R E V I E W
Open Access
Universal bounds and blow-up estimates
for a reaction-diffusion system
Nejib Mahmoudi
**Correspondence: [email protected] Laboratoire Équations aux Dérivées Partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, Tunis, 2092, Tunisia
Abstract
This paper is concerned with nonnegative solutions of the reaction-diffusion system:
ut–u=vp+
μ
1ur, vt–v=uq+μ
2vs.In a suitable range of parameters, we prove (initial and final) blow-up rates, as well as universal bounds for global solutions. This is done in connection with new
Liouville-type theorems in a half-space, that we establish.
MSC: Primary 35B44; secondary 35K57; 35K58
Keywords: semilinear parabolic systems; reaction-diffusion systems; doubling property; Liouville-type theorem; blow-up rate; universal bound
1 Introduction
In this paper, we study (initial and final) blow-up rates, as well as universal bounds for global solutions, for a class of semilinear reaction-diffusion systems, in connection with Liouville-type theorems. Our study is motivated by [], where Poláčiket al.developed a general method for obtaining universal initial and final blow-up rates for the scalar equa-tionut–u=up(p> ), based on rescaling arguments and Liouville-type theorems, com-bined with a key doubling property. In this context, the Liouville-type theorem means the nonexistence of nontrivial, nonnegative and bounded solutions defined for all negative and positive times on the whole spaceRn, or on a half-spaceRn
+={x∈Rn;x> }. We here consider the system:
ut–u=vp+μur,
vt–v=uq+μvs, ()
wherep,q,r,s> andμ,μ≥. We use the following notation for the scaling exponents:
α= p+
pq– , β= q+
pq– . ()
Let us recall that, even in the scalar case, the optimal exponent for the Liouville-type prop-erty is not presently known (see the monograph [] and the recent work []). In the case of systems, as far as we know, the only nonexistence result inRn
+={x∈Rn;x> }is a
consequence of the Fujita-type theorem in [], Theorem , p.. The latter asserts the stronger property of nonexistence inR+×Rn(instead ofR×Rn) for the Cauchy prob-lem
⎧ ⎪ ⎨ ⎪ ⎩
ut–u=vp, t> ,x∈Rn, vt–v=uq, t> ,x∈Rn,
u(,x) =u, v(,x) =v, x∈Rn,
()
whereu,vare nonnegative, continuous, and bounded functions, under the assumption
max(α,β)≥n/ withα,βare given by (). As a consequence, Cui [] proved a Fujita-type theorem for the Cauchy problem associated with the system () forμ=μ= , assuming that at least one of the following five conditions (i)-(v) is satisfied:
(i) r≤ + /n, (ii) s≤ + /n, (iii) max(α,β)≥n/, ()
(iv) max(α,β) <n/, p≤ + /n and r<np/(np– ), ()
(v) max(α,β) <n/, q≤ + /n and s<nq/(nq– ). ()
(Note that max(α,β) <n/ impliesp,q> /n.) See [], Theorem ., p.. The condi-tions (i)-(v) are optimal; see []. It will turn out that problem () withμ=μ= reveals a number of interesting, qualitatively new phenomena, in comparison with the unperturbed model system (), such as the existence of non-simultaneous blowing-up solutions. See [, ].
There are also Fujita-type results for half-spaces (see []). However, their optimal ex-ponents are always smaller than the corresponding exex-ponents in the whole space, leading to more stringent conditions in the applications (to blow-up estimates). On the contrary, for Liouville-type results with scalar equations, it was shown in [] that nonexistence in a half-space can be derived as a consequence of nonexistence in the whole space,without requiring stronger restrictions on the exponents(and actually the restriction becomes even weaker). This was done by adapting a monotonicity argument, based on moving planes, introduced by Dancer [] for elliptic equations. One of our main concerns here is to ex-tend the result from [] to system (). Namely, we establish the following theorem.
Theorem . Let p,q,r,s> ,p≤q,andμ,μ≥be such that at least one of the following five conditions(a)-(e)is satisfied:
(a) β≥n–
,
(b) μ> and r≤ + n– ,
(c) μ> and s≤ + n– ,
(d) β<n–
, μ> , p≤ +
n– and r<
(n– )p (n– )p– ,
(e) β<n–
, μ> , q≤ +
n– and s<
whereβis given by(). (Note thatβ< (n– )/implies p,q> /(n– ).)Then the system
⎧ ⎪ ⎨ ⎪ ⎩
ut–u=vp+μ
ur, t∈R,x∈Rn+, vt–v=uq+μvs, t∈R,x∈Rn
+, u=v= , t∈R,x∈∂Rn+,
()
has no nontrivial,nonnegative,bounded,and classical solution.
By applying the method of [], as an application of Theorem . and of the result of [], we then obtain the following universal initial and final blow-up rates for system (), as well as universal bound for global solutions.
Theorem . Letbe a(uniformly C)smooth domain ofRn.Let p,q,r,s> ,p≤q,be such that
r≤p(q+ )
p+ ifμ> , s≤ q(p+ )
q+ ifμ> ,
and one of the following five conditions is satisfied:
β≥n
,
μ> , r= p(q+ )
p+ and r≤ + n,
μ> , s= q(p+ )
q+ and s≤ + n,
μ> , r= p(q+ )
p+ , β< n
, p≤ +
n and r< np np– ,
μ> , s= q(p+ )
q+ , β< n
, q≤ +
n and s< nq nq– ,
whereβis given by().
Then there exists a constant C=C(p,q,r,s,μ,μ,) > such that,for any T∈(,∞] and any nonnegative,classical solution(u,v)of
⎧ ⎪ ⎨ ⎪ ⎩
ut–u=vp+μ
ur, <t<T,x∈, vt–v=uq+μvs, <t<T,x∈, u=v= , <t<T,x∈∂,
()
we have
u(t,x)≤C +t–α+ (T–t)–α, <t<T,x∈ ()
and
v(t,x)≤C +t–β+ (T–t)–β, <t<T,x∈. ()
Remark . Theorem . improves on previously known results (see [–]) in three directions:
(i) the constantCis independent of(u,v);
(ii) can be an any (smooth) domain;
(iii) no monotonicity conditions are assumed either in space or in time.
The rest of the paper is organized as follows. In Section , we prove Theorem .. Next, we prove Theorem ., in Section .
2 Liouville-type theorem: proof of Theorem 1.1
In this section, we are concerned with the proof of Theorem .. It is a consequence of [], Theorem , p., [], Theorem ., p. and the following theorem.
Theorem . Let p,q,r,s> ,μ,μ≥.Then we have the following statements:
(a) The components of each positive,bounded,and classical solution(u,v)of()are increasing inx:
∂xu(t,x) > and ∂xv(t,x) > , t∈R,x∈R
n +.
(b) If there exists a positive,bounded,and classical solution of(),then there exists a positive,bounded,and classical solution of
ut–u=vp+μur, t∈R,x∈Rn–,
vt–v=uq+μvs, t∈R,x∈Rn–. () Proof Part(a). We putf(u,v) =vp+μurandg(u,v) =uq+μvs. Forλ> , let
Tλ= x∈Rn; <x<λ
.
As in [], for a functionhdefined onRn
+, lethλandVλhbe the functions defined onTλby
hλ(x) =hλ–x,x, <x<λ,x= (x, . . . ,x
n)∈Rn–, Vλh(x) =hλ(x) –h(x), x=
x,x∈Tλ.
Let (u,v) be a positive, bounded, and classical solution of (). For eachλ, (Vλu,Vλv)
satisfies the following system:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Vλut–Vλu=Cλ(t,x)Vλv+Cλ(t,x)Vλu, t∈R,x∈Tλ,
Vλvt–Vλv=Cλ(t,x)Vλu+Cλ(t,x)Vλv, t∈R,x∈Tλ,
Vλu=Vλv= , t∈R,x=λ,x∈Rn–, Vλu,Vλv> , t∈R,x= ,x∈Rn–,
()
where
Cλ(t,x) =
Cλ(t,x) =
fuu(t,x) +suλ(t,x) –u(t,x), ds,
Cλ(t,x) =
guu(t,x) +suλ(t,x) –u(t,x), ds,
Cλ(t,x) =
gv,v(t,x) +svλ(t,x) –v(t,x)ds,
fu=μrur–, fv=pvp–, gu=quq–, gv=μsvs–.
We claim that
Vλu≥ and Vλv≥ inR×Tλfor eachλ> . ()
With () at hand, by the maximum principle [], Proposition ., p. and (), we obtain
Vλu> and Vλv> inR×Tλfor eachλ> .
Moreover, sinceVλu(t,λ,x) =Vλv(t,λ,x) = , from the Hopf maximum principle, it
fol-lows that
∂Vλu(t,λ,x)
∂x < and
∂Vλv(t,λ,x) ∂x < .
Therefore, since
∂ ∂xVλu
t,λ,x= –∂u(t, λ–λ,x
)
∂x –
∂u(t,λ,x)
∂x = –
∂u(t,λ,x)
∂x ,
we obtain
∂u(t,λ,x)
∂x > for eachλ> .
Similarly, we prove that∂v(t,∂xλ,x)
> , for eachλ> . To complete the proof of Part (a), it is
therefore sufficient to prove the claim (). We recall first the following lemma of Dancer [].
Lemma . Given any positive constants l,λsatisfyingλl<π,there exists a smooth function h onTλsuch that
⎧ ⎪ ⎨ ⎪ ⎩
h+lh= , x∈Tλ,
h(x) > , x∈Tλ,
h(x)→ ∞, |x| → ∞, x∈Tλ.
()
We split the rest of the proof in two steps. Step.Proof of()for smallλ.
Lethbe given by Lemma .. Sincehis a positive and smooth function onTλsuch that
h(x)→ ∞, as|x| → ∞, there existsε> such thath≥ε. Fix a positive constantγ and set
l:= sup
t∈R,x∈Rn
+
which is finite, by the boundedness of (u,v). Define the function
(Vλu,Vλv) :=
eγtV
λu/h,eγtVλv/h
,
wherehis given by Lemma . forλ> sufficiently small (so thatλl<π). With () at hand, a simple computation shows that
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
Vλut–Vλu–∇hh· ∇Vλu– (γ +Cλ(t,x) –l)Vλu–Cλ(t,x)Vλv
= , t∈R,x∈Tλ,
Vλvt–Vλv–∇hh· ∇Vλv– (γ +Cλ(t,x) –l)Vλv–Cλ(t,x)Vλu
= , t∈R,x∈Tλ,
Vλu,Vλv≥, t∈R,x∈∂Tλ,
Vλu(t,x),Vλv(t,x)→, t∈R,x∈Tλ,|x| → ∞.
()
Moreover, () implies that
γ +Cλ(t,x) +Cλ(t,x) –l≤, γ+Cλ(t,x) +Cλ(t,x) –l≤. ()
ForM> to be fixed later, we put
W:= –Vλu–M, Z:= –Vλv–M.
Using (), we have
Wt–W≤C|∇W|+
γ+Cλ
(t,x) –l
(W+M) +Cλ
(t,x)(Z+M)
≤C|∇W|+γ+Cλ(t,x) –lW+Cλ(t,x)Z ()
and
Zt–Z≤C|∇Z|+γ +Cλ(t,x) –lZ+Cλ(t,x)W. ()
By the last two properties in (), we haveW+(t,·) :=max(,W)(t,·),Z+(t,·) :=max(,Z)(t, ·)∈H
(Tλ) for allt∈R. Multiplying () withW+, integrating by parts and usingCλ≥, it follows that
d dt Tλ
W+≤–
Tλ
|∇W+|+C
Tλ
|∇W+|W+
+
Tλ
γ +Cλ(t,x) –lW++
Tλ
Cλ(t,x)Z+W+≤K
Tλ
W++Z+
for some constantsC,K> . Arguing similarly onZ+and adding up, we obtain
d dt
Tλ
W++Z+≤K
Tλ
W++Z+. ()
We now setA:=ε–max(u∞,v∞), whereε=infTλh> , and, for any givent∈R, we
choose
M=Aeγt≥max
sup
x∈Tλ
|Vλu|(t,x),sup
x∈Tλ
|Vλv|(t,x)
ThenW+(t,·)≡Z+(t,·)≡ and it follows from () thatW,Z≤ in (t,∞)×Tλ.
Con-sequently, for allt,t∈Rwitht<t, we have
sup
x∈Tλ
–Vλu(t,x)
h(x) ≤Me
–γt=Ae–γ(t–t).
Lettingt→–∞, we obtainVλu≥ everywhere, and similarlyVλv≥. We conclude
that () holds forλsmall.
Step.Proof of ()for largeλ(hence for allλ). Let
λ=sup μ> |() holds for allλ∈(,μ)
. ()
By Step , it follows thatλ> . We assume by contradiction thatλ<∞. Then there exists a sequenceλk≥λsuch thatλk→λand the set
Fk:= (t,x)∈R×Tλk|min
Vλku(t,x),Vλkv(t,x)
<
is nonempty. Set
mk:=sup maxut,y,x,vt,y,x|t∈R,y∈(, λk),x∈Rn–, ∃x∈(,λk)/t,x,x∈Fk.
We may assume that either:
(i) mk≥εfor someε> (by passing to a subsequence if necessary);
(ii) mk→.
If case (i) holds, there exist some sequencestk∈R,xk
∈(,λk),yk∈(, λk), andξk∈Rn– such that
minVλku
tk,xk,ξk,Vλkv
tk,xk,ξk< and maxutk,yk,ξk,vtk,yk,ξk≥ε.
By passing to a subsequence, we may assume thatxk
→aandyk →b for somea,b∈ [,λ]. Next, we consider the functions
uk(t,x) :=ut+tk,x,x+ξk, vk(t,x) :=vt+tk,x,x+ξk, t∈R,x,x∈Rn.
For allk, (uk,vk) is a positive, bounded, and classical solution of () satisfying
minVλkuk
,xk, ,Vλkvk
,xk, =minVλku
tk,xk,ξk,Vλkv
tk,xk,ξk<
and
maxuk,yk, ,vk,yk, ≥ε.
Moreover, by definition ofλit follows thatVλuk,Vλvk≥ inR×Tλ. Sinceukandvk
(still denoted (uk,vk)) converges inCloc,(R×Rn
+) to a nonnegative and bounded solution (u,v) of (). The above properties of (uk,vk) imply that
minVλu(,a, ),Vλv(,a, )
≤, maxu(,b, ),v(,b, )≥ε
andVλu,Vλv≥ inR×Tλ.
Next we claim thatu andvare positive everywhere inR×Tλ. Indeed, assume for
contradiction thatu(t,x) orv(t,x) vanishes for somet∈Randx∈Tλ. Due to the
coupled structure of the system, by the strong maximum principle, it follows thatu≡ v≡ in (–∞,t]×Tλ, hencet< . But then, using the boundedness ofuandvand the
maximum principle again, we deduceu≡v≡ on [t,∞)×Tλ: a contradiction.
Now, since (Vλu,Vλv) solves the corresponding problem () andVλu,Vλv≥, it
follows thatVλu,Vλv> inR×Tλ. In particular we necessarily havea=λ. Moreover,
by the Hopf maximum principle, it follows that
∂xu(,λ, ) = –∂xVλu(,λ, ) > .
Similarly, we obtain ∂xv(,λ, ) > . Consequently, ∂xu(,x, ) and ∂xu(,x, ) are
bounded below by a positive constant on an interval aroundλ and this remains valid if uandvare replaced, respectively, byuk andvkforksufficiently large. That is, there existsδ> such that
∂xu
tk,x,ξk
= –∂xuk(,x, ) > , x∈[λ–δ,λ+δ]. ()
Similarly, we obtain ∂xv(t
k,x,ξk) > , for x ∈[λ
–δ,λ+δ]. However, since λk – xk >xk both belong to [λ–δ,λ+δ] for largek, this contradicts the assumption that
min(Vλku(tk,x k
,ξk),Vλkv(tk,x k
,ξk)) < . Therefore the assumption (i) leads to a contradic-tion.
Now consider case (ii). We go back to the problem () withλ=λkandksufficiently large. By assumption, gu(,v),fu(,v),fv(u, ),gv(u, ) = and the definitions of Cλk
i , i= , . . . , , andmkimply that, for
lk:=max
sup
(t,x)∈Fk
Cλk
(t,x) +C
λk (t,x)
, sup
(t,x)∈Fk
Cλk
(t,x) +C
λk (t,x)
,
we have
lim sup
k→∞ lk≤.
Fixkso large thatl:=lk+γ <λ–
k π, whereγ is any small positive constant. Setλ=λk, apply Lemma . and lethbe the resulting function. As above, the function (Vλu,Vλv) :=
(eγtV
λu/h,eγtVλv/h) satisfies the problem (). Since
γ +Cλ
(t,x) +Cλ(t,x) –l≤, γ+Cλ(t,x) +Cλ(t,x) –l≤ onFk,
it follows that inequality () (resp., ()) is satisfied on the set{(t,x)∈R×Tλ,W(t,x) > }
(), still applies and yieldsVλku,Vλkv≥, contradicting the nonemptiness ofFk.
There-fore the two possibilities (i) and (ii) lead to a contradiction. Thus,λ=∞. This completes the proof of the claim, hence the proof of Part (a).
Part(b). Let (u,v) be a positive, bounded, and classical solution of (). Fork= , , . . . , we consider the functions
ukt,x,x:=ut,x+k,x, vkt,x,x:=vt,x+k,x, t,x,x∈k,
wherek=R×(–k,∞)×Rn–. Since (u,v) is a positive, bounded, and classical solution of (), then for allk, (uk,vk) is a positive, bounded, and classical solution (on its domains) of the following system:
∂tuk–uk=vpk+μurk,
∂tvk–vk=uqk+μvsk.
Moreover,uk=vk= on∂k. FromLm parabolic estimates, there exists a subsequence ((uk,vk))k≥converges inCloc,(R×Rn) to a nonnegative, bounded pair of functions (u,v). Moreover, (u,v) is a classical solution of
ut–u=vp+μur, vt–v=uq+μvs.
Sinceuk,vkare increasing inx(by Part (a)), then
ukt,x,x=ut,x+k,x≥ut,x+k,x> for allk≥k,
withx+k> . Therefore, by passing to limits, we obtainu(t,x,x)≥u(t,x+k,x) > , for all (t,x)∈R×Rn. Similarly, we obtainv> . Therefore also by Part (a),u,vare increas-ing inx. Moreover, letx<x∈R, we haveuk(t,x,x) <uk(t,x,x),x+k> , passing to limits, we obtainu(t,x,x)≤u(t,x,x). Moreover,uk+E(x)(t,x,x) >uk(t,x,x), passing to
limits, we obtainu(t,x,x)≥u(t,x,x). Therefore,u(t,x,x) =u(t,x,x) for allx,x∈R, which means thatuis independent ofx. Similarly, we prove thatvis independent ofx.
This completes the proof of Theorem ..
We are now in position to prove Theorem ..
Proof of Theorem . We assume by contradiction that the system () has a positive, bounded, and classical solution (u,v). By Theorem .(b), there exists a positive, bounded, and classical solution of () onR×Rn–; a contradiction with [], Theorem , p. or
[], Theorem ., p..
3 Blow-up rates via Fujita and Liouville-type theorems: proof of Theorem 1.2 In this section, we are concerned with the proof of Theorem .. We will use the following key doubling lemma from [].
real k> .If there exists y∈D such that
M(y)dist(y,) > k,
then there exists x∈D such that
M(x)dist(x,) > k, M(x)≥M(y),
and
M(z)≤M(x) for all z∈D∩BXx,kM–(x).
We now turn to the proof of Theorem ..
Proof of Theorem. We assume by contradiction that the theorem fails. Then there exist sequencesTk∈(,∞), (uk,vk) being a solution of () in (,Tk)×,yk∈, andσk∈(,Tk) such that the functions
Mk:=u/k α+v /β
k , k= , , . . . , ()
satisfy
Mk(σk,yk) > k +d–k (σk), ()
wheredk(t) = (min(t,Tk–t))/. Then
Mk(σk,yk) > kdk–(σk) ()
and
Mk(σk,yk) > k. ()
We will use Lemma . withX=Rn+equipped with the parabolic distancedP, defined by
dP(t,x), (σ,y)=|t–σ|/+|x–y| for all (t,x), (σ,y)∈Rn+,
=k= [,Tk]×,D=Dk= (,Tk)×, and=k={,Tk} ×. Let us mention that
dk(t) =dP
(t,x),k
, (t,x)∈k.
Indeed, let (t,x)∈k, we have
dP(t,x),k
= inf
(σ,y)∈k
|t–σ|/+|x–y|
=|x–x|+inf(t– )/, (Tk–t)/
As mentioned above by Lemma . withX=Rn+ equipped with the parabolic distance dPand (), it follows that there existxk∈,tk∈(,Tk) such that
Mk(tk,xk) > kd–k (tk), ()
Mk(tk,xk)≥Mk(σk,yk) > k, ()
and
Mk(t,x)≤Mk(tk,xk) for all (t,x)∈Dk∩Bk(tk,xk),kM–(tk,xk), ()
where
Bk
(tk,xk),kM–(tk,xk)
= (t,x)∈Rn+;|x–xk|+|t–tk|/≤kM–(tk,xk)
.
In the rest of the proof, we use the notation
λk:=M–k (tk,xk).
By (), we obtain
λk<
kk→→∞. ()
Moreover, we observe that
tk–k λ
k ,tk+
kλ k
×
∩
|x–xk|<kλk
⊂Dk∩Bk. ()
Indeed, we have (∩ {|x–xk|<kλk
})⊂. Also by (), we obtain
|t–tk|<k λ
k <k
λ
k<dk(tk) =min
tk, (Tk–tk),
hencet∈(,Tk). Finally,
|x–xk|+|t–tk|/≤kλk +
kλk =kλk.
Therefore, (t,x)⊂Bk. Now, we rescale the functionsuk,vkby setting
wk(σ,y) :=λkαuktk+λkσ,xk+λky, zk(σ,y) :=λkβvktk+λkσ,xk+λky,
where (tk+λkσ,xk+λky)∈(tk–k
λ
k ,tk+
kλk
)×(∩ {|x–xk|< kλk
}), which imply that (σ,y)∈Dk:= (–k,k)×(λ–
k (–xk)∩ {|y|<k}). The pair of functions (wk,zk) solves the system
⎧ ⎪ ⎨ ⎪ ⎩
∂σwk–wk=z
p k+λ
α(–r)+
k wrk, (σ,y)∈Dk,
∂σzk–zk=wqk+λ β(–s)+
k zsk, (σ,y)∈Dk, wk=zk= , y∈λ–
k (∂–xk),|y|<k,|s|< k
.
Moreover,
w/α
k (, ) +z /β
k (, ) =λku/k α(tk,xk) +λkv/
β
k (tk,xk) =λkλ–k = . ()
By (), we obtain
w/α
k +z /β
k
(σ,y) =λku/k α
tk+λkσ,xk+λky
+λkv/
β
k
tk+λkσ,xk+λky
=λkMktk+λkσ,xk+λky
≤λkMk(tk,xk) = for all (σ,y)∈Dk.
Therefore, sincewk,zk≥, we obtain
≤wqk(σ,y)≤αq, ≤zpk(σ,y)≤βp for all (σ,y)∈Dk. ()
Also, sincer≤p(q+ )/(p+ ),s≤q(p+ )/(q+ ), andλk→, we obtain
≤λkα(–r)+wrk(σ,y)≤αr, ≤λkβ(–s)+zsk(σ,y)≤βs ()
for all, (σ,y)∈Dk. Letρk:=dist(xk,∂). Then either the sequence (ρk/λk)kis bounded or unbounded. By passing to a subsequence, we may assume that either:
(a) ρk
λk → ∞
or (b) ρk
λk → c≥.
If case (a) holds, sinceuk,vk∈C,((,Tk)×)∩C((,Tk)×), we obtain
wk,zk∈C,(Dk)∩C
–k
, k
×
λ–k (–xk)∩
|y|<k
.
Moreover,
wk,zk= on
–k
, k
×
λ–k (∂–xk)∩
|y|<k
.
Setδ:= ifr=p(q+ )/(p+ ),δ:= ifr<p(q+ )/(p+ ), andδ:= ifs=q(p+ )/(q+ ),
δ:= ifs<q(p+ )/(q+ ). By using interiorLmparabolic estimates, it follows that there exists a subsequence (wk,zk) that converges inClocα (R×Rn), <α< , to a pair of functions (w,z) a nonnegative, bounded, and classical solution of the problem
wσ–w=zp+μδwr, (σ,y)∈R×Rn, zσ –z=wq+μδzs, (σ,y)∈R×Rn.
Moreover,w/α(, ) +z/β(, ) = ; a contradiction with [], Theorem , p. or [],
Theorem ., p.. (Note that these results remain valid with any positive coefficients in front of the termswrandzs, instead .)
that the image of the boundary∂will be contained in the hyperplaneθ= ,xkbecomes ,xkbecomesθk:= (ρk, , , . . . , ) and the image ofUk will contain the set{θ :|θ|<ε} for someε> . We may assume thatε,εare independent ofkand the local charts are uniformly bounded inC. In these new coordinates, the system forϕ=ϕk(t,θ) =uk(t,x) andφ:=φk(t,θ) :=vk(t,x) becomes
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ϕt–
i,jai,j ∂
ϕ
∂θi∂θj –
ibi∂ϕ∂θi=φ p+μ
ϕr, t> ,|θ|<ε,θ> ,
φt–
i,jai,j
∂φ ∂θi∂θj –
ibi
∂φ ∂θi=φ
q+μ
ϕs, t> ,|θ|<ε,θ> ,
ϕ=φ= , t> ,|θ|<ε,θ= ,
()
whereai,j(t,θ) =
l
∂θi
∂xl
∂θj
∂xl =D·
tD, withD=D
k= (∂θi/∂xj)i,j, andbi(t,θ) =θi, henceA= Ak:= (aij)i,jare uniformly elliptic. Also, since∂is uniformlyC, it follows that theakijare uniformly bounded and thebki inL∞. Moreover, sinceD() is a Euclidean transformation, it follows thatAj() =D()·tD() = Id. Then the rescaled functionswk,zkdefined by
wk(σ,y) :=λkαϕk
tk+λkσ,θk+λky
, zk(σ,y) :=λkβφk
tk+λkσ,θk+λky
,
where (σ,y)∈ {(σ,y) :|σ|<k/;|y–θk
λk|<
ε λk,y> –
ρk
λk}. Then (wk,zk) solves the system
⎧ ⎨ ⎩
∂σwk–
i,jai,j
∂wk
∂yi∂yj –λk
ibi
∂wk
∂yi =z p k+μλ
α(–r)+ k wrk,
∂σzk–
i,jai,j
∂zk
∂yi∂yj –λk
ibi
∂zk
∂yi =w q
k+μλkα(–r)+zsk,
where (σ,y)∈ {(σ,y) :|σ|<k/;|y–θk
λk|<
ε λk,y> –
ρk
λk}. Also
wk=zk= for|σ|<k/,y– θk
λk
< ε
λk
,y= –ρk
λk .
As in the first case, by using interior-boundedLmparabolic estimates, we conclude that there exists a subsequence (wk,zk) that converges in Cα
loc, <α < , to a nonnegative bounded and classical solution of the problem
⎧ ⎪ ⎨ ⎪ ⎩
wσ–w=zp+μδwr, (σ,y)∈R×Hc, zσ–z=wq+μδzs, (σ,y)∈R×Hc, w=z= , y=c,
whereHc:={y∈Rn;y> –c}. Moreover,w/α(, ) +z/β(, ) = ; a contradiction with Theorem .. This finishes the proof of Theorem ..
Competing interests
The author declares to have no competing interests.
Received: 19 August 2015 Accepted: 19 November 2015
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