Finite Time Blow Up and Stability of a Semilinear Equation with a Time Dependent Lévy Generator

Stochastic Models, 22:735–752, 2006Copyright © Taylor & Francis Group, LLCISSN: 1532-6349 print/1532-4214 onlineDOI: 10.1080/15326340600878636

FINITE TIME BLOW UP AND STABILITY OF A SEMILINEAREQUATION WITH A TIME DEPENDENT LÉVY GENERATOR

José Alfredo López Mimbela � Centro de Investigación en Matemáticas,Guanajuato, México

Aroldo Pérez Pérez � Universidad Juárez Autónoma de Tabasco,Cunduacán, Tabasco

� We give sufficient conditions for finite time blow up and for existence of global non-negativemild solutions to semilinear equations of the form �u(t , x)/�t = L(t)u(t , x) + �u�(t , x), t >s ≥ 0, u(s, x) = �(x), x ∈ �d , where � > 0 and � > 1 are constants, � ≥ 0 is bounded andmeasurable, and L(t) is the generator of a Lévy process in �d , t ≥ 0. We apply our resultsto equations with operators of the form L(t) = k(t)�� and L(t) = 2−1 ∑d

i ,j=1 kij (t)�2/�xi�xj ,

where �� is the generator of the symmetric �-stable process, and k, kij ≥ 0 , 1 ≤ i, j ≤ d, arecontinuous functions.

Keywords Blow up of semilinear equations; Evolution systems; Global solutions; Lévygenerator.

Mathematics Subject Classification Primary 60J80, 60G51; Secondary 35B35, 35K57.

1. INTRODUCTION AND BACKGROUND RESULTS

Consider a reaction-diffusion equation of the form

�u�t

= Au + �u�, u(0, x) = �(x), x ∈ E , (1)

where A is the infinitesimal generator of a strong Markov process, Eis a locally compact space, � > 0 and � > 1 are constants, and � ≥ 0 isbounded and measurable. Reaction-diffusion equations of the prototype(1) frequently arise as mathematical models in biology, populationdynamics, heat conduction, physic and engineering, and have been

Received January 2005; Accepted September 2005Address correspondence to José Alfredo López Mimbela, Area de Probabilidad Y Estadistica,

Centro de Investigación en Matemáticas, Apartado Postal 402, 36000 Guanajuato, México; E-mail:[email protected]

736 López Mimbela and Pérez

investigated sustainedly in the last three decades due to the rich andmathematically appealing structure associated to their qualitative behavior;see Refs.[3,18,19] and the references therein. It is well known that for any non-trivial initial value � there exists a number T� ∈ (0,∞] such that (1) has aunique solution u on [0,T�) × E , which is bounded on [0,T ] × E for any0 < T < T�, and if T� < ∞, then ‖u(t , ·)‖∞ → ∞ as t ↑ T�. When T� = ∞we say that u is a global solution, and when T� < ∞ we say that u blows upin finite time or that u is nonglobal.

Blow up properties of semilinear equations have been intensely studiedmainly in the analytic literature; see Refs.[1,6,10,19] for surveys. In theclassical case of the Laplacian A = � in E = �d , Fujita[9] proved that, ford(�− 1)< 2, any nontrivial positive solution of (1) is nonglobal, whereasif d(�− 1)> 2, then the solution of (1) is global provided the initial valuesatisfies � ≤ �qr for some r > 0 and some sufficiently small constant � > 0,where qr , r > 0, are the transition densities of Brownian motion. In thissense, dc := 2/(� − 1) is the critical dimension for blow up of (1). Thecase of the fractional power A = −(−�)�/2 of the Laplacian, 0 < � ≤ 2, wasinvestigated by Nagasawa and Sirao[17] and Sugitani[24], who showed that inthis setting dc = �/(� − 1).

A parallel probabilistic treatment of Eq. (1) was initiated by Nagasawaand Sirao[17], and developed further by López-Mimbela[11], López-Mimbela and Wakolbinger[14,15], Birkner et al.[4,5], and López-Mimbela andPrivault[12,13]; see also Ref.[7]. In the case of integer exponents � ≥ 2 it wasproved in Refs.[11,15] that the mild solution to (1) can be represented interms of an expectation functional of a related branching particle system,namely,

u(t , x) = �[e St

∏y∈Bx (t)

�(y)], x ∈ E , t ≥ 0� (2)

Here (Bx(t))t≥0 is a branching particle system in E (with exponentialindividual lifetimes, branching numbers �, and particle motions withgenerator A), starting from an ancestor at x ∈ E , and St denotes the totaltime length of the ancestor’s offspring tree up to time t ≥ 0.

The representation (2) allows for the following heuristic explanation:Assume that � is sufficiently small and that the individuals in thepopulation are sufficiently mobile. Then, with large probability a givenparticle can rapidly reach those regions where the initial value is close to 0.At branching, such a particle contributes with � − 1 additional factors �(y)to the product inside the RHS of (2), which, because of the decay of �,counteracts the tendency to explosion of the tree-length exponential. Also,if the particles are sufficiently mobile and the initial value decays quicklyenough, larger exponents � favor nonexplosion of solutions. In the case

Blow up and Stability of a Semilinear Equation 737

of the Laplacian A = �, which is the generator of Brownian motionwith variance parameter 2, the representation (2) renders an intuitiveexplanation of Fujita’s conditions quoted above since large mobility ofBrownian particles goes along with big dimensions.

In Ref.[17] it was shown that the mild solution of (1) (with an integer� ≥ 2) is nonglobal provided

T (t)�(x) ≥(

1�t(� − 1)

)1/(�−1)

(3)

for some x ∈ E and t ≥ 0, where �T (t)t≥0 denotes the semigroup withinfinitesimal generator A. Moreover, it was proved that the condition

(� − 1)�∫ ∞

0

(supy∈E

T (s)�(y))�−1

ds < 1 (4)

ensures supx∈E u(t , x) < ∞ for all t ≥ 0. In this paper we consider thesemilinear non-autonomous Cauchy problem

�u(t , x)�t

= L(t)u(t , x) + �u�(t , x), t > s ≥ 0, (5)

u(s, x) = �(x), x ∈ �d ,

where �, � are as in (1), 0 ≤ � ∈ C0(�d) (where C0(�d) is the space ofcontinuous functions on �d vanishing at infinity), and L(t), t ≥ 0, are Lévygenerators of the form

L(t)f (x) = 12

d∑i ,j=1

aij(t , x)�2f (x)�xi�xj

+d∑

i=1

bi(t , x)�f (x)�xi

,

+∫�d

(f (x + y) − f (x) − 〈y, f (x)〉

1 + |y|2)�(t , x , dy),

f ∈ Dom(L(t)), t ≥ 0� (6)

Here a : [0,∞) × �d → S+d is bounded and continuous (where S+

d is thespace of symmetric, non-negative definite, square real matrices of order d),b : [0,∞) × �d → �d is bounded and measurable, and �(t , x , ·) is a Lévymeasure such that

∫�|y|2(1 + |y|2)−1�(t , x , dy) is bounded and continuous in

(t , x) for all Borel sets � ⊂ �d\�0.From the probabilistic perspective, the operators L(t), t ≥ 0, represent

the generators of the most general stochastically continuous, �d -valuedMarkov processes with independent increments. In applications, thesegenerators allow for non-local integro-differential or pseudodifferential


diffusive terms that have been used, e.g., in models of anomalousgrowth of molecular interfaces involving hopping and trapping (Ref.[16]),and in hydrodynamic models with modified diffusivity (Ref.[2]). Variousdifferential equations involving special issues of (6) arise in fields likestatistical physics, hydrodynamics and molecular biology[21].

By adapting the above criteria to our time-inhomogeneous context, weshall prove that conditions similar to (3) and (4) are sufficient, respectively,for finite time blow up and for existence of a global solution of (5). Weapply these results to the case of L(t) = k(t)��, t ≥ 0, where k : [0,∞) →[0,∞) is continuous and does not vanish identically, and the operator �� isthe generator of the d -dimensional symmetric �-stable process, 0 < � ≤ 2.Assuming that

∫ t0 k(s)ds ∼ t as t → ∞ for some > 0, we show that in this

case the critical dimension for blow up of (5) is

dc = �

(� − 1),

meaning that if d < dc , then, apart from u ≡ 0, there are no positive globalsolutions to (5), whereas if d > dc , then the solution u of (5) is globalprovided u(0, ·) is sufficiently small. We also show that the case = 0, whichcorresponds to an integrable k, yields finite time blow up of u for anynontrivial initial value.

Similar conclusions hold for L(t) = k(t)(�� + ∑di=1 bi�/�xi), t ≥ 0,

where k is as above, � ∈ (0, 1) ∪ (1, 2] and b = (b1, � � � , bd) ∈ �d .We emphasize that the factor k(t) is able to modify the diffusivity of the

motion process as to alter drastically the blow up behavior of Equation (5).For the special case of k considered here, it turns out that integrability ofk already excludes existence of global solutions, regardless of the spatialdimension and the stability exponent �. Intuitively, in this case the blowup behavior of Equation (5) (i.e., finite time blow up vs. existence ofglobal solutions) parallels that of du(t)/dt = u�(t) with u(0) > 0. Similarly,if

∫ t0 k(s)ds ∼ t as t → ∞ with > 0, then the blow up behavior of (5)

is analogous to that of the equation �u/�t = ��/ u + �u�, with u(0, ·) � 0.Thus, again at a heuristic level, in the case of > 0 it is the stabilityexponent that results altered. Since such exponent determines the mobilityof stable motions, the factor k(t) can diminish or enhance the diffusivity ofthe motion, according to 0 < < 1 or > 1.

We also investigate the case of L(t) = 2−1∑d

i ,j=1 kij(t)�2/�xi�xj , where

the matrix-valued function k := �kij , 1 ≤ i , j ≤ d is continuous andpositive-definite, and show under suitable assumptions on k that indimensions d > 1 + 2/(� − 1) the solution of (5) is global provided theinitial value satisfies �(x) ≤ �(1 + |x |)−d+1, x ∈ �d , for a sufficiently small� > 0. In the case d = 1, � = 2 and 0 < � ≤ k(t) ≤ � < ∞, t ≥ 0, where �,� are constants, we prove finite time blow up for any nonvanishing � ≥ 0.


2. CRITERIA FOR FINITE TIME BLOW UP, AND FOREXISTENCE OF GLOBAL SOLUTIONS

Let a, b, and � be as in (6), then (see Ref.[22]) the martingale problemfor �L(t)t≥0 is well posed in C∞

c (�d) (where Cnc (E) is the space of

continuous functions � : E → �, having compact support and possessingcontinuous derivatives of order n, 1 ≤ n ≤ ∞), i.e., there exists an �d -valued Markov process X ≡ �X (t)t≥0, uniquely determined in law, suchthat for every � ∈ C∞

c (�d) and each s ≥ 0,

M (t) := �(X (t)) −∫ t

sL(r )�(X (r ))dr , t ≥ s,

is a martingale. Let �P (s, x , t , dy)t≥s≥0 denote the transition probabilityfunction corresponding to the process X . The family of operators�U (t , s)t≥s≥0 defined by

U (t , s)�(x) =∫

�(y)P (s, x , t , dy), x ∈ �d , (7)

where � is any bounded measurable function, constitutes the evolutionsystem generated by �L(t)t≥0.

Let �Y (t)t≥0 denote the time-homogeneous Markov process on [0,∞)with transition probabilities

P (t , r , �) = �r−t(�), � ∈ �([0,∞)), t , r ≥ 0,

whose generator Q is given by Qf = −f ′, f ∈ C 1c ([0,∞)). Let D be the

linear span of functions g ∈ C0([0,∞) × �d) (where C0([0,∞) × �d) isthe space of continuous functions on [0,∞) × �d vanishing at infinity) ofthe form

g (t , x) = f (t)�(x), f ∈ C 1c ([0,∞)), � ∈ C∞

c (�d)�

Then

Z = �Z (t)t≥0 ≡ �(Y (t),X (t))t≥0

is a Markov process with values in [0,∞) × �d whose generator A is theclosure of the linear operator A|D , where A is given by

Ag (r , x) = −�g (r , x)�r

+ L(r )g (r , x), r ≥ 0, x ∈ �d ,

for all functions g (t , x) which are differentiable in t , g (t , ·) ∈ Dom(L(t)),t ≥ 0, and Ag ∈ C0([0,∞) × �d); see Ref.[8], p. 253.


Note that if �P (t , (r , x), dz)t≥0 are the transition probabilitiescorresponding to the process Z , then

P (t , (r , x),C × �) ={P (r − t , x , r , �)�r−t(C), if r > t ≥ 0,P (0, x , r , �)�0(C), if 0 ≤ r ≤ t ,

(8)

for any C ∈ �([0,∞)) and � ∈ �(�d). Thus, from (7) and (8) it followsthat the semigroup �T (t)t≥0 with generator A is given by

(T (t)g )(r , x) ={U (r , r − t)g (r − t , x), 0 ≤ t < r , x ∈ �d ,U (r , 0)g (0, x), 0 ≤ r ≤ t

(9)

where g ∈ C0([0,∞) × �d). Clearly, �T (t)t≥0 is a positivity-preserving,strongly continuous semigroup of contractions whose generator isconservative, hence Z is a strong Markov process on E = [0,∞) × �d .

Now, let us consider the semilinear problem

�

�tw(t , z) = Aw(t , z) + �w�(t , z), w(s, z) = �(z), t > s ≥ 0, z ∈ E , (10)

where A is the generator of the semigroup defined by (9). From (2) weknow that, for any integer � ≥ 2, the mild solution of (10) admits therepresentation

w(t , (r , x)) = E[e St−s

∏z∈B(r ,x)(t)

�(z)], t ≥ s ≥ 0, (r , x) ∈ E � (11)

Lemma 2.1. Let �(·, ·) ∈ C0([0,∞) × �d) be non-negative. If for some(r , x) ∈ E and t > s ≥ 0,

(� − 1)�(t − s)(T (t − s)�(r , x))�−1 ≥ 1,

then

E[e St−s

∏z∈B(r ,x)(t)

�(z)]

= ∞�

Proof. See Ref.[14], p. 804.

Theorem 2.1. Let � ≥ 2 be an integer. If for some x ∈ �d and t > s ≥ 0,

(� − 1)�(t − s)(U (t , s)�(x))�−1 ≥ 1, (12)

then the mild solution of (5) blows up in finite time.


Proof. The mild solution of (10) is given by

�(t , (r , x)) = T (t − s)�(r , x)

+ �

∫ t

sT (t − �)��(�, (r , x))d�, (r , x) ∈ E , t ≥ 0�

Thus,

�(t , (t , x)) = T (t − s)�(t , x) + �

∫ t

sT (t − �)��(�, (t , x))d�,

and from (9) we know that

w(t , (t , x)) = U (t , s)�(s, x) + �

∫ t

sU (t , �)��(�, (�, x))d��

Choosing � ∈ C0([0,∞) × �d) so that �(s, ·) ≡ u(s, ·) we see that u(t , x) :=�(t , (t , x)) is the mild solution of (5). Hence, from (11) we have that u(t , x)can be expressed as

u(t , x) = E[e St−s

∏z∈B(t ,x)(t)

�(z)], t ≥ s ≥ 0, (t , x) ∈ E � (13)

By combining (9) and (12) we obtain

(� − 1)�(t − s)(T (t − s)�(t , x))�−1 ≥ 1,

and then by Lemma 2.1,

E[e St−s

∏z∈B(t ,x)(t)

�(z)]

= ∞�

Thus, the conclusion follows from (13).

Using (4), a sufficient condition for existence of global solutions of (5)can be deduced for � ∈ �2, 3, � � � . In order to derive a criterion valid forall powers � > 1, we shall instead adapt the method of proof of Ref.[25] toour time-inhomogeneous setting.

Theorem 2.2. Let � ≥ 0, � > 1 and s ≥ 0 be such that

(� − 1)�∫ ∞

s‖U (t , s)�‖�−1

∞ dt < 1� (14)


Then the mild solution of (5) is global and satisfies

u(t , x) ≤ U (t , s)�(x)

[1 − (� − 1)�∫ ts ‖U (r , s)�‖�−1

∞ dr ] 1�−1

, t ≥ s, x ∈ �d �

Proof. Let us define

B(t) =[1 − (� − 1)�

∫ t

s‖U (r , s)�‖�−1

∞ dr]− 1

�−1

, t ≥ s�

Then B(s) = 1 and

ddtB(t) = − 1

� − 1

[1 − (� − 1)�

∫ t

s‖U (r , s)�‖�−1

∞ dr]− 1

�−1−1

× [−(� − 1)�‖U (t , s)�‖�−1∞

]= �‖U (t , s)�‖�−1

∞ B�(t),

which gives

B(t) = 1 + �

∫ t

s‖U (r , s)�‖�−1

∞ B�(r )dr � (15)

Let v : [0,∞) × �d → [0,∞) be a continuous function such that v(t , ·) ∈C0(�d), t ≥ 0, and

U (t , s)�(x) ≤ v(t , x) ≤ B(t)U (t , s)�(x), t ≥ s, x ∈ �d �

Then

�v(t , x) := U (t , s)�(x) + �

∫ t

sU (t , r )v�(r , x)dr

satisfies

�v(t , x) ≤ U (t , s)�(x) + �

∫ t

sB�(r )U (t , r )(U (r , s)�(x))�dr

≤ U (t , s)�(x) + �

∫ t

sB�(r )U (t , r )U (r , s)�(x)‖U (r , s)�‖�−1

∞ dr

= U (t , s)�(x)[1 + �

∫ t

s‖U (r , s)�‖�−1

∞ B�(r )dr]

= B(t)U (t , s)�(x),


where we used (15) in the last equality. Therefore,

U (t , s)�(x) ≤ �v(t , x) ≤ B(t)U (t , s)�(x), t ≥ s, x ∈ �d �

Now put u0(t , x) = U (t , s)�(x) and un+1(t , x) = �un(t , x), n = 0, 1, � � � .Using that u0(t , x) ≤ u1(t , x) for s ≤ t , x ∈ �d , and the fact that theevolution family �U (t , s)t≥s≥0 preserves positivity, we obtain that

�u(t , x) ≤ � �(t , x)

whenever u(t , x) ≤ �(t , x) for all t ≥ 0 and x ∈ �d . By induction it followsthat

un(t , x) ≤ un+1(t , x), n ≥ 0,

rendering

u(t , x) ≡ lim supn→∞

un(t , x) ≤ B(t)U (t , s)�(x) < ∞

for all t ≥ s and x ∈ �d . An application of the monotone convergencetheorem yields that u(t , x) satisfies

u(t , x) = U (t , s)�(x) + �

∫ t

sU (t , r )u�(r , x)dr , t ≥ s, x ∈ �d �

Hence u is a mild global solution of (5).

3. THE CASE L(t) = k(t)��

We shall denote by W ≡ �W (t)t≥0 the d -dimensional sphericallysymmetric �-stable process, 0 < � ≤ 2. Recall that W is a Lévy process in �d

whose infinitesimal generator is the fractional power −(−�)�/2 =: �� of theLaplacian. The transitions densities �p(t , ·)t>0 of W are radially symmetricand enjoy the self-similarity property p(t , x)= t−d/�p(1, t−1/�x), x ∈�d , t > 0.

In this section we suppose that L(t) = k(t)��, t ≥ 0, where k : [0,∞) →[0,∞) is continuous and does not vanish identically. The semilinearequation (5) thus becomes

�u(t , x)�t

= k(t)��u(t , x) + �u�(t , x), t > s ≥ 0,

u(s, x) = �(x), x ∈ �d �

(16)

We define

K (t , s) :=∫ t

sk(r )dr , 0 ≤ s ≤ t , (17)


and denote by �(·) the Lévy measure corresponding to the infinitelydivisible distribution �(·) = P [W (1) ∈ ·]. Then ��(t , ·) := K (t , 0)�(·)t≥0 is afamily of Lévy measures satisfying

�(0, ·) = 0,

�(s, �) ≤ �(t , �) whenever 0 ≤ s ≤ t for every � ∈ �(�d),

�(s, �) → �(t , �) for every � ∈ �(�d) as s → t �

Hence (see, e.g., Ref.[20]) there exists an additive process in law �X (t)t≥0,uniquely determined up to identity in law, such that the probabilitymeasure �(t , ·) = P [X (t) ∈ ·] has Lévy measure �(t , ·), t ≥ 0. Moreover,�X (t)t≥0 is a Markov process having transition probability functionP (s, x , t , �) given by

P (s, x , t , �) = P [X (t) − X (s) ∈ � − x], t ≥ s ≥ 0, x ∈ �d , � ∈ �(�d)�

Since �(·) has Lévy measure �(·), the probability �K (t ,0)(·) has Lévymeasure K (t , 0)�(·) ≡ �(t , ·). Therefore P [X (t) ∈ ·] = �(t , ·) = �K (t ,0)(·) =P [W (K (t , 0)) ∈ ·], and thus

P (s, x , t , �) = P [W (K (t , 0)) − W (K (s, 0)) ∈ � − x]= P [W (K (t , 0) − K (s, 0)) ∈ � − x]= P [W (K (t , s)) ∈ � − x]= (S(K (t , s))1�)(x),

where �S(t)t≥0 denotes the semigroup with generator �� and 1� is theindicator function of � . Since the function (t , x) �→ S(K (t , s))�(x), t ≥ s,solves

�w(t , x)�t

= k(t)��w(t , x), t > s, x ∈ �d ,

w(s, x) = �(x),

it follows that for any � : �d → � bounded and measurable,

U (t , s)�(x) =∫�d

�(y)P (s, x , t , dy) = S(K (t , s))�(x)� (18)

3.1. Finite Time Blow Up of Positive Solutions

Let Br (x) denote the open ball in �d of radius r > 0 centered at x ∈�d ; we put Br ≡ Br (0). Without loss of generality, let t0 > 0 be such thatK (t0, 0) = ∫ t0

0 k(r )dr ≡ � > 0.


Lemma 3.1.1. Assume that � ≥ 2 is an integer, and that for some t > t0

Kd(�−1)

� (t , 0) < (� − 1)�t

(p(1, z)

∫B�1�

�(y)dy

)�−1

, (19)

for any z ∈ �B2. Then the mild solution of (16) blows up in finite time.

Proof. In order to verify the conditions of Theorem 2.1 we can assume,due to translation invariance of (16), that the initial value in (16) does notidentically vanish on B

�1�. Using (18), self-similarity of stable densities and

the fact that p(t , ·) is radially decreasing, t > 0, we obtain for x ∈ B(K (t ,0))

1�

and z ∈ �B2,

U (t , 0)�(x) = S(K (t , 0))�(x)

= E0[�(W (K (t , 0)) + x)]= E0

[�(K

1� (t , 0)

(W (1) + K − 1

� (t , 0)x))]

≥∫B1

�(K

1� (t , 0)y

)P

[W (1) ∈ dy − K − 1

� (t , 0)x]

=∫B1

�(K

1� (t , 0)y

)p(1, y − K − 1

� (t , 0)x)dy

≥ p(1, z)∫B1

�(K

1� (t , 0)y

)dy

= p(1, z)K − d� (t , 0)

∫B(K (t ,0))

1�

�(y)dy�

Hence, for any t ≥ t0,

U (t , 0)�(x) ≥ p(1, z)K − d� (t , 0)

∫B�1�

�(y)dy,

and for t big enough,

(� − 1)�t(U (t , 0)�(x))�−1 ≥ (� − 1)�t

p(1, z)∫B�1�

�(y)dy

�−1

K − d(�−1)� (t , 0)�

Condition (12) now follows from (19).

A direct consequence of the lemma above is the following explosionresult.


Proposition 3.1.1. a) Assume 0 ≤ � ∈ C0(�d) does not vanish identically,and ∫ ∞

0k(r )dr < ∞�

Then the mild solution of (16) blows up in finite time for all � ∈ (0, 2] and allintegers d ≥ 1, � ≥ 2.

b) Let � ≥ 2 be an integer. Assume that there exist C > 0 and > 0 suchthat

0 < K (t , 0) ≤ Ct

for all t large enough, and the initial condition 0 ≤ � ∈ C0(�d) does not vanishidentically. If d (� − 1) < �, then the mild solution of (16) blows up in finite time.

Notice that in the case k ≡ 1, Proposition 3.1.1.b yields the well knownfact that the solution of (16) blows up in finite time when d < �/(� − 1)and � ≥ 0 does not vanish identically; see, e.g., Ref.[17].

3.2. Existence of Global Solutions

We consider again the problem (16), but now � > 1 is not restricted tobe an integer.

Proposition 3.2.1. If∫ ∞

0K − d(�−1)

� (t , 0)dt ≡ M < ∞ (20)

and ( ∫�d

�(y)dy)�−1

<1

(� − 1)�p�−1(1, 0)M, (21)

then the mild solution of (16) is global.


Proof. Using (18) and scaling properties of stable densities wededuce that

U (t , 0)�(x) =∫�d

�(y)p(K (t , 0), x − y)dy

≤ p(1, 0)K − d� (t , 0)

∫�d

�(y)dy,

hence

(� − 1)�∫ ∞

0‖U (t , 0)�‖�−1

∞ dt ≤ (� − 1)�p�−1

(1, 0)( ∫

�d�(y)dy

)�−1

M �

Thus, (14) holds when (20) and (21) are valid. An application ofTheorem 2.2 finishes the proof.

Corollary 3.2.1. Assume condition (20), and let � > 0 be such that

��−1 <[(� − 1)�p

�−1(1, 0)M

]−1�

If there exists l > 0 such that

0 ≤ �(x) ≤ �p(l , x), x ∈ �d ,

then the solution of (16) is global.

Proposition 3.2.2. If there exist l > 0 and � > 0 such that

Ml ≡∫ ∞

0(l + K (t , 0))− d(�−1)

� dt < ∞,

��−1 <[(� − 1)�p

�−1(1, 0)Ml

]−1,

0 ≤ �(x) ≤ �p(l , x), x ∈ �d , (22)


Proof. Notice that

U (t , 0)�(x) =∫�d

p(K (t , 0), x − y)�(y)dy

≤ �

∫�d

p(K (t , 0), x − y)p(l , y)dy = �p(l + K (t , 0), x)�


The assumption on � gives

(� − 1)�∫ ∞

0‖U (t , 0)�‖�−1

∞ dt ≤ (� − 1)��−1p�−1

(1, 0)Ml < 1�

It follows from Theorem 2.2 that the mild solution of (16) is global.

Corollary 3.2.2. Assume that ct ≤ K (t , 0) for certain positive constants c , ,and all t ≥ 0. If � satisfies (22) and

d (� − 1)�

> 1,


When k ≡ 1 and d(� − 1) > �, the last corollary yields the well knownresult that the solution of (16) is global provided that, for some l > 0 and� > 0 small enough,

0 ≤ �(x) ≤ �p(l , x), x ∈ �d ,

see Ref.[17].

Remark 3.2.1. Propositions 3.2.1 and 3.2.2 remain valid if the operator�� in (16) is substituted by the generator

��,b ≡ �� +d∑

i=1

bi�

�xi,

where � ∈ (0, 1) ∪ (1, 2] and b = (b1, � � � , bd) ∈ �d . Indeed, let �Z (t)t≥0 bethe Lévy process corresponding to the generator ��,b , and let �W (t)t≥0

be the Lévy process corresponding to ��. Due to

�(·) ≡ P [Z (1) ∈ ·] = P [W (1) ∈ ·] ∗ �b(·)

we have

�Z (t) D= �W (t) + tb,

where � ∗ � denotes convolution of the measures � and �, and D= meansequality in distribution. Thus, the evolution system corresponding to thegenerators �k(t)��,bt≥0 is given by

U (t , s)�(·) = S(K (t , s))�(· + K (t , s)b), � ∈ C0(�d),


where �S(t)t≥0 is the semigroup of generator �� and K (t , s) is defined in(17). The proofs of Propositions 3.2.1 and 3.2.2 (with ��,b instead of ��)now proceed as before.

4. THE CASE L(t) = 2–1∑d

i,j=1kij(t)�2/�xi�xj

Let us assume again that k : [0,∞) → [0,∞) is continuous and doesnot vanish identically, and let K (t , s), t ≥ s ≥ 0 defined as in (17). FromProposition 3.1.1 it immediately follows that for integral � ≥ 2, the mildsolution of

�u(t , x)�t

= k(t)2

�u(t , x) + �u�(t , x), t > 0, x ∈ �d

u(0, x) = �(x),(23)

blows up in finite time, provided 0 < K (t , 0) ≤ Ct for all t large enough,d (� − 1) < 2, and the initial condition 0 ≤ � ∈ C0(�d) does not vanishidentically. In particular, (23) blows up if d = 1, = 1, and � = 2. Anonexplosion result can be proved for dimensions d ≥ 2 and � > 1 notnecessarily integral, as we shall see bellow.

Let S+d denote the space of d × d matrices with real coordinates, which

are symmetric and non-negative definite.

Proposition 4.1. Let k : [s,∞) → S+d be continuous and such that �|�|2≤

〈�, k(t)�〉 ≤ �|�|2, where t ≥ s ≥ 0, � ∈ �d and 0 < � < � < ∞. Let u(t , x) bethe mild solution of the semilinear problem

�u(t , x)�t

= 12

d∑i ,j=1

kij(t)�2

�xi�xju(t , x) + �u�(t , x), t > s ≥ 0,

u(s, x) = �(x), x ∈ �d ,

(24)

where � > 0, � > 1, d ≥ 2, and 0 ≤ � ∈ C0(�d). If (d−1)(�−1)2 > 1 and �(x) ≤

�(1 + |x |)−d+1 for a sufficiently small � > 0, then u(t , x) is global.

Proof. Let

K (t , s) = (Kij(t , s))1≤i ,j≤d , 0 ≤ s ≤ t ,

where Kij(t , s) := ∫ ts ki ,j(r )dr , i , j = 1, � � � , d . For 0 ≤ s < t and x , y ∈ �d we

define

p(s, x , t , y) = 1

(2�)d2 (detK (t , s))

12exp

[−12〈y − x ,K −1(t , s)(y − x)〉

]�


Then

P (s, x , t , �) :=∫�

p(s, x , t , y)dy, 0 ≤ s < t , x ∈ �d , � ∈ �(�d),

is a transition probability function, and

v(t , x) ≡ U (t , s)�(x) =∫�d

�(y)P (s, x , t , dy)

solves the homogeneous equation

�v(t , x)�t

= 12

d∑i ,j=1

kij(t)�2

�xi�xjv(t , x)

v(s, x) = �(x),

see Ref.[23]. It is easy to verify that

p(s, x , t , y) ≤(�

�

) d2

p(t − s, y − x),

where

p(t − s, y − x) = 1

(2��(t − s))d2exp

[− |y − x |22�(t − s)

]�

In order to apply Theorem 2.2 we fix s ≥ 0 and l ∈ (s,∞). Then,∫ ∞

s‖U (t , s)�‖�−1

∞ dt =∫ l

s‖U (t , s)�‖�−1

∞ dt +∫ ∞

l‖U (t , s)�‖�−1

∞ dt =: I1 + I2�

(25)

Using the continuity of (t , s) �→ U (t , s)�, � ∈ C0(�d), and the fact that, foreach t ≥ s ≥ 0, U (t , s) is a positivity-preserving linear bounded operator,we obtain

I1 =∫ l

s‖U (t , s)�‖�−1

∞ dt ≤ ��−1

∫ l

s‖U (t , s)(1 + |·|)−d+1‖�−1

∞ dt ≤ ��−1M1

for some constant M1 > 0. For the second integral we have

I2 =∫ ∞

l‖U (t , s)�‖�−1

∞ dt =∫ ∞

l

∥∥∥∥ ∫�d

�(y)p(s, x , t , y)dy∥∥∥∥�−1

∞dt

≤(�

�

) d(�−1)2

��−1

∫ ∞

l

∥∥∥∥ ∫�d(1 + |y|)−d+1p(t − s, y − x)dy

∥∥∥∥�−1

∞dt


=(�

�

) d(�−1)2

��−1

∫ ∞

l

1

(2��(t − s))d(�−1)

2

×∥∥∥∥ ∫

�d(1 + |y|)−d+1e− |y−x |2

2�(t−s) dy∥∥∥∥�−1

∞dt �

Using that the function

w(r , x) =∫�d(1 + |y|)−d+1 exp

[−|y − x |2

2�(r )

]dy, r > 0,

is radially symmetric in x , and that ‖w(r , ·)‖∞ = w(r , 0), we obtain

I2 ≤(�

�

) d(�−1)2

��−1

∫ ∞

l

1

(2��(t − s))d(�−1)

2

[∫�d(1 + |y|)−d+1e− |y|2

2�(t−s) dy]�−1

dt

=(�

�

) d(�−1)2

(�

�d2

)�−1 ∫ ∞

l

1

(2�(t − s))d(�−1)

2

[∫�d(1 + (2�(t − s))

12 |z|)−d+1

× exp(−|z|2)(2�(t − s))d2 dz

]�−1

dt

=(�

�

) d(�−1)2

(�

�d2

)�−1∫ ∞

l

[∫�d(1 + (2�(t − s))

12 |z|)−d+1 exp(−|z|2)dz

]�−1

dt �

≤(�

�

) d(�−1)2

(�

�d2

)�−1∫ ∞

l

1

(2�(t − s))(d−1)(�−1)

2

[∫�d

|z|−d+1exp(−|z|2)dz]�−1

dt

=(�

�

) d(�−1)2

(�

�d2

)�−1 ∫ ∞

l

1

(2�(t − s))(d−1)(�−1)

2

[∫ ∞

0exp(−r 2)dr

]�−1

dt �

Since (d−1)(�−1)2 > 1, it follows that there exists a constant M2 > 0 such that

I2 ≤ M2��−1. These estimates together with (25) render∫ ∞

s‖U (t , s)�‖�−1

∞ dt ≤ (M1 + M2)��−1�

Choosing � > 0 sufficiently small we obtain (14).

ACKNOWLEDGMENT

The authors are grateful to an anonymous referee for a carefulreading and detailed revision of the original manuscript. His/Her valuablecomments contributed to improve the presentation of our work.


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