = as(0) + 2esK(s) − Ew(0)(es+ e−s− 2) + Z s 0 2erwp+1(r)dr + 4 p + 1 Z s 0 (e−s− e−r)e2rwp+1(r)dr, esas(s) = as(0) + 2esK(s) − Ew(0)(es+ e−s− 2) + 2 Z s 0 [1 + 2 p + 1(e r−s_{− 1)]e}r_wp+1_(r)dr = as(0) + 2esK(s) − Ew(0)(es+ e−s− 2) + 2 p + 1 Z s 0 [p − 1 + 2er−s]erwp+1(r)dr, as(s) = as(0)e−s+ 2( Ew(0) 2 (1 − e −2s_{) +} 2 p + 1e −2sZ s 0 e2rwp+1(r)dr) − Ew(0)(1 + e−2s− 2e−s) + 2 p + 1 Z s 0 (p − 1 + 2er−s)er−swp+1(r)dr = as(0)e−s+ Ew(0)(1 − e−2s) + 4 p + 1e −2sZ s 0 e2rwp+1(r)dr − Ew(0)(1 + e−2s− 2e−s) + 2 p + 1 Z s 0 (p − 1 + 2er−s)er−swp+1(r)dr = (as(0) + 2Ew(0))e−s− 2Ew(0)e−2s + 2 p + 1 Z s 0 (p − 1 + 4er−s)er−swp+1(r)dr. Therefore, (1.9) follows.

Now to prove (1.10). According to (1.5) and (1.6), we obtain that esas(s) = as(0) +

Z s

0

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= as(0) + Z s 0 2erp + 1 2 (Ks+ 2K − Ew(0)) + Ks (r)dr = as(0) − (p + 1)Ew(0)(es− 1) + Z s 0 er((p + 3)Ks+ 2(p + 1)K)(r)dr and

as(s) = −(p + 1)Ew(0) + (as(0) + (p + 1)Ew(0))e−s +

Z s

0

er−s((p + 3)Ks+ 2(p + 1)K)(r)dr = −(p + 1)Ew(0) + ((p + 1)Ew(0) + as(0))e−s

+ (p + 3)K(s) + (p − 1) Z s

0

er−sK(r)dr.

To show (1.11), we use (1.6) and the definition J (s) := Jw(s) = a(s)− p+1 2 , Js= −p+1₂ a(s)− p+1 2 −1a_s, ass(s) = −((p + 1)Ew(0) + as(0))e−s+ (p + 3)Ks(s) + (p − 1)K(s) − (p − 1) Z s 0 er−sK(r)dr = −((p + 1)Ew(0) + as(0))e−s+ (p + 3)Ks(s) + (p − 1) Z s 0 er−sK(r)dr. Jss(s) = − p + 1 2 a(s) −p+1 2 −2 a(s)ass(s) − p + 3 2 as(s) 2 = −p + 1 2 a(s) −p+1 2 −2

2a(s)(−((p + 1)Ew(0) + as(0))e−s) − (p + 1)(p − 1)a(s)−p+1₂ −1Z s 0 er−sK(r)dr = −(p + 1)J (s)1+p+12 − ((p + 1)Ew(0) + as(0))e−s + (p − 1) Z s 0 er−sK(r)dr.

The above formulation is equivalent to assertion (1.11). Thus Lemma 1.2 is proved. Definition 1.3. A function g : R → R has a blow-up rate q means that g exists only in finite time, that is, there is a finite number T∗ such that

lim t→T∗g(t)

−1_{= 0} and there exists a non-zero β ∈ R with

lim t→T∗(T

∗_{− t)}q_{g(t) = β,} in this case β is called the blow-up constant of g.

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Lemma 1.4. If g(t) and h(t, r) are continuous with respect to their variables and the limit limt→TR

g(t) 0 h(t, r)dr exists, then lim t→T Z g(t) 0 h(t, r)dr = Z g(T ) 0 h(T, r)dr.

2. Nonexistence of global solution when Ew(0) < 0

In this section we want to show there is not global solution for (1.1) under negative energy Ew(0) < 0.

Theorem 2.1. If T is the life-span of u and u is the positive solution of the problem (1.1) with Ew(0) < 0, then T is finite. This means that the global solution of (1.1) does not exist for (u1− u0)2< _p+12 u

p+1 0 .

Proof. We consider the cases as(0) > 0 and as(0) ≤ 0. In the first case, using (1.10) of Lemma 1.2 we obtain that

as(s) = −(p + 1)Ew(0)(1 − e−s) + as(0)e−s+ (p + 3)K(s) + (p − 1)

Z s

0

er−sK(r)dr > 0 for all s ≥ 0 and

a(s) ≥ −(p + 1)Ew(0)(s − 1 + e−s) + as(0)(1 − e−s) ≥ 1 for s ≥ s0:= 1 + −1 (p + 1)Ew(0) . According to Lemma 1.1, (1.6) for s ≥ s0,

(ass+ as)(s) = 2 w(s)p+1−a(s) 4 + 2ws(s)2+ w(s)2 2 ≥ 2a(s)p+12 −a(s) 4 + 2|wsw|(s), ass(s) ≥ 2a(s) a(s)p−12 −1 4 =a(s) 2 a(s)p−12 − 1 +3 2a(s) p+1 2 ≥3 2a(s) p+1 2 _;

thus we obtain that there exists s1:= 1 − _E_w1₍₀₎2 2 p+3 p+1 a(s0) > s0 so that (a2_s(s))s= 2as(s)ass(s) ≥ 3a(s)p+12 _a_s_{(s) =} 6 p + 3(a(s) p+3 2 ₎_s_, a2_s(s) ≥ 6 p + 3a(s) p+3 2 + a2 s(s0) − 6 p + 3a(s0) p+3 2 ≥ 3 p + 3a(s) p+3 2 . Since for s ≥ s1,

a(s) ≥ −(p + 1)Ew(0)(s − 1 + e−s) ≥ −(p + 1)Ew(0)(s − 1), a(s)p+32 ≥ (p + 1) p+3 2 − E_w(0) p+3 2 _{(s − 1)}p+3₂ _{≥ 2a(s} 0) p+3 2 ,

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as(s) ≥ r 3 p + 3a(s) p+3 4 , −4 p − 1(a(s) 1−p 4 )_s= a(s)− p+3 4 a_s≥ r 3 p + 3, −4 p − 1a(s) 1−p 4 ≥ r 3 p + 3(s − s1) − 4 p − 1a(s1) 1−p 4 _, a(s)1−p4 ≤ −p − 1 4 r 3 p + 3(s − s1) − 4 p − 1a(s1) 1−p 4 ; therefore there exists

S∗≤ S₁∗= s1+ r p + 3 3 4 p − 1a(s1) 1−p 4 such that a(s)1−p4 → 0 for s → S∗.

This means that the solution for of (1.1) does not exist for all t ≥ 1 and the life-span T∗ _{of u is finite with T}∗ _{≤ ln S}∗

1. Whereas, if as(0) ≤ 0 using (1.10) of Lemma 1.2 again, we obtain that

as(s) = −(p + 1)Ew(0)(1 − e−s) + as(0)e−s+ (p + 3)K(s) + (p − 1)

Z s

0

er−sK(r)dr > 0 for all large s ≥ s2, where s2 is given by

s2= ln(1 +

as(0) (p + 1)Ew(0)

) and

a(s) ≥ −(p + 1)Ew(0)(s − s2+ e−s− e−s2) + as(0)(e−s2− e−s) ≥ 1 for all s ≥ s3, s3 can be obtained by

− (p + 1)Ew(0)(s3+ e−s3) − as(0)e−s3 = 1 − (p + 1)Ew(0)(s2+ e−s2) − as(0)e−s2 According to Lemma 1.1, (1.6) for s ≥ s3,

(ass+ as)(s) = 2 w(s)p+1−a(s) 4 + 2ws(s)2+ w(s)2 2 ≥ 2a(s)p+12 −a(s) 4 + 2|wsw|(s), ass(s) ≥ 2a(s)(a(s) p−1 2 −1 4) = a(s) 2 (a(s) p−1 2 − 1) +3 2a(s) p+1 2 ≥ 3 2a(s) p+1 2 ;

thus we obtain that there exists s4> s3 such that for s ≥ s4, a(s4) p+3 2 = −p + 3 3 a 2 s(s3) + 2a(s3) p+3 2 ,

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(a2s(s))s= 2as(s)ass(s) ≥ 3a(s)p+12 _a_s_(s) = 6 p + 3(a(s) p+3 2 ₎_s_, a2_s(s) ≥ 6 p + 3a(s) p+3 2 + a2 s(s3) − 6 p + 3a(s3) p+3 2 ≥ 3 p + 3a(s) p+3 2 , as(s) ≥ r 3 p + 3a(s) p+3 4 ∀s ≥ s_4, −4 p − 1(a(s) 1−p 4 )_s= a(s)− p+3 4 a_s≥ r 3 p + 3 ∀s ≥ s4, −4 p − 1a(s) 1−p 4 ≥ r 3 p + 3(s − s4) − 4 p − 1a(s4) 1−p 4 ∀s ≥ s_4, a(s)1−p4 ≤ −p − 1 4 r 3 p + 3(s − s4) − 4 p − 1a(s4) 1−p 4

therefore, there exists

S₂∗≤ s4+ r p + 3 3 4 p − 1a(s4) 1−p 4 so that a(s)1−p4 → 0 for s → S∗ 2.

This means that the solution for the problem (1.1) does not exist for all t ≥ 1 and the life-span T∗ of u is finite with T∗≤ ln S∗

2.

3. Nonexistence of global solution when Ew(0) > 0

In this section we want to show there is no global solution of (1.1) under positive energy Ew(0) > 0 when one of the following conditions holds

u0(u1− u0) ≥ 0, (3.1) u0(u1− u0) < 0, u1(u1− u0) > 2 p + 1u p+1 0 . (3.2)

We have the result for nonexistence of global solution of (1.1).

Theorem 3.1. If T is the life-span of u and u is the positive solution of the problem (1.1) with Ew(0) > 0 and u0(u1− u0) ≥ 0, then T is finite. This means that the global solution of (1.1) does not exist for (u1− u0)2>_p+12 u

p+1

0 , u0(u1− u0) ≥ 0. If T is the life-span of u and u is the positive solution of the problem (1.1) with Ew(0) > 0, u0(u1− u0) < 0 and Ew(0)+ u0(u1− u0) > 0, then T is finite. This means that the global solution of (1.1) does not exist for u1(u1− u0) >_p+12 u

p+1 0 . Remark 3.2. Under positive energy Ew(0) > 0, u0(u1−u0) < 0 with u1(u1−u0) ≤

2 p+1u

p+1

0 we conjecture that solutions of (1.1) exist globally, and the asymptotic behavior of such solutions is similar to the function

c + c −p_t1−p p(p − 1) as t → ∞, but we are unable to prove this rigorously.

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Proof. Using Lemma 1.2, (1.9), Ew(0) > 0 and u0(u1− u0) ≥ 0 we obtain esas(s) ≥ (as(0) + 2Ew(0)) − 2Ew(0)e−s+ 2(p − 1) p + 1 Z s 0 era(r)wp−1(r)dr, A(s) = Z s 0 erwp+1(r)dr, As(s) = esa(s) p+1 2 , a(s) = (e−sAs(s)) 2 p+1_, as(s) = 2 p + 1(e −s_A s(s)) 2 p+1−1_e−s_(A ss− As)(s), 2 p + 1(e −s_A s(s)) 2 p+1−1_(A_ss_{(s) − A}_s_(s)) ≥ (as(0) + 2Ew(0)) − 2Ew(0)e−s+ 2(p − 1) p + 1 A(s), 2 p + 1(e −s_A s(s)) 2 p+1e−s(A_ss(s) − A_s(s)) ≥ (as(0) + 2Ew(0))e−2sAs(s) − 2Ew(0)e−3sAs(s) +2(p − 1) p + 1 e −2s_A s(s)A(s), 2 p + 3((e −s_A s(s)) p+3 p+1 − a p+3 2 0 ) ≥as(0) + 2Ew(0) + p − 1 p + 1A(s) es− 2Ew(0) e−3sA(s) ≥ 0 for some large s4, s ≥ s5, since as(0) + 2Ew(0) > 0. Therefore, for s ≥ s5,

(e−sAs(s)) p+3 p+1 ≥ a p+3 2 0 + p + 3 2 as(0) + 2Ew(0) + p − 1 p + 1A(s) es− 2Ew(0) e−3sA(s), As(s) ≥a p+3 2 0 + p + 3 2 as(0) + 2Ew(0) + p − 1 p + 1A(s) es− 2Ew(0) e−3sA(s) p+1 p+3 ≥ (1 2) p+1 p+3_a p+1 2 0 e s + (1 2) p+1 p+3 p + 3 2 as(0) + 2Ew(0) +p − 1 p + 1A(s) es− 2Ew(0) p+1p+3 e−3(p+1)p+3 s_A(s) p+1 p+3_es = (1 2) p+1 p+3_a p+1 2 0 e s_{+ (}1 2) p+1 p+3₍p + 3 2 ) p+1 p+3 as(0) + 2Ew(0) +p − 1 p + 1A(s) es− 2Ew(0) p+1p+3 e−p+32ps_A(s) p+1 p+3_, A(s) ≥ (1 2) p+1 p+3_a p+1 2 0 (e s_{− e}s5_{) + A(s} 5).

By the same arguments as in the proof of Theorem 2.1, the assertions in can be

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4. Nonexistence of global solution when Ew(0) = 0

In this section we want to show there is no global solution of (1.1) under zero energy Ew(0) = 0, with u0(u1− u0) > 0.

Theorem 4.1. If T is the life-span of u and u is the positive solution of (1.1) with Ew(0) = 0 and u0(u1− u0) > 0, then T is finite. This means that the global solution of (1.1) does not exist for (u1− u0)2= _p+12 u

p+1

0 , u0(u1− u0) > 0. Remark 4.2. If Ew(0) = 0, u0(u1− u0) ≤ 0 we conjecture that solutions of (1.1) exist globally and have the same asymptotic behavior as that stated in Remark 3.2. Yet, again, we do not have a rigorous proof.

Proof. Using the Lemma 1.2, (1.10), Ew(0) = 0 and u0(u1− u0) > 0 we obtain as(s) = as(0)e−s+ (p + 3)K(s) + (p − 1) Z s 0 er−sK(r)dr > 0, a(s) = a(0) + as(0)(1 − e−s) + (p + 3) Z s 0 K(r)dr + (p − 1) Z s 0 Z r 0 eη−rK(η)dηdr, a(s) = a(0) + as(0)(1 − e−s) + (p + 3) Z s 0 K(r)dr + (p − 1) Z s 0 (e−r Z r 0 eηK(η)dη)dr = a(0) + as(0)(1 − e−s) + (p + 3) Z s 0 K(r)dr + (p − 1)− e−s Z s 0 erK(r)dr + Z s 0 K(r)dr = a(0) + as(0)(1 − e−s) + Z s 0 (2(p + 1) − (p − 1)er−s)K(r)dr ≥ a(0) + as(0)(1 − e−s) + (p + 3) Z s 0 K(r)dr. By Lemma 1.2, (1.7), a(s) ≥ a(0) + as(0)(1 − e−s) + (p + 3) Z s 0 K(r)dr = a(0) + as(0)(1 − e−s) + p + 3 p + 1 Z s 0 2e−2r Z r 0 e2ηwp+1(η)dηdr = a(0) + as(0)(1 − e−s) − p + 3 p + 1e −2sZ s 0 e2ηwp+1(η)dη +p + 3 p + 1 Z s 0 wp+1(r)dr, a(s) ≥ a(0) + as(0)(1 − e−s) + p + 3 p + 1 Z s 0 (1 − e2r−2s)wp+1(r)dr = a(0) + as(0)(1 − e−s) + p + 3 p + 1e −2sZ s 0 (es+ er)(es− er)wp+1(r)dr, a(s) ≥ a(0) + as(0)(1 − e−s) + p + 3 p + 1e −sZ s 0 (es− er_)wp+1_(r)dr = a(0) + as(0)(1 − e−s) + p + 3 p + 1e −sZ s/2 0 + Z s s/2 (es− er_)wp+1_(r)dr,

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Z s/2 0 (es− er_)wp+1_{(r)dr ≥ a(0)}p+1₂ Z s/2 0 (es− er_)dr = a(0)p+12 1 +s 2e s_{− e}s 2, Z s s/2 (es− er_)wp+1_{(r)dr ≥ a(}s 2) p+1 2 s 2e s_{+ e}s 2 − es; thus a(s) ≥ a(0) + as(0)(1 − e−s) + p + 3 p + 1e −sn_a(0)p+1 2 (1 + s 2e s_{− e}s 2) + a(s 2) p+1 2 (s 2e s_{+ e}s 2 − es) o . Further, for s ≥ s6:= 1 2 + p + 1 p + 3(1 − a(0) − as(0)(1 − e −s_))a(0)p+1−2 we also have a(s) ≥ a(0) + as(0)(1 − e−s) + p + 3 p + 1 Z s 0 (1 − e2r−2s)wp+1(r)dr ≥ a(0) + as(0)(1 − e−s) + p + 3 p + 1a(0) p+1 2 (s −1 − e −2s 2 ) ≥ a(0) + as(0)(1 − e−s) + p + 3 p + 1a(0) p+1 2 _{(s −}1 2) ≥ 1, (ass+ as)(s) = 2 w(s)p+1−a(s) 4 + 2ws(s)2+ w(s)2 2 ≥ 2a(s)p+12 −a(s) 4 + 2|wsw|(s), ass(s) ≥ 2a(s) a(s)p−12 −1 4 =a(s) 2 a(s)p−12 − 1 +3 2a(s) p+1 2 ≥3 2a(s) p+1 2 _;

thus there exists s7> s6 such that for s ≥ s7, a(s7) p+3 2 _{= −}p + 3 3 a 2 s(s6) + 2a(s6) p+3 2 _, (a2_s(s))s= 2as(s)ass(s) ≥ 3a(s)p+12 a_s(s) = 6 p + 3(a(s) p+3 2 )_s, a2_s(s) ≥ 6 p + 3a(s) p+3 2 _{+ a}2 s(s6) − 6 p + 3a(s6) p+3 2 ≥ 3 p + 3a(s) p+3 2 _, as(s) ≥ r 3 p + 3a(s) p+3 4 ∀s ≥ s₇,

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−4 p − 1(a(s) 1−p 4 )_s= a(s)− p+3 4 a_s≥ r 3 p + 3 ∀s ≥ s7, −4 p − 1a(s) 1−p 4 ≥ r 3 p + 3(s − s7) − 4 p − 1a(s7) 1−p 4 ∀s ≥ s₇ a(s)1−p4 ≤ −p − 1 4 r 3 p + 3(s − s7) − 4 p − 1a(s7) 1−p 4 ; therefore there exists

S₃∗≤ s7+ r p + 3 3 4 p − 1a(s7) 1−p 4 such that a(s)1−p4 → 0 for s → S∗ 3.

This means that the solution of (1.1) does not exist for all t ≥ 1 and the life-span T∗ of u is finite with T∗≤ ln S∗

3.

Remark 4.3. If we reconsider the solution behavior of the problem (1.1) on (0, 1], as one may use the same transformations as given in Fundamental Lemmas and obtain problems (1.2)–(1.4) on the interval −∞ < s ≤ 0. On the other hand, by changing variables τ = −s, w(s) = X(τ ), equation (1.2) yields

Xτ τ − Xτ = Xp, τ ∈ (0, ∞), (4.1)

X(0) = X0= w(0) = w0= u0, (4.2)

Xτ(0) = X1= −w1= u0− u1. (4.3)

In [7] we estimated the life-span τ∗ of the positive solution X of (4.1) in three different cases:

(a) X1= 0, X0> 0: τ∗≤ ek1, for a suitable k1. (b) X1> 0, X0> 0: (i) EX(0) ≥ 0, τ∗≤ ek2, k2:= _p−12 q p+1 2 X 1−p 2 0 . (ii) EX(0) < 0, τ∗≤ ek3, k3:= _p−12 X0 X1 .

Therefore, the solutions of (1.1) on (0, 1] can not defined on this interval but blow up in the interior under such circumstances.

Acknowledgments. We want to thank Prof. Klaus Schmitt for his comments and improvements on writing. We want to thank Prof. Long-Yi Tsai and Prof. Tai-Ping Liu for their continuous encouragement and their discussions of this work, We want to thank NSC and Grand Hall for their financial support and the referee for his interesting and helpful comments on this work.

References

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Meng-Rong Li

Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan E-mail address: liwei@math.nccu.edu.tw

Tsung-Jui Chiang-Lin

Graduate Institute of Finance, National Taiwan University of Science an Technology, Taipei, Taiwan

E-mail address: D9918005@mail.ntust.edu.tw Young-Shiuan Lee

Department of Statistics, National Chengchi University, Taipei, Taiwan E-mail address: 99354501@nccu.edu.tw

Daniel Wei-Chung Miao

Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei, Taiwan