R E S E A R C H
Open Access
Global and blow-up solutions for nonlinear
parabolic problems with a gradient term
under Robin boundary conditions
Juntang Ding
**Correspondence:
School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, P.R. China
Abstract
In this paper, we study the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:
⎧ ⎪ ⎨ ⎪ ⎩
(b(u))t=∇ ·(g(u)∇u) +f(x,u,|∇u|2,t) inD×(0,T),
∂u
∂n+
γ
u= 0 on∂
D×(0,T),u(x, 0) =u0(x) > 0 inD,
whereD⊂RN(N≥2) is a bounded domain with smooth boundary
∂
D. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for ‘blow-up time’, and an upper estimate of ‘blow-up rate’ are specified under some appropriate assumptions on the functionsf,g,band initial valueu0.MSC: 35K55; 35B05; 35K57
Keywords: global solution; blow-up solution; parabolic problem; Robin boundary condition; gradient term
1 Introduction
In this paper, we study the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(b(u))t=∇ ·(g(u)∇u) +f(x,u,q,t) inD×(,T), ∂u
∂n+γu= on∂D×(,T),
u(x, ) =u(x) > inD,
(.)
whereq:=|∇u|,D⊂RN (N≥) is a bounded domain with smooth boundary∂D,∂/∂n
represents the outward normal derivative on∂D,γ is a positive constant,uis the initial value,Tis the maximal existence time ofu, andDis the closure ofD. SetR+:= (, +∞). We assume, throughout the paper, thatb(s) is aC(R+) function,b(s) > for anys∈R+,g(s) is a positiveC(R+) function,f(x,s,d,t) is a nonnegativeC(D×R+×R+×R+) function, andu(x) is a positiveC(D) function. Under the above assumptions, the classical theory [] of parabolic equation assures that there exists a unique classical solutionu(x,t) with
someT> for problem (.) and the solution is positive overD×[,T). Moreover, the regularity theorem [] impliesu(x,t)∈C(D×(,T))∩C(D×[,T)).
Many papers have studied the global and blow-up solutions of parabolic problems with a gradient term (see, for instance, [–]). Some authors have discussed the global and blow-up solutions of parabolic problems under Robin boundary conditions and have got a lot of meaningful results (see [–] and the references cited therein). Some special cases of problem (.) have been treated already. Zhang [] dealt with the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut=∇ ·(g(u)∇u) +f(u) inD×(,T), ∂u
∂n+γu= on∂D×(,T),
u(x, ) =u(x) > inD,
whereD⊂RN (N≥) is a bounded domain with smooth boundary∂D. By constructing
auxiliary functions and using maximum principles, the sufficient conditions characterized by functionsf,g anduwere given for the existence of a blow-up solution. Zhang [] investigated the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(b(u))t=u+f(u) inD×(,T), ∂u
∂n+γu= on∂D×(,T),
u(x, ) =u(x) > inD,
whereD⊂RN (N≥) is a bounded domain with smooth boundary∂D. By constructing some auxiliary functions and using maximum principles, the sufficient conditions were obtained there for the existence of global and blow-up solutions. Meanwhile, the upper estimate of a global solution, the upper bound of ‘blow-up time’ and the upper estimate of ‘blow-up rate’ were also given. Ding [] considered the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(b(u))t=∇ ·(g(u)∇u) +f(u) inD×(,T), ∂u
∂n+γu= on∂D×(,T),
u(x, ) =u(x) > inD,
whereD⊂RN (N≥) is a bounded domain with smooth boundary∂D. By constructing
some appropriate auxiliary functions and using a first-order differential inequality tech-nique, the sufficient conditions were obtained for the existence of global and blow-up so-lutions. For the blow-up solution, an upper and a lower bound on blow-up time were also given.
In this paper, we study problem (.). Since the functionf(x,u,q,t) contains a gradient termq=|∇u|, it seems that the methods of [–] are not applicable for problem (.). In this paper, by constructing completely different auxiliary functions with those in [– ] and technically using maximum principles, we obtain some existence theorems of a global solution, an upper estimate of the global solution, the existence theorems of a blow-up solution, an blow-upper bound of ‘blow-blow-up time’, and an blow-upper estimates of ‘blow-blow-up rate’. Our results extend and supplement those obtained [–].
2 Global solution
The main result for the global solution is the following theorem.
Theorem . Let u be a solution of problem(.).Assume that the following conditions
(i)-(iv)are satisfied:
(i) for anys∈R+,
sb(s)≥, sb(s) –sb(s)≤,
g(s) b(s)
≤,
g(s)
g(s) b(s)
+
b(s)
+ g
g(s) b(s)
+
b(s)≤;
(.)
(ii) for any(x,s,d,t)∈D×R+×R+×R+,
ft(x,s,d,t)≤, fd(x,s,d,t)
b(s)
+
b(s)
≤,
f(x,s,d,t)b(s) g(s) s–
f(x,s,d,t)b(s) g(s) ≤;
(.)
(iii)
+∞
m
b(s)
es ds= +∞, m:=min D
u(x); (.)
(iv)
α:=max
D
∇ ·(g(u)∇u) +f(x,u,q, )
eu > , q:=|∇u|
. (.)
Then the solution u to problem(.)must be a global solution and
u(x,t)≤H–αt+Hu(x,t), (x,t)∈D×R+, (.) where
H(z) :=
z
m
b(s)
es ds, z≥m, (.)
and H–is the inverse function of H.
Proof Consider the auxiliary function
P(x,t) :=b(u)ut–αeu. (.)
Now we have
∇P=but∇u+b∇ut–αeu∇u, (.)
and
Pt=b(ut)+b(ut)t–αeuut
=b(ut)+b
g bu+
g b|∇u|
+ f
b t–αe
uu t
=b(ut)+
g–b
g
b utu+gut+
g–b
g
b ut|∇u|
+g+ fq
∇u· ∇ut+
fu–
bf b –αe
u u
t+ft. (.)
It follows from (.) and (.) that
g
bP–Pt=
bg b +
bg b –g
u
t|∇u|+
b
g
b – g
– f
q ∇u· ∇ut
+
b
g
b –g
u
tu–α
g be
u|∇u|–αg be
uu–b(u
t)
+
bf
b –fu+αe
u u
t–ft. (.)
By (.), we have
u=b
gut– g g|∇u|
–f
g. (.)
Substitute (.) into (.), to get
g
bP–Pt=
bg b –
bg b –g
+(g)
g ut|∇u| +
b
g
b – g
– f
q ∇u· ∇ut
–(b
) g g b
(ut)+
fg g –
bf
b –fu ut+
αg
be
u–αg
be
u |∇u|
+αf
be
u–f
t. (.)
With (.), we have
∇ut=
b∇P–
b
but∇u+α eu
b∇u. (.)
Next, we substitute (.) into (.) to obtain
g bP+
g b
+ fq b
∇u· ∇P–Pt
=
bg b +
bg b –g
+(g)
g –
(b)g (b) +
bfq
b ut|∇u|
+
α b
g
(b)e
u–αg
be
u–αg
be
u– αfq
be
u |∇u|
–(b
) g g b
(ut)+
fg
g –
bf
b –fu ut+α f be
u–f
In view of (.), we have
ut=
bP+α
eu
b. (.)
Substituting (.) into (.), we get
g bP+
g b
+ fq b
∇u· ∇P
+ g g g b
+ fq
b
|∇u|+ g (b)
fb
g u
P–Pt
= –αeu
g g g b + b + g g b + b
+ fq b + b
|∇u|
–(b
) g g b
(ut)–α
geu
(b)
fb
g u–
fb g
–ft. (.)
The assumptions (.) and (.) guarantee that the right-hand side of (.) is nonnegative, i.e.,
g bP+
g b
+ fq b
∇u· ∇P
+ g g g b
+ fq
b
|∇u|+ g (b)
fb
g u
P–Pt
≥ inD×(,T). (.)
By applying the maximum principle [], it follows from (.) thatPcan attain its non-negative maximum only forD× {}or∂D×(,T). ForD× {}, by (.), we have
max
D
P(x, ) =max
D
b(u)(u)t–αeu
=max
D
∇ ·g(u)∇u
+f(x,u,q, ) –αeu
=max
D
eu ∇ ·
(g(u)∇u) +f(x,u,q, )
eu –α
= .
We claim thatPcannot take a positive maximum at any point (x,t)∈∂D×(,T). In fact, suppose thatPtakes a positive maximum at a point (x,t)∈∂D×(,T), then
P(x,t) > and
∂P
∂n
(x,t)
> . (.)
With (.) and (.), we have
∂P
∂n=b
u
t ∂u
∂n+b
∂ut ∂n –αe
u∂u ∂n= –γb
uu
t+b
∂u
∂n t+γ αue
u
= –γbuut+b(–γu)t+γ αueu= –γ
ubut+γ αueu
= –γub
bP+α
be
u +γ αueu
= –γ(ub
)
b P+γ αe
uub– (ub)
Next, by using the fact that (sb(s))≥,sb(s) – (sb(s))≤ for anys∈R+, it follows from (.) that
∂P
∂n
(x,t)
≤,
which contradicts with inequality (.). Thus we know that the maximum ofP inD× [,T) is zero,i.e.,
P≤ inD×[,T),
and
b(u)
eu ut≤α. (.)
For each fixedx∈D, integration of (.) from totyields
t
b(u)
eu utdt= u(x,t)
u(x)
b(s)
es ds≤αt, (.)
which implies thatumust be a global solution. Actually, if thatublows up at finite time T, then
lim
t→T–u(x,t) = +∞.
Passing to the limit ast→T–in (.) yields
+∞
u(x)
b(s)
es ds≤αT
and
+∞
m
b(s) es ds=
u(x)
m
b(s) es ds+
+∞
u(x)
b(s) es ds≤
u(x)
m
b(s)
es ds+αT< +∞,
which contradicts with assumption (.). This shows thatuis global. Moreover, it follows from (.) that
u(x,t)
u(x)
b(s) es ds=
u(x,t)
m
b(s) es ds–
u(x)
m
b(s) es ds=H
u(x,t)–Hu(x)≤αt.
SinceHis an increasing function, we have
u(x,t)≤H–αt+Hu(x).
3 Blow-up solution
The following theorem is the main result for the blow-up solution.
Theorem . Let u be a solution of problem(.).Assume that the following conditions
(i)-(iv)are fulfilled:
(i) for anys∈R+,
sb(s)≥, sb(s) –sb(s)≥,
g(s) b(s)
≥,
g(s)
g(s) b(s)
+
b(s)
+ g
g(s) b(s)
+
b(s)≥;
(.)
(ii) for any(x,s,d,t)∈D×R+×R+×R+,
ft(x,s,d,t)≥, fd(x,s,d,t)
b(s)
+
b(s)
≥,
f(x,s,d,t)b(s) g(s) s–
f(x,s,d,t)b(s) g(s) ≥;
(.)
(iii)
+∞
M
b(s)
es ds< +∞, M:=max D
u(x); (.)
(iv)
β:=min
D
∇ ·(g(u)∇u) +f(x,u,q, )
eu > , q:=|∇u|
. (.)
Then the solution u of problem(.)must blow up in finite time T,and
T≤
β +∞
M
b(s)
es ds, (.)
u(x,t)≤G–β(T–t), (x,t)∈D×[,T), (.)
where
G(z) :=
+∞
z
b(s)
es ds, z> , (.)
and G–is the inverse function of G.
Proof Construct the following auxiliary function:
ReplacingPandαwithQandβin (.), respectively, we get
g bQ+
g b
+ fq b
∇u· ∇Q
+ g g g b
+ fq
b
|∇u|+ g (b)
fb
g u
Q–Qt
= –βeu
g g g b + b + g g b + b
+ fq b + b
|∇u|
–(b
) g g b
(ut)–β
geu
(b)
fb
g u–
fb g
–ft. (.)
Assumptions (.) and (.) imply that the right-hand side in equality (.) is nonpositive, i.e.,
g bQ+
g b
+ fq b
∇u· ∇Q
+ g g g b
+ fq
b
|∇u|+ g (b)
fb
g u
Q–Qt
≤ inD×(,T). (.)
With (.), we have
min
D
Q(x, ) =min
D
b(u)(u)t–βeu
=min
D
∇ ·g(u)∇u
+f(x,u,q, ) –βeu
=min
D
eu ∇ ·
(g(u)∇u) +f(x,u,q, )
eu –β
= . (.)
SubstitutingPandαwithQandβin (.), respectively, we have
∂Q
∂n = –γ (ub)
b Q+γ βe
uub– (ub)
b on∂D×(,T). (.)
Combining (.)-(.) with the fact that (sb(s))≥,sb(s) – (sb(s))≥ for anys∈R+, and applying the maximum principles again, it follows that the minimum ofQinD×[,T) is zero. Thus
Q≥ inD×[,T),
and
b(u)
eu ut≥β. (.)
At the pointx∗∈D, whereu(x∗) =M, integrate (.) over [,t] to get
t
b(u)
eu utdt= u(x∗,t)
M
b(s)
which implies thatumust blow up in finite time. Actually, ifuis a global solution of (.), then for anyt> , (.) shows
+∞
M
b(s) es ds≥
u(x∗,t)
M
b(s)
es ds≥βt. (.)
Lettingt→+∞in (.), we have
+∞
M
b(s)
es ds= +∞,
which contradicts with assumption (.). This shows thatumust blow up in finite time t=T. Furthermore, lettingt→Tin (.), we get
T≤
β +∞
M
b(s) es ds.
By integrating inequality (.) over [t,s] ( <t<s<T), for each fixedx, we obtain
Gu(x,t)≥Gu(x,t)–Gu(x,s)=
+∞
u(x,t) b(s)
es ds– +∞
u(x,s) b(s)
es ds
=
u(x,s)
u(x,t) b(s)
es ds= s
t
b(u)
eu utdt≥β(s–t).
Hence, by lettings→T, we have
Gu(x,t)≥β(T–t).
SinceGis a decreasing function, we obtain
u(x,t)≤G–β(T–t).
The proof is complete.
4 Applications
Whenb(u)≡uandf(x,u,q,t)≡f(u), the results stated in Theorem . are valid. When g(u)≡ and f(x,u,q,t)≡f(u) or f(x,u,q,t)≡ f(u), the conclusions of Theorems . and . still hold true. In this sense, our results extend and supplement the results of [–].
In what follows, we present several examples to demonstrate the applications of the ab-stract results.
Example . Letube a solution of the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut=u+++uu|∇u|+e –u(e–u+eq)
+u (e
–t+|x|) inD×(,T),
∂u
∂n+ u= on∂D×(,T),
whereq=|∇u|,D={x= (x,x,x)| |x|< }is the unit ball ofR. The above problem can be transformed into the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(ueu)
t=∇ ·(( +u)eu∇u) + (e–u+ eq)(e–t+|x|) inD×(,T), ∂u
∂n+ u= on∂D×(,T),
u(x, ) = –|x| inD.
Now
b(u) =ueu, g(u) = ( +u)eu, f(x,u,q,t) =e–u+ eqe–t+|x|, u(x) = –|x|, γ = .
In order to determine the constantα, we assume
s:=|x|,
then ≤s≤ and
α=max
D
∇ ·(g(u)∇u) +f(x,u,q, ) eu
=max
D
|x|– |x|– + +|x|exp– + |x|+exp– + |x| =max
≤s≤
s– s– + ( +s)exp(– + s) +exp(– + s)
= ..
It is easy to check that (.)-(.) hold. By Theorem .,umust be a global solution, and
u(x,t)≤H–αt+Hu(x)
= – +
.t+ +u(x)
= – +
.t+ –|x|.
Example . Letube a solution of the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut=u–u(+u)|∇u|+u(e
u–e–q)
+u ( +t|x|) inD×(,T), ∂u
∂n+ u= on∂D×(,T),
u(x, ) = –|x| inD,
whereq=|∇u|,D={x= (x,x,x)| |x|< }is the unit ball ofR. The above problem may be turned into the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(u+lnu)t=∇ ·(( +u)∇u) + (eu– e–q)( +t|x|) inD×(,T), ∂u
∂n+ u= on∂D×(,T),
Now we have
b(u) =u+lnu, g(u) = +
u, f(x,u,q,t) =
eu– e–q +t|x|, u(x) = –|x|, γ = .
By setting
s:=|x|,
we have ≤s≤ and
β=min
D
∇ ·(g(u)∇u) +f(x,u,q, ) eu
=min
D
–|x|+ |x|– ( –|x|)exp( –|x|)+
–exp–|x|–
=min ≤s≤
–s+ s– ( –s)exp( –s)+
–exp(–s– )
= ..
Again it is easy to check that (.)-(.) hold. By Theorem .,umust blow up in finite timeT, and
T≤
β +∞
M
b(s) es ds=
.
+∞
+ s
esds= .,
u(x,t)≤G–β(T–t)=G–.(T–t), where
G(z) =
+∞
z
b(s) es ds=
+∞
z
+ s
esds, z≥,
andG–is the inverse function ofG.
Remark . We can see from Example . that when the equation has a gradient term
with exponential increase, the functionsgandbincrease exponentially to ensure that the solution of (.) blows up. It follows from Example . that when the equation has a gra-dient term with exponential decay, the appropriate assumptions on the functionsgandb can guarantee the solution of (.) to be global.
Competing interests
The author declares that he has no competing interests.
Author’s contributions All results belong to Juntang Ding.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 61074048 and 61174082) and the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011).
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