Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems

Manuscirpt submitted to Website: http://AIMsciences.orgAIMS’ JournalsVolume X, Number 0X, XX 200X pp. X–XX

DIRICHLET BOUNDARY CONDITIONS CAN PREVENTBLOW-UP IN REACTION-DIFFUSION EQUATIONS AND

SYSTEMS

Marek Fila

Department of Applied Mathematics and StatisticsComenius University

842 48 Bratislava, Slovakia

Hirokazu Ninomiya

Department of Applied Mathematics and InformaticsRyukoku University

Seta, Otsu 520-2194, Japan

and

Juan Luis Vazquez

Departamento de MatematicasUniversidad Autonoma de Madrid

28049 Madrid, Spain

(Communicated by Aim Sciences)

Abstract. This paper examines the following question: Suppose that we havea reaction-diffusion equation or system such that some solutions which are ho-mogeneous in space blow up in finite time. Is it possible to inhibit the occur-rence of blow-up as a consequence of imposing Dirichlet boundary conditions,or of other effects where diffusion plays a role? We give examples of equationsand systems where the answer is affirmative.

1. Introduction. One of the most remarkable properties that distinguish nonlin-ear evolution problems from the linear ones is the possibility of eventual occurrenceof singularities starting from perfectly smooth data, or more accurately, from classesof data for which a theory of existence, uniqueness and continuous dependence canbe established for small time intervals. The simplest form of spontaneous singulari-ties in nonlinear problems appears when the variable or variables tend to infinity astime approaches a certain finite limit T > 0. This is what we call blow-up in finitetime.

Blow-up happens in an elementary form in the theory of ordinary differentialequations (ODE’s). The typical ODE for blow-up is

Ut = Up. (1)

In the linear case p = 1, solutions exist globally in time. On the contrary, if p > 1all solutions with positive initial data blow up in finite time. We see that thenonlinearity plays an important role in the appearance of blow-up. We can be more

2000 Mathematics Subject Classification. Primary: 35K57, 35K50; Secondary: 35B35.Key words and phrases. blow-up, reaction-diffusion, Dirichlet conditions prevent blow-up.

1

2 M. FILA, H. NINOMIYA AND J.L. VAZQUEZ

precise in that respect: let us consider the solutions of the ordinary differentialequation

Ut = f(U), (2)where f(s) is a positive function defined for s ≥ a. The solutions of the ODE withinitial data U(0) ≥ a are given by

t =∫ U(t)

U(0)

ds

f(s).

Therefore, these solutions blow up to infinity in a finite time if and only if∫ ∞ du

f(u)< ∞. (3)

The phenomenon of blow-up has been much studied in a space-time setting whereu = u(x, t) and a diffusion effect is added to the reaction term f(U). One may thinkthat the diffusivity should have a stabilizing effect. It is the purpose of this paperto show that diffusion can even prevent blow-up for certain equations and also forsystems. On the other hand, it is well known that diffusion can sometimes induceblow-up for systems, see [3], [4], [7], for instance.

2. Statement of results. We start our investigation with the typical scalar reac-tion-diffusion equation:

ut = ∆u + f(u) for x ∈ Ω, t > 0, (4)

and assume that f : [0,∞) → [0,∞) is smooth and f(u) > 0 for u > 0. We alsoassume that f satisfies the ODE blow-up condition (3). Notice that the solutions of(4) that are space-homogeneous, i.e., only functions of time, are just the solutionsof (2).

We wonder what effect may the diffusion term have on classes of non-homoge-neous solutions, u = u(x, t). If the equation is equipped with the Neumann bound-ary condition

∂u

∂n= 0 on ∂Ω, t > 0, (5)

where n is the outer normal vector of ∂Ω, the maximum principle tells us thatsolutions also blow up in finite time.

The situation is different for the problem posed in a bounded domain with ho-mogeneous Dirichlet boundary data. We propose to investigate the following typeof question:

Question. Given a function f : [0,∞) → [0,∞), f(u) > 0 for u > 0 and satisfyingthe ODE blow-up condition (3), decide whether or not the solutions of (4) withbounded initial data and the Dirichlet boundary condition

u(x, t) = 0 on ∂Ω, t > 0, (6)

also blow up in finite time.Several situations may arise, showing how strongly diffusion plus the homoge-

neous Dirichlet boundary condition affect blow-up.

Option 1. All solutions with nontrivial nonnegative initial data blow up in finitetime as in the ODE case.

Option 2. All solutions with large enough initial data blow up in finite time butsolutions with small initial data do not.

PREVENTING BLOW-UP 3

Option 3. No solution with bounded initial data blows up in finite time.

Classes of f ’s for which one of the two first options holds are easy to construct.Option 1 holds if for instance f(u) ≥ λ1(Ω)u + up with p > 1 where λ1(Ω) is thefirst eigenvalue of −∆ with the zero Dirichlet boundary condition. This can beseen using Kaplan’s eigenfunction method. Linearization around zero yields thatOption 2 is true if lim supu→0 f(u)/u < λ1(Ω).

On the contrary, the existence of reaction terms f for which the last option holdshas been an open problem. Here, we actually construct examples of f for which allsolutions stay bounded in time and stabilize to bounded stationary states.

Theorem 1. There is a smooth function f : [0,∞) → [0,∞), f(u) > 0 for u > 0,such that the following holds:(i) all solutions of (2) with U(0) > 0 blow up in finite time;(ii) when the reaction-diffusion equation (4) is posed in a bounded domain Ω ⊂ RN ,N ≥ 1, with bounded initial data and homogeneous boundary conditions (6), thenall solutions exist and remain bounded for all t ≥ 0. Moreover, the same resultis true even for the solutions of equation (4) satisfying nonzero boundary data,0 ≤ u(x, t) ≤ C for x ∈ ∂Ω, t ≥ 0.

This means that linear diffusion with the Dirichlet boundary condition can com-pletely prevent blow-up of solutions with bounded data in some reaction-diffusionmodels. In the case of nonlinear diffusion, this is well known, see [6], Chapter VII,for example. Our main point here is to show that linear diffusion is strong enoughto stop blow-up.

The idea of the construction of function f for Theorem 1 is to start with a typicalblow-up function satisfying (3), like f0(u) = cup, with c > 0, p > 1, and thenmodifying it in an infinite number of intervals In = (an, bn) with an < bn < an+1,an →∞, so that the obtained f will be small enough in subintervals of In in orderto serve as a buffer for the possible growth of the solutions of the Cauchy-Dirichletproblem. However, the modification has to be positive and (3) must still hold, andthese simultaneous requirements make the construction quite delicate. The mainargument used in the control of the growth of the solutions of the Cauchy-Dirichletproblem is the strong maximum principle. Comparison with the supersolutions weconstruct does not apply to solutions which are homogeneous in space, like the ODEsolutions, and this explains the difference of behaviour that we were looking for.

As a possible precedent to this result, we may consider the work of Galaktionovand Vazquez [1] which investigates the conditions for so-called complete and incom-plete blow-up for the more general reaction-diffusion equation

ut = φ(u)xx + f(u),

posed in the whole line x ∈ R with t ≥ 0; φ is continuous and increasing and f iscontinuous and positive. There is a result that gives some light on the question thatwe are discussing. Indeed, in Section 3 of paper [1] a class of pairs (φ, f) is found forwhich (i) f satisfies the ODE blow-up condition (3), but (ii) there exists a travellingwave U(x, t) = F (x − ct) such that F (s) is a monotone decreasing function of theargument s ∈ R and

F (−∞) = +∞, F (+∞) = 0.

An easy comparison argument shows then that all solutions of the Dirichlet problemut = ∆φ(u)+f(u), posed in a bounded domain Ω with bounded initial data and zeroboundary condition, are global and cannot blow up in finite time. The construction


of such pairs is rather delicate, and they were called the ‘pathological class’ in thepaper.

There are two main differences that make this result less appealing than Theo-rem 1. On one hand, it is not excluded that the solutions grow up, i.e., that theybecome unbounded as t → ∞. On the other hand, the conditions on φ and fexclude linear diffusion, see Appendix in [1].

We have the following extension of Theorem 1 to the Cauchy problem in RN .

Theorem 2. For every p > 1 there is a smooth function fp : [0,∞) → [0,∞),fp(u) > 0 for u > 0, lim supu→∞ u−pfp(u) > 0 such that all solutions of (2) withU(0) > 0 blow up in finite time, while solutions of (4) with Ω = RN , (N ≥ 1) existfor all t ≥ 0 provided u(x, 0) = u0(x) ≥ 0 is bounded and there are i ∈ 1, . . . , Nand M > 0 such that

|xi|1

p−1 u0(x) ≤ M, x = (x1, . . . , xN ) ∈ RN .

The conclusion is weaker than in Theorem 1 since we do not claim that allsolutions remain bounded. As in [1], we use travelling waves to construct suitablesupersolutions.

Next, we consider the two-component system:

ut = d1∆u + f(u, v), x ∈ Ω, t > 0,vt = d2∆v + g(u, v), x ∈ Ω, t > 0.

(7)

Mizoguchi, Ninomiya and Yanagida [3] found nonlinear functions such that thecorresponding system of ordinary differential equations:

Ut = f(U, V )Vt = g(U, V ). (8)

possesses a globally stable equilibrium (0, 0) while some solutions of (7) with thehomogeneous Neumann boundary condition blow up in a finite time (see also [4, 7]).

Here we present nonlinear functions f and g such that some solutions of (8) blowup in a finite time, while all solutions of (7) with the Dirichlet boundary condition

u(x, t) = v(x, t) = 0, x ∈ ∂Ω, t > 0, (9)

converge to (0, 0) in H1(Ω)×H1(Ω).

Theorem 3. There is a solution of (8) with

f(u, v) := |u− v|p−1(u− v),g(u, v) := |u− v|p−1(u− v)− v,

(10)

which blows up in finite time if p > 1. However, all solutions of (7), (9) withthe same nonlinear term (10) exist globally in time, stay uniformly bounded andconverge to (0, 0) in H1(Ω)×H1(Ω) as t →∞, provided that (N −2)p < N +2 and

(d1 − d2)λ1(Ω) > 1 (11)

where λ1(Ω) is the first eigenvalue of −∆ with the Dirichlet boundary condition.

In the proof of this theorem we use a method which is completely different fromthe idea described before. Instead of constructing supersolutions, we employ aLyapunov functional associated with problem (7), (9) to control the solutions.

We prove Theorems 1, 2 and 3 in Sections 3, 4 and 5, respectively.


3. Proof of Theorem 1. To prove Theorem 1 we shall use the following lemma.

Lemma 1. Given λ > 0 and an increasing sequence an such that a1 > 1,limn→∞ an = ∞, there are a smooth function f : [0,∞) → [0,∞) with f(u) > 0for u > 0, and a sequence bn such that

an < bn < an+1, n = 1, 2, . . . ,∫ ∞

1

du

f(u)< ∞, (12)

∫ bn

an

du√F (bn)− F (u)

≥√

2aλn, (13)

where F ′ = f .

Proof. Take any C1-function g : [0,∞) → [0,∞), g(u) > 0 for u > 0 such that∫ ∞

1

ds

g(s)< ∞, g(s) > 1 for s ≥ 1. (14)

Choose four sequences αn ⊂ (0, 1), bn, cn and dn (an < bn < cn < an+1)specified later (see (18) – (20)). Then, we construct an auxiliary function g bymodifying the function g on the intervals on [an, bn] and [bn, cn] in the followingway (see Figure 1)

g(u) =

dn +g(an)− dn

(bn − an)αn(bn − u)αn for an ≤ u ≤ bn,

dn +g(cn)− dn

cn − bn(u− bn) for bn ≤ u ≤ cn.

an

bn

cn

an+1 b

n+1

u

g(u)

Figure 1. Graph of g(u)


With this modification, we have∫ bn

an

du

g(u)=

∫ bn

an

(dn +

g(an)− dn

(bn − an)αn(bn − u)αn

)−1

du

≤∫ bn

an

(g(an)− dn

(bn − an)αn(bn − u)αn

)−1

du

=bn − an

(1− αn)(g(an)− dn), (15)

and∫ cn

bn

du

g(u)=

∫ cn

bn

(dn +

g(cn)− dn

cn − bn(u− bn)

)−1

du

=cn − bn

g(cn)− dnlog

g(cn)dn

. (16)

Let

G(u) = dn(u− bn)− g(an)− dn

(αn + 1)(bn − an)αn(bn − u)αn+1

on the interval [an, bn]. Then(G

)′ = g and∫ bn

an

du√G(bn)− G(u)

=∫ bn

an

(dn(bn − u) +

g(an)− dn


)−1/2

du

→∫ bn

an

(g(an)


)−1/2

du

as dn → 0. Thus we obtain

limdn→0

∫ bn

an

du√G(bn)− G(u)

=2

1− αn

√(αn + 1)(bn − an)

g(an).

We choose dn ∈ (0, 1/2) small enough so that∫ bn

an

du√G(bn)− G(u)

≥ 11− αn

√bn − an

g(an). (17)

We now choose the sequences βn, αn, bn and cn. Firstly, we take apositive sequence βn such that

∑n

βn < ∞, βn < a2λn , 4β2

ng(an)a−2λn < an+1 − an. (18)

We also define

αn := 1− βna−2λn > 0, bn := an + 2β2

ng(an)a−2λn < an+1. (19)

By (16) and the fact that dn < 1/2 ≤ g(s)/2 for s ≥ 1, we can choose cn so closeto bn that ∫ cn

bn

du

g(u)≤ 2(cn − bn) log

g(cn)dn

≤ βn. (20)


Then, we have ∫ cn

an

du

g(u)≤ 2g(an)βn

g(an)− dn+ βn ≤ 5βn,

and ∫ bn

an

du√G(bn)− G(u)

≥√

2aλn.

Thus, (12) and (13) are satisfied for g. Take a smooth function f such that

12g(u) ≤ f(u) ≤ g(u). (21)

We can easily check (12). Since∫ bn

u

f(s)ds = F (bn)− F (u),

we haveF (bn)− F (u) ≤ G(bn)− G(u),

by integrating the second inequality in (21) over [bn, u]. This guarantees that f alsosatisfies (13).

The existence of supersolutions immediately follows from the previous lemma.

Lemma 2. Let f be as in Lemma 1. Then there is a solution un of

unxx + f(un) = 0 in − aλn < x < aλ

n, (22)

unx(0) = 0, un(x) ≥ an for − aλn < x < aλ

n. (23)

Proof. Since the solution of the initial value problem

unxx + f(un) = 0,

unx(0) = 0, un(0) = bn,

is given by ∫ bn

un(x)

du√F (bn)− F (u)

=√

2|x|,

the assertion follows from (13).

Proof of Theorem 1. Assume that Ω is a bounded domain. Let u(x, t) be a solutionof (4) with u(x, 0) = u0(x). The function un defined in Lemma 2 becomes a super-solution for (4). For any bounded initial function u0(x) there is a positive integern such that

x1 | x = (x1, x) ∈ Ω ⊂ [−aλn, aλ

n],u0(x) < an ≤ un(x1) for x ∈ Ω.

There is no problem in comparing the data on the lateral boundary. The maximumprinciple implies that u(x, t) ≤ un(x1) for t > 0.


4. Proof of Theorem 2. Now we take g(u) := up with p > 1 and we first constructa travelling wave solution u(x, t) = w(ξ) of the equation

ut = uxx + up

with ξ := x− t. This means that w satisfies the ODE

−wξ = wξξ + wp. (24)

Lemma 3. There is a positive constant ξ0 and a solution w of (24) in [−ξ0,∞)such that

w(−ξ0) = 0,

wξ(ξ) < 0, ξ ∈ (0,∞),w(ξ) > 0, ξ ∈ [−ξ0,∞),

limξ→∞

ξ1

p−1 w(ξ) = (p− 1)−1

p−1 .

Proof. Introducing v = wξ, we have

wξ = v,

vξ = −v − wp.

The linearization at (0, 0) is (0 10 −1

),

which implies that there exists a center manifold which is tangential to the vector(1, 0). This orbit enters the fourth quadrant for some ξ = ξ1 and remains there forξ > ξ1. Since

dv

dw= −1− wp

v→ 0 as ξ →∞,

we obtain that(w1−p(ξ))ξ → p− 1 as ξ →∞.

By an appropriate translation in ξ, the proof has been completed.

Proof of Theorem 2. First, we consider the case Ω = R. Take p > 1, λ > (p− 1)/2and an increasing sequence an with a1 > 1 and limn→∞ an = ∞. In the proof ofLemma 1, use g(u) = up and choose βn satisfying (18) and

βn ≤ min

n−2, 2−1g(an)−1/2aλn

= min

n−2, 2−1aλ−p/2

n

. (25)

Defining αn and bn by (19), we can construct a nonlinear function f as in Lemma1. Note that f(u) ≤ up for u ≥ 1. By Lemma 2, there exists a solution un(x) of(22) and (23).

We shall use the travelling wave w from Lemma 3 to construct a supersolutionoutside of [−aλ

n, aλn]. First, we observe that

ws(x, t) := s2/(p−1)w(s(x− st))

is also a travelling wave solution with speed s > 0 and ws(st, t) = s2/(p−1)w(0).Choosing

sn :=(

bn

w(0)

)(p−1)/2

,


we have

wsn(snt, t) = bn.

Since w is monotone decreasing in ξ > 0, there are positive constants ξn such that

w(snξn) =w(0)an

bn.

By (19) and (25), limn→∞ an/bn = 1. Thus, snξn converges to 0. Putting

tn :=aλ

n − ξn

sn,

we have

wsn(aλn, tn) = s2/(p−1)

n w(sn(aλn − sntn))

=bn

w(0)w(snξn)

= an.

By the definition of bn and sn, we then have

tn =aλ

nw(0)(p−1)/2

(an + 2β2ng(an)a−2λ

n )(p−1)/2− ξn

sn

≥ 2(1−p)/2w(0)(p−1)/2aλ−(p−1)/2n − ξn

sn→∞ as n →∞.

Let

xn(t) := infx | 0 ≤ x ≤ aλn, wsn(y, t) ≤ un(y) for y ∈ [x, aλ

n] for 0 ≤ t ≤ tn.

Take

ψn(x, t) :=

wsn(−x,−t) for x ≤ −xn(t),un(x) for − xn(t) < x < xn(t),wsn(x, t) for xn(t) ≤ x.

Thus, ψn is a supersolution of (4) in R for 0 ≤ t ≤ tn.The asymptotic behavior of w(ξ) as ξ → ∞ (Lemma 3) yields that u(x, 0) ≤

ψn(x, 0) for large n, hence

u(x, t) ≤ ψn(x, t) in R× [0, tn].

Since tn →∞ as n →∞, this implies global existence in time.For the higher dimensional case, ψn(xi, t) becomes a supersolution.

5. Proof of Theorem 3. In the previous two sections we used the maximumprinciple which is not applicable for most systems. Now we proceed differently.

Proof of Theorem 3. Consider the solution (U(t), V (t)) of (8). We show that somesolutions blow up in finite time. (cf. [5, Example 6.2]). Indeed,

(U, V ) : |U − V |p−1(U − V ) >

p

p− 1V, pV > U > V > 0

is positively invariant under the flow of (8). The function W (t) := U(t) − V (t)satisfies

Wt = V, Vt = |W |p−1W − V,


which impliesWtt + Wt = |W |p−1W.

It follows in a similar manner as in the proof of Lemma 4.3 in [3] that W (t) blowsup in a finite time if W (0) is sufficiently large.

Next we consider the solution (u(x, t), v(x, t)) of (7) and (9). Set

w(x, t) := u(x, t)− v(x, t).

Then (w, v) satisfies

wt = d1∆w + (d1 − d2)∆v + v,vt = d2∆v − v + |w|p−1w.

(26)

By (11), the positive self-adjoint operator −(d1 − d2)∆− 1 has an inverse operatorK,

K := (−(d1 − d2)∆− 1)−1,

satisfying

‖K‖L(L2(Ω),L2(Ω)) ≤1

(d1 − d2)λ1(Ω)− 1.

The first equation in (26) can be rewritten as

v = K(d1∆w − wt).

By the second equation of (26), we have

K(d1∆wt − wtt) = d2K∆(d1∆w − wt)−K(d1∆w − wt) + |w|p−1w. (27)

Define the norm

‖ϕ‖−1 = ‖K1/2ϕ‖L2(Ω) for ϕ ∈ L2(Ω).

Multiplying (27) by wt and integrating in x over Ω, we have

−d1‖(−∆)1/2wt‖2−1 −12

d

dt‖wt‖2−1

=d1d2

2d

dt‖∆w‖2−1 + d2‖(−∆)1/2wt‖2−1

+d1

2d

dt‖(−∆)1/2w‖2−1 + ‖wt‖2−1 +

1p + 1

d

dt

∫

Ω

|w|p+1dx,

which implies

d

dtL(t) = −(d1 + d2)‖(−∆)1/2wt‖2−1 − ‖wt‖2−1 ≤ 0, (28)

where

L(t) :=12‖wt‖2−1 +

d1d2

2‖∆w‖2−1 +

d1

2‖(−∆)1/2w‖2−1 +

1p + 1

∫

Ω

|w|p+1dx.

Thus, we obtain that ‖w‖Lp+1(Ω) is bounded above. Then |w|p−1w is boundedin L(p+1)/p(Ω) and by the second equation of (26) we see that v is bounded inW 2,(p+1)/p(Ω). By the first equation of (26) we obtain the boundedness of w inW 2,(p+1)/p(Ω) ⊂ Lq(Ω), q > p + 1. By a boot-strap argument for w and v, we seethat w and v are bounded in L∞(Ω). Hence the solution can be extended for allpositive t and stays uniformly bounded.


It remains to prove that (u(·, t), v(·, t)) → (0, 0) in H1(Ω) × H1(Ω) as t → ∞.The inequality (28) also implies that L(t) is monotone decreasing and that

∫ ∞

0

((d1 + d2)‖(−∆)1/2wt‖2−1 + ‖wt‖2−1

)dt < ∞. (29)

We shall show by contradiction that L(t) → 0 as t →∞. Suppose that

limt→∞

L(t) = `∞ > 0. (30)

Multiplying (27) by w and integrating in x yield

−d1

2d

dt‖(−∆)1/2w‖2−1 −

∫

Ω

wKwttdx

= d1d2‖∆w‖2−1 +d2

2d

dt‖(−∆)1/2w‖2−1 + d1‖(−∆)1/2w‖2−1

+12

d

dt‖w‖2−1 +

∫

Ω

|w|p+1dx.

Note thatd2

dt2‖w‖2−1 = 2‖wt‖2−1 + 2

∫

Ω

wKwttdx.

Thus we have12

d2

dt2‖w‖2−1 +

d1 + d2

2d

dt‖(−∆)1/2w‖2−1 +

12

d

dt‖w‖2−1

= ‖wt‖2−1 − d1d2‖∆w‖2−1 − d1‖(−∆)1/2w‖2−1 −∫

Ω

|w|p+1dx.

It follows from (29) and (30) that∫ ∞

0

(‖wt‖2−1 − d1d2‖∆w‖2−1 − d1‖(−∆)1/2w‖2−1 −

∫

Ω

|w|p+1dx)dt = −∞.

Then there exist a positive time T and a positive constant δ such that12

d

dt‖w‖2−1 +

d1 + d2

2‖(−∆)1/2w‖2−1 +

12‖w‖2−1 ≤ −δ < 0 for t ≥ T.

This implies that ‖w‖−1 becomes negative for large t which is a contradiction.Hence we obtain `∞ = 0.

If X = L(p+1)/p(Ω) × L(p+1)/p(Ω) and p > 1, (N − 2)p < N + 2, then problem(7), (9) generates a local semiflow in Xα if α ∈ (0, 1) is close enough to 1 (sothat Xα ⊂ H1(Ω) ×H1(Ω)), cf. [2]. It follows from the definition of L(t) that wconverges to 0 in Lp+1(Ω). Applying Theorem 4.3.6 in [2] to the second equationof (26), we see that v converges to 0 in W 2,(p+1)/p−ε for small ε > 0. ApplyingTheorem 4.3.6 in [2] to system (7), (9) one has (u(·, t), v(·, t)) → (0, 0) in Xα ast →∞.

Acknowledgements. The first author was partially supported by VEGA Grant1/0259/03 (Slovakia). The second author was partially supported by Grant-in-Aidfor Encouragement of Young Scientists No. 15740076 (Japan). The third authorwas partially supported by MCYT Project BFM2002-04572-C02-02 (Spain). Thesecond and third authors wish to thank the Comenius University of Bratislava for itskind hospitality during the First Euro-Japanese Workshop on Blow-Up, September2004, where this work was finished. The authors also thank Hans Weinberger for ahelpful suggestion.


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Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems

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