GLOBAL EXISTENCE FOR A QUASILINEAR REACTION-DIFFUSION SYSTEM

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GLOBAL EXISTENCE FOR A QUASILINEAR REACTION-DIFFUSION SYSTEM

ALAN V. LAIR Received 21 July 2004

We show that the reaction-diﬀusion systemut=∆ϕ(u) +f(v),vt=∆ψ(v) +g(u), with homogeneous Neumann boundary conditions, has a positive global solution on Ω× [0,∞) if and only if^∞ds/ f(F⁻¹(G(s)))= ∞(or, equivalently,^∞ds/g(G⁻¹(F(s)))= ∞), whereF(s)=_s

0f(r)dr andG(s)=_s

0g(r)dr. The domainΩ⊆R^N (N≥1) is bounded with smooth boundary. The functionsϕ,ψ, f, andg are nondecreasing, nonnegative C([0,∞)) functions satisfyingϕ(s)ψ(s)f(s)g(s)>0 fors >0 andϕ(0)=ψ(0)=0. Ap- plied to the special case f(s)=s^pandg(s)=s^q, p >0,q >0, our result proves that the system has a global solution if and only ifpq≤1.

1. Introduction

We consider the reaction-diﬀusion system

u_t=∆ϕ(u) +f(v), v_t=∆ψ(v) +g(u) onQ_∞≡Ω×(0,∞),

∂_νϕ(u)=∂_νψ(v)=0 on∂Ω×(0,∞), u(x, 0)=u0(x)≥0, v(x, 0)=v0(x)≥0 onΩ,

(1.1)

whereΩis a bounded domain inR^Nwith a smooth boundary∂Ω,∂_ν=∂/∂νis the deriv- ative in the directionνof the outward normal to∂Ω, and the functionsϕ,ψ, f, andgare nondecreasing, nonnegativeC([0,∞)) functions satisfying

ϕ(s)ψ(s)f(s)g(s)>0 fors >0, ϕ(0)=ψ(0)=0. (1.2) We show that the problem (1.1) has a global solution if and only if f andgsatisfy

_∞

ds

fF⁻¹G(s)^{= ∞}

or equivalently, _∞

ds

gG⁻¹F(s)^{= ∞}

, (1.3)

whereF(s)≡_s

0 f(ξ)dξ andG(s)≡_s

0g(ξ)dξ. This, in turn, is exactly the necessary and suﬃcient condition needed to guarantee the existence of a global solution to the initial

International Journal of Mathematics and Mathematical Sciences 2005:11 (2005) 1809–1818 DOI:10.1155/IJMMS.2005.1809

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value problem for the system (seeLemma 2.2.)

y(t)= fz(t), z(t)=gy(t), 0< t <∞,

y(0)=a≥0, z(0)=b≥0, a+b >0. (1.4) In the special case f(s)=s^pandg(s)=s^q(p >0,q >0), condition (1.3) becomespq≤1.

Reaction-diﬀusion systems have been studied for decades (see, e.g., [4,5,10,11], and their references). The particular problem of determining conditions under which such systems have global solutions has been the object of studies for almost as long. (See [6,7, 8,9,10,11,15,16,17,18,19] and their references.) For the system (p >0,q >0)

ut=∆u+v^p,

vt=∆v+u^q, (1.5)

it is well known that the existence of global solutions in general depends on more than just the values of the exponents p and q. In particular, when homogeneous Dirichlet boundary conditions are imposed, it is well known [6,8] that forpq≤1, the system has only global solutions, but ifpq >1, the system will have a global solution for “small” initial data butnot for “large” initial data. A similar phenomenon occurs for the Cauchy problem [5,7]. However, our results (see Theorems2.1and3.2) show that this cannot occur with homogeneous Neumann boundary data, where blowup (i.e., no global solution) depends exclusively on the reaction terms and occurs (for (1.5)) precisely ifpq >1.

We show that this is true also in the presence of nonlinear diffusion. Thus the existence of a global solution is also independent of the diffusion term, although the diffusion rate may well determine thenatureof blowup as in the scalar case (see [12,14]). On the other hand, with homogenous Dirichlet boundary data, Galaktionov et al. [10,11] have shown that the quasilinear system

ut=∆u^ν⁺¹+v^p,

vt=∆v^µ+1+u^q (1.6)

has only global solutions ifpq <(1 +µ)(1 +ν), but forpq≥(1 +µ)(1 +ν), the existence of global solutions depends on the initial data and the size of the domain. In the present case, this does not occur. Indeed applyingTheorem 3.2to the system (1.6) with homogeneous Neumann boundary conditions, we find that a global solution exists if and only ifpq≤1.

We note also that some authors (e.g., [9,16]) have been concerned with whether a diﬀusion-free system can have a global solution while the corresponding diﬀusive system does not. Obviously, this cannot occur with the present system.

2. Smooth constitutive functions

Before establishing the general case, we first consider the case where the constitutive functions and the initial and boundary data are smooth. Thus we prove the following theorem.

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Theorem2.1. Letu0andv0be nonnegativeC^∞(Ω)functions, at least one of which is non- trivial. Assume that the functionsϕ,ψ,f, andgare nonnegative nondecreasingC^∞([0,∞)) functions satisfying (1.2) andϕψ>0. Then problem (1.1) has a nonnegative classical solu- tion if and only if f andgsatisfy condition (1.3).

Before proving this, we establish a preliminary lemma.

Lemma2.2. Suppose that f andgare nonnegative nondecreasing functions on[0,∞)sat- isfying f(s)g(s)>0fors >0. Then the system of ordinary diﬀerential equation (1.4) has a nonnegative classical solution if and only if (1.3) holds.

Proof. Necessity. Without loss of generality, assume thata >0, and problem (1.4) has solution (y,z). Then yg(y)=zf(z), which gives d/dt[G(y)−F(z)]=0. Thus there is a constant K so that G(y)=F(z) +K, and clearly from the initial values of y and z, we getK=G(a)−F(b). LetF(z) =F(z) +K and note that y(t)=G⁻¹(F(z(t))), and hencez(t)=g(y(t))=g(G⁻¹(F(z(t)))) ≥g(G⁻¹(F(b))) =g(a)>0, which implies that limt→∞z(t)= ∞. Now, ifb >0, we get

d dt

_z(t)

b

ds

gG⁻¹ F(s)⁼1 which implies that _z(t)

b

ds

gG⁻¹ F(s)⁼t. (2.1) Lettingt→ ∞, we establish condition (1.3). Ifb=0, then for everyε >0, we get

d dt

_z(t)

ε

ds

gG⁻¹ F(s)⁼1 (2.2)

which, after integrating fromδ >0 tot, gives _z(t)

ε

ds

gG⁻¹ F(s)⁼t−δ− _ε

z(δ)

ds

gG⁻¹ F(s). (2.3)

Lettingδ→0 gives

_z(t)

ε

ds

gG⁻¹ F(s)⁼t− _ε

0

ds

gG⁻¹ F(s), (2.4)

which implies that the integral on the right converges, and hence lettingε→0, we establish that (2.1) holds forb=0. The proof now continues as in the caseb >0.

Suﬃciency. Now suppose that (1.3) holds. DefineF(s) =_s

bf(t)dtandG(s) =_s

ag(t)dt.

We need to prove that the problem (1.4) has a classical solution. Once again, we assume thata >0. Define

H(s)= _s

a

dσ

f F⁻¹ G(σ). (2.5)

Clearly,H(a)=0,H(s)>0 fors >0. ThusH is one-to-one and from (1.3), which holds forF replaced byFandGreplaced byG, we know that H([a,∞))=[0,∞). Now define y: [0,∞)→[a,∞) byy(t)=H⁻¹(t) and definez(t)=F⁻¹(G(y(t))). We now show that

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y,zsatisfy (1.4). Clearly,y(0)=aandH(y(t))=t. ThusH(y(t))y(t)=1 so thaty(t)= f(F⁻¹(G(y(t)))) = f(z(t)). Likewise, z(0)=F⁻¹(G(y(0))) =F⁻¹(G(a)) =F⁻¹(0)=b, and

z= G(y)y F F⁻¹ G(y)⁼

G(y)y

F(z) ⁼g(y). (2.6)

This completes the proof.

Proof ofTheorem 2.1. Necessity. Suppose that problem (1.1) has a nonnegative clas- sical solution (u,v). From [13, Theorem 5.1], there exist T0>0 anda >0 such that min{u(x,t),v(x,t)}> aonΩ×[T0,∞). We now consider the system

α(t)= fβ(t)

2 , β(t)=gα(t)

2 fort > T0, αT0

=βT0

=a 2.

(2.7)

We will show that this system has a solution, and then invokeLemma 2.2to yield that (1.3) holds, which will complete the proof of necessity. Clearly, the system (2.7) has a solution on some, perhaps small, interval. Lett0> T0 be the supremum of all valuesτ such that a solution exists on [T0,τ). Ift0= ∞, then the system (2.7) has a solution and (1.3) holds as a result ofLemma 2.2. Thus suppose thatt0<∞. We will first show that

α(t)< u(x,t), β(t)< v(x,t) onΩ× T0,t0

. (2.8)

Thus suppose that there exists (x,T) ∈Ω×[T0,t0), where at least one of the two inequalities (2.8) fails to hold. Letζ∈C²(Ω) such that∂_νζ <0 on∂Ωandζ≥1 onΩ. Clearly, inequalities (2.8) hold fortnear T0 since they hold for t=T0. HenceT > T0. Define W(x,t)=u(x,t)−εζ(x) andZ(x,t)=v(x,t)−εζ(x), whereε >0 is chosen small so that the following conditions hold for alls∈[0,s0],s0≡max_Ω_×_[T₀_,T_](u+v):

ζ(x)<m0

2ε,

−ϕ(s)∆ζ(x)−εϕ(s)∇ζ(x)²≤m0

4ε,

−ψ(s)∆ζ(x)−εψ(s)∇ζ(x)²≤m0

4ε,

(2.9)

where m0=min{a,g(a/2),f(a/2)}. Then since W(x,T0)−α(T0)=u(x,T0)−εζ(x)− α(T0)≥a−εζ(x)−a/2>0 and similarly forZ(x,T0)−β(T0), we getW(x,T0)> α(T0) andZ(x,T0)> β(T0) onΩ. Now let

t1=supτ∈

T0,T|W(x,t)≥α(t),Z(x,t)≥β(t)∀(x,t)∈Ω×

T0,τ. (2.10)

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Clearlyt1> T0and att=t1, eitherW−αorZ−βis zero for somex0∈Ω. Without loss of generality, we assume that it isZ−β; that is, min_x_∈_ΩZ(x,t1)−β(t1)=Z(x0,t1)−β(t1)= 0 andW(x,t1)−α(t1)≥0 onΩ. Since∂_νZ=∂_νv−ε∂_νζ= −ε∂_νζ >0 on∂Ω×(T0,t0), we must havex0∈Ω, and hence∇Z(x0,t1)=0 and∆Z(x0,t1)≥0. Thus, at (x0,t1), we have the following:

0≥Z_t−β=v_t−β=∆ψ(v) +g(u)−g(α) 2

=ψ(v)∆v+ψ(v)|∇v|²+g(u)−g(α) 2

=ψ(v)∆(Z+εζ) +ψ(v)|∇Z+ε∇ζ|²+g(u)−g(α) 2

≥εψ(v)∆ζ+ε²ψ(v)|∇ζ|²+g(u)−g(α) 2

≥ −m0

4 +g(u)−g(α) 2

≥ −m0

4 +g(u) 2 +

g(u)−g(α) 2

≥ −m0

4 +g(α) 2 ^{≥ −}

m0

4 +g(a/2) 2 >0.

(2.11)

We thus arrive at a contradiction. Therefore inequalities (2.8) hold. Hence, the solution of (2.7) can be extended to an interval [0,t₀^∗), wheret^∗₀ > t0. This contradicts the fact that t0is the supremum of all such values. Therefore, our assumption thatt0<∞cannot hold.

Thus (2.7) has a global solution, and thereforeLemma 2.2implies that (1.3) holds.

Suﬃciency. Now suppose that (1.3) holds. From [2, page 17], we know that there ex- ists a maximal timeT0∈(0,∞] such that problem (1.1) has a (unique) solution, (u,v), and furthermore, ifuandvremain bounded and positive onΩ×(0,T) for allT< T0, thenT0= ∞. Clearly u and v are positive. Thus we need only to show that they are bounded onΩ×(0,T). To do this, we let (y,z) satisfy the system (whose solution ex- ists byLemma 2.2)

y(t)=2fz(t), z(t)=2gy(t), 0< t <∞,

y(0)=z(0)=M+ 1, (2.12)

whereM= u0∞,Ω+v0∞,Ω. We first show that

0≤u(x,t)< y(t), 0≤v(x,t)< z(t) onQT. (2.13) Thus suppose that there exists an (x0,T)∈QT such that (2.13) does not hold. Clearly, inequalities (2.13) hold fortsmall sinceu(x, 0)≤M < y(0) andv(x, 0)≤M < z(0). Hence T >0. Define the functionζas in the proof of necessity and let p(x,t)=u(x,t) +εζ(x) andq(x,t)=v(x,t) +εζ(x) onΩ×[0,T], whereε >0 is chosen small so that each of the

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following hold onΩfor everys∈[0,s0], wheres0=max_Ω_×_[0,T](u+v):

ζ(x)<1

ε, (2.14)

−ϕ(s)∆ζ(x) +εϕ(s)∇ζ(x)²≤M0

ε , (2.15)

−ψ(s)∆ζ(x) +εψ(s)∇ζ(x)²≤M0

ε , (2.16)

whereM0=min{g(M+ 1), f(M+ 1)}. From (2.14), it is clear that

p(x, 0)−y(0)<0, q(x, 0)−z(0)<0, ∀x∈Ω. (2.17) By our assumption concerningT, there existsτ0∈(0,T] such that either max_x_∈_Ωp(x,τ0)

=y(τ0) or max_x_∈_Ωq(x,τ0)=z(τ0). Let

t0=supτ∈[0,T]|p(x,t)≤y(t), q(x,t)≤z(t)∀(x,t)∈Ω×[0,τ]. (2.18)

Then att=t0, eitherp−yorq−zis zero for somex0∈Ω. Without loss of generality, we assume that max_x_∈_Ωp(x,t0)−y(t0)=p(x0,t0)−y(t0)=0, and hence max_x_∈_Ωq(x,t0)− z(t0)≤0. Notice that∂_νp=ε∂_νζ <0 on∂Ω×(0,t0) so thatx0∈Ω, and hence∇p(x0,t0)

=0 and∆p(x0,t0)≤0. We now have, at (x0,t0), the following:

0≤pt−y=ut−y=∆ϕ(u) +f(v)−2f(z)

=ϕ(u)∆(p−εζ) +ϕ(u)∇(p−εζ)²+f(v)−2f(z)

≤ −εϕ(u)∆ζ+ε²ϕ(u)|∇ζ|²+f(v)−f(z)−f(z)

≤M0−f(z)≤ f(M+ 1)−f(z)<0,

(2.19)

which provides a contradiction. Thus no sucht0exists. Hence p < yandq < zonΩ× [0,T] which, in turn, yields u < yand v < z on Ω×[0,T]. Thus there is noT where (2.13) fails to hold, and hence it holds onQTand therefore holds for allT< T0giving

T0= ∞. This completes the proof.

3. Nonsmooth constitutive functions

We now consider the case where the data and constitutive functions are not smooth.

In this case, it is well known that the system (1.1) does not, in general, have a classical solution even in the case of a single equation (see, e.g., [1]). Therefore, we will consider a weak formulation of a solution motivated by B´enilan et al. [3] and similar to that of [14].

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Definition 3.1. The sequence of problems

∂un

∂t ⁼∆ϕn

un

+fn

vn

, ∂vn

∂t ⁼∆ψn

vn

+gn

un

inQ_∞,

∂_νϕ_nu_n=∂_νψ_nv_n=0 on∂Ω×[0,∞), un(x, 0)=u0,n(x), vn(x, 0)=v0,n(x) onΩ

(3.1)

is called asequence of approximating problemsfor (1.1) if

ϕ_n,ψ_n,f_n,g_n∈C^∞[0,∞), u_0,n,v_0,n∈C^∞(Ω), ϕn(0)=ψn(0)=0, fn(0)≥0, gn(0)≥0,

ϕ_n>0, ψ_n>0, f_n>0, g_n>0,

nlim→∞ϕ_n−ϕ_∞_,S+ψ_n−ψ_∞_,S+f_n−f_∞_,S+g_n−g_∞_,S=0,

nlim→∞u0,n−u0

∞,Ω+v0,n−v0

∞,Ω

=0,

(3.2)

for every compact subsetSof [0,∞). Furthermore, a sequence{(un,vn)}of classical solutions to the approximating problems (3.1) is called asequence of approximating solutions to problem (1.1). Finally, a nonnegative function pair (u,v) defined onQ_∞is ageneralized solutionof problem (1.1) if there exists a sequence{(un,vn)}of approximating solutions which, for everyT >0, converges to (u,v) weakly inL¹(Q_T) and

supn

u_n_∞_,Q

T+v_n_∞_,Q

T

<∞. (3.3)

We prove the following theorem.

Theorem3.2. Letu0andv0be positiveC(Ω)functions and assume that the functionsϕ,ψ, f, andgare nondecreasing, nonnegativeC([0,∞))functions satisfying (1.2). Then problem (1.1) has a generalized solution if and only if f andgsatisfy condition (1.3).

Since much of the proof that follows is like that ofTheorem 2.1above, we merely point out important diﬀerences.

Proof. Necessity. Suppose that problem (1.1) has a generalized solution (u,v). Let (un, v_n) be a sequence of approximating solutions, thus satisfying (3.1). Sinceu0andv0are strictly positive onΩand the sequence {(u0,n,v0,n)} converges uniformly onΩ, there exists a subsequence, which, for convenience, we will assume is the sequence itself, for which there exists a positive constantasuch that min{u_n(x,t),v_n(x,t)}> aonΩ×[0,∞).

(We note that in the smooth case (Theorem 2.1), the initial data did not need to be strictly positive. However, for nonsmooth data and constitutive functions, it is unknown whether a generalized solution with nonnegative, nontrivial initial data ever becomes strictly positive at a later time.) The proof may now proceed as withTheorem 2.1(withT0=0) to prove

αn(t)< un(x,t), βn(t)< vn(x,t) onΩ×[0,∞), (3.4)

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where (α_n,β_n) is the solution to system (2.7) with f andg replaced with f_nandg_n, respectively, and hence for allT >0,

αn(t) +βn(t)≤sup

k

uk

∞,QT+vk

∞,QT

<∞ n∈N, 0≤t≤T. (3.5)

We can now use this inequality to show that (2.7) has a solution on [0,∞). To do this, we note that there is an interval, perhaps small, on which a solution (α,β) to (2.7) exists. In fact, from the proof ofLemma 2.2, it is clear that a solution exists on the interval [0,t0), wheret0=H(∞) with

H(s)= _s

a/2

dσ

f F⁻¹ G(σ) (3.6)

andF, Gdefined as in the proof ofLemma 2.2with bothaandbreplaced bya/2. How- ever, since the sequences{f_n},{g_n}converge uniformly on compact subsets of [0,∞), so do the sequences{Fn},{Gn}, whereFn(s)=_s

a/2fn(σ)dσ andGn(s)=_s

a/2gn(σ)dσ. It is then straightforward to show thatF_n⁻¹,G⁻_n¹converge uniformly on compact subsets of [0,∞), and thereforeH_n⁻¹converges uniformly on compact subsets of [0,t0). Therefore, (αn,βn)→(α,β) asn→ ∞uniformly on compact subsets of [0,t0), and hence (α,β) must satisfy

α(t) +β(t)≤sup

k

uk

∞,QT+vk

∞,QT

<∞, 0≤t≤T < t0. (3.7)

Therefore, the functionsαandβmust be defined on [0,∞). Indeed, the only way thatα andβcan fail to exist att0 is for lim_t_→_t⁻₀α(t)= ∞(similarly forβ), which is impossible because of (3.7). Thereforeαandβexist on [0,t0] and can be extended to a larger interval [0,t0+ε), which contradicts the fact thatt0was the extent of the existence. Therefore, we must havet0= ∞so that (1.3) holds.

Suﬃciency. Now suppose that (1.3) holds. We show that problem (1.1) has a nonnega- tive generalized solution. We choose sequences{fn},{gn},{ϕn},{ψn},{u0,n}, and{v0,n} as specified in the definition of a generalized solution. Such sequences are not diﬃcult to construct using mollifiers and the properties of the functions f,g,ϕ,ψ,u0, andv0. Furthermore, the sequences{fn},{gn}may be (and are) chosen so that for eachn, they satisfy (1.3) with f andg replaced by f_nandg_n, respectively. Let (u_n,v_n) be the smooth solution of (3.1), and let (yn,zn) be the solution of

y_n(t)=2fn

zn(t), z_n(t)=2gn

yn(t), 0< t <∞,

y_n(0)=z_n(0)=M+ 1, (3.8)

whereM=sup_n(u_0,n∞,Ω+v_0,n∞,Ω). It is then clear that, as in the proof of (2.12), 0≤u_n(x,t)< y_n(t), 0≤v_n(x,t)< z_n(t) onQ_∞. (3.9) Also, since (y_n,z_n) converges locally uniformly to (y,z), we know that (y_n,z_n) is locally

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bounded so that (3.3) holds. To complete the proof, we will prove that the sequence {(u_n,v_n)}has a subsequence{(U_n,V_n)}defined on Q_∞ and obviously satisfying (3.3), which converges weakly inL¹(QT) to a function pair (u,v) for allT >0. To do this, we note that (3.3) implies that{(un,vn)}is pointwise bounded onQT which, in turn, implies that for eachk∈N, theL²(Q_k) norm (and everyL^p(Q_k) norm for p≥1) of the sequence{(un,vn)}is bounded. In particular, theL²(Q1) norm is bounded independent ofnso the sequence{(un,vn)}has a weakly convergent subsequence inL²(Q1). We de- note this subsequence by{(u_n,1,v_n,1)}, and we let (P1,R1) be its weakL²(Q1) limit. Like- wise, the sequence{(un,1,vn,1)}is bounded in theL²(Q2) norm, and hence has a subsequence{(un,2,vn,2)}which is weakly convergent to a function pair (P2,R2) inL²(Q2).

Clearly (P1,R1)=(P2,R2) on Q1. We continue the process to produce for each k∈N the sequence{(un,k,vn,k)}, a subsequence of{(un,k−1,vn,k−1)}, which is weakly convergent to (Pk,Rk) inL²(Qk) and (Pk,Rk)=(Pk−1,Rk−1) onQk−1. Clearly the sequence (Pk,Rk) converges weakly inL²(Q_T) for allT >0 to the function pair (u,v) defined onQ_∞ by (u,v)=(Pj,Rj) onQj, j∈N. In addition, it is easy to prove that the sequence of diago- nal entries of the double-indexed sequence{(un,k,vn,k)}, namely{(un,n,vn,n)}, converges weakly inL²(Q_T), and hence weakly inL¹(Q_T) to (u,v) for allT >0. Thus the desired sequence{(Un,Vn)}of approximating solutions which converges to (u,v) is{(un,n,vn,n)}, and therefore (u,v) is a generalized solution of (1.1). This completes the proof.

An open problem. We note that there is an important diﬀerence regarding the initial data in the hypothesis of Theorem 2.1, the smooth case, andTheorem 3.2, the nonsmooth case. In the latter, the initial data is required to be strictly positive, whereas in the for- mer it needs only to be nonnegative and nontrivial. This leaves open the problem: can Theorem 3.2be extended to the case whereu0andv0are merely nonnegative with at least one of them nontrivial? With smooth constitutive functions, the solution will, in time, be- come strictly positive with only nonnegative nontrivial initial data. It is unknown whether this will ever occur in the nonsmooth case. However, it may be possible that a diﬀerent proof can be devised, as in the scalar case [14], whereTheorem 3.2can be extended.

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Alan V. Lair: Department of Mathematics and Statistics, Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, OH 45433-7765, USA

E-mail address:[email protected]