R E S E A R C H
Open Access
Global existence and blow-up for a class of
nonlinear reaction diffusion problems
Juntang Ding
**Correspondence:
School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, P.R. China
Abstract
This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented.
MSC: 35K57; 35K55; 35B05
Keywords: reaction diffusion problem; global existence; blow-up
1 Introduction
The global existence and blow-up for nonlinear reaction diffusion equations have been widely studied in recent years (see, for instance, [–]). In this paper, we consider the fol-lowing problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(g(x,u))t=∇ ·(a(u)b(x)∇u) +f(u) inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) =u(x) > inD,
(.)
whereD⊂RN(N≥) is a bounded domain with smooth boundary∂D,Dis the closure of
D,∂/∂nis the outward normal derivative on∂D,Tis the maximal existence time ofu. Set
R+:= (, +∞). We assume, throughout this paper, thata(s) is a positiveC(R+) function,
b(x) is a positiveC(D) function,f(s) is a positiveC(R+) function,g(x,s) is aC(D×R+)
function,gs(x,s) > for any (x,s)∈D×R+, andu(x) is a positiveC(D) function. Under
the above assumptions, it is well known from the classical parabolic equation theory [] and maximum principle [] that there exists a unique local positive solution for problem (.). Moreover, by the regularity theorem [],u(x,t)∈C(D×(,T))∩C(D×[,T)).
Many authors discussed the global existence and blow-up for nonlinear reaction dif-fusion equations with Neumann boundary conditions and obtained a lot of interesting results [–]. Some special cases of (.) have been studied already. Lair and Oxley []
investigated the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut=∇ ·(a(u)∇u) +f(u) inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) =u(x) > inD,
whereD⊂RN (N≥) is a bounded domain with smooth boundary∂D. The necessary and sufficient conditions characterized by functionsaandf were given for the global ex-istence and blow-up solution. Zhang [] dealt with the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(g(u))t=u+f(u) inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) =u(x) > inD,
whereD⊂RN (N≥) is a bounded domain with smooth boundary∂D. The sufficient conditions were obtained there for the existence of global and blow-up solutions. Gaoet al.[] considered the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(g(u))t=∇ ·(a(u)∇u) +f(u) inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) =u(x) > inD,
whereD⊂RN (N≥) is a bounded domain with smooth boundary∂D. The sufficient conditions were developed for the existence of global and blow-up solutions. Meanwhile, the upper estimate of the global solution, the upper bound of the ‘blow-up time’, and the upper estimate of the ‘blow-up rate’ were also given.
In this paper, we study reaction diffusion problem (.). Note that f(u), g(x,u) and
a(u)b(x) are nonlinear reaction, nonlinear diffusion and nonlinear convection, respec-tively. Since the diffusion functiong(x,u) depends not only on the concentration variable
ubut also on the space variablex, it seems that the methods of [, ] are not appli-cable for the problem (.). In this paper, by constructing completely different auxiliary functions from those in [, ] and technically using maximum principles, we obtain the conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also given. Our results extend and supplement those obtained in [, ].
We proceed as follows. In Section we study the blow-up solution of (.). Section is devoted to the global solution of (.). A few examples are given in Section to illustrate the applications of the obtained results.
2 Blow-up solution
Theorem . Let u(x,t)be a solution of the problem(.).Assume that the data of the problem(.)satisfies the following conditions:
(i) for any(s,t)∈D×R+,
a(s) –a(s)≤,
gs(x,s)
a(s)
s ≤,
a(s)f(s) – (a(s)f(s))
a(s)
–a(s)f(s) – (a(s)f(s))
a(s) ≤;
(.)
(ii) the constant
α=min
D
eu
f(u)gu(x,u) ∇ ·
a(u)b(x)∇u
+f(u)
> ; (.)
(iii) the integration
+∞
M es
f(s)ds< +∞, M=maxD
u(x). (.)
Then u(x,t)must blow up in a finite time T and
T≤
α
+∞
M es
f(s)ds (.)
as well as
u(x,t)≤–α(T–t), (.)
where
(z) =
+∞
z es
f(s)ds, z> , (.)
and–is the inverse function of.
Proof Consider the auxiliary function
Q(x,t) = euut–αf(u). (.)
Now we have
∇Q= euut∇u+ eu∇ut–αf∇u, (.)
Q= euut|∇u|+ eu∇u· ∇ut+ euutu+ euut–αf|∇u|–αfu, (.)
and
Qt= eu(ut)+ eu(ut)t–αfut
= eu(ut)+ eu
ab gu
u+a
b
gu|∇
u|+ a
gu∇
b· ∇u+ f
gu
= eu(ut)+
ab gu
–abguu
g
u
euutu+
ab gu
euut+
ab gu –a bg uu g u
euut|∇u|
+ a
b
gu
eu(∇u· ∇ut) +
a gu
–aguu
g
u
euut(∇b· ∇u) +
a gu
eu(∇b· ∇ut)
+
f gu
–fguu
g
u
eu–αf
ut. (.)
It follows from (.) and (.) that
ab gu
Q–Qt=
abguu
g u –a b gu +ab gu
euut|∇u|+
ab
gu – a
b
gu
eu(∇u· ∇ut)
+ abguu g u –a b gu +ab gu
euutu–α
abf gu |∇
u|–αabf
gu
u– eu(ut)
+ aguu g u –a gu
euut(∇b· ∇u) –
a gu
eu(∇b· ∇ut)
+ fguu g u – f gu
eu+αf
ut. (.)
By the first equation of (.), we have
u= gu
abut– a
a|∇u|
–
b∇b· ∇u– f
ab. (.)
Substitute (.) into (.) to obtain
ab gu
Q–Qt
=
(a)b agu –a b gu –a b gu +ab gu
euut|∇u|+
ab
gu – a
b
gu
eu(∇u· ∇ut)
+ a gu gu a u
eu(ut)–
a gu
euut(∇b· ∇u) +α
af gu
(∇b· ∇u) +
af agu – f gu – f gu
euut
+
αa
bf
gu
–αabf
gu
|∇u|– a
gu
eu(∇b· ∇ut) +αff
gu
. (.)
It follows from (.) that
∇ut= e–u∇Q–ut∇u+αe–uf∇u. (.)
Substituting (.) into (.), we get
ab gu Q+ b gu
a–a∇u+ a
gu ∇b
· ∇Q–Qt
=
(a)b
agu +a b gu –a b gu –ab gu
euut|∇u|+
αabf
gu –αa
bf
gu
–αabf
gu
|∇u|
+ a gu gu a u
eu(ut)+
af agu – f gu –f gu
euut+α
ff gu
With (.), we have
ut= e–uQ+αe–uf. (.)
Substitute (.) into (.) to get
ab gu
Q+
b
gu
a–a∇u+ a
gu∇
b
· ∇Q
+
ab gu
a a
–a
a +
|∇u|+ a
gu
f a
+f
a
Q–Qt
=αab
gu
af– (af)
a
–af– (af)
a
|∇u|
+ a
gu
gu
a
u
eu(ut)+α f
agu
a–a. (.)
The assumptions (.) ensure that the right-hand side of (.) is nonpositive; that is,
ab gu
Q+
b
gu
a–a∇u+ a
gu∇
b
· ∇Q
+
ab gu
a a
–a
a +
|∇u|+ a
gu
f a
+f
a
Q–Qt
≤ inD×(,T). (.)
Now, by (.), we have
∂Q
∂n = e
u∂ut ∂n + e
uu t
∂u
∂n–αf ∂u
∂n= e
u
∂u
∂n
t
= on∂D×(,T). (.)
Furthermore, it follows from (.) that
min
D
Q(x, ) =min
D
eu
gu(x,u)
∇ ·a(u)b(x)∇u
+f(u)
–αf(u)
=min
D
f(u)
eu
f(u)gu(x,u) ∇ ·
a(u)b(x)∇u
+f(u)
–α
= . (.)
Combining (.)-(.) and applying the maximum principle [], we find that the mini-mum ofQinD×[,T) is zero. Hence,
Q≥ inD×[,T);
that is,
eu
At the pointx∈D, whereu(x) =M, integrating (.) over [,t], we get
t
eu
f(u)utdt=
u(x,t)
M es
f(s)ds≥αt.
By the assumption (.), we know thatu(x,t) must blow up in finite timet=T, moreover,
T≤
α
+∞
M es
f(s)ds.
For each fixedx, integrating the inequality (.) over [t,s] ( <t<s<T), we obtain
u(x,t)≥u(x,t)–u(x,s)=
+∞
u(x,t)
es
f(s)ds–
+∞
u(x,s)
es
f(s)ds
=
u(x,s)
u(x,t)
es
f(s)ds=
s t
eu
f(u)utdt≥α(s–t).
Lettings→T, we have
u(x,t)≥α(T–t),
which implies
u(x,t)≤–α(T–t).
The proof is complete.
3 Global solution
In this section we establish sufficient conditions on the data of the problem (.) in order to ensure that the solution has global existence. Under these conditions, we derive an explicit upper estimate of the global solution. The main results of this section are the following theorem.
Theorem . Let u(x,t)be a solution of the problem(.).Assume that the data of the
problem(.)satisfies the following conditions:
(i) for any(s,t)∈D×R+,
a(s) –a(s)≥,
gs(x,s)
a(s)
s ≥,
a(s)f(s) – (a(s)f(s))
a(s)
–a(s)f(s) – (a(s)f(s))
a(s) ≥;
(.)
(ii) the constant
β=max
D
eu
f(u)gu(x,u)
∇ ·a(u)b(x)∇u
+f(u)
(iii) the integration
+∞
m es
f(s)ds< +∞, m:=minD
u(x). (.)
Then u(x,t)must be a global solution and
u(x,t)≤βt+u(x)
, (.)
where
(z) =
z m
es
f(s)ds, z≥m, (.)
and–is the inverse function of.
Proof Consider an auxiliary function
P(x,t) = euut–βf(u). (.)
In (.), by replacingQandαbyPandβ, respectively, we have
ab gu P+ b gu
a–a∇u+ a
gu∇
b
· ∇P
+ ab gu a a –a
a +
|∇u|+ a
gu f a +f a
P–Pt
=βab
gu
af – (af)
a
–af – (af)
a
|∇u|
+ a gu gu a u
eu(ut)+β f
agu
a–a. (.)
It follows from (.) that the right-hand side of (.) is nonnegative; that is,
ab gu P+ b gu
a–a∇u+ a
gu∇
b
· ∇P
+ ab gu a a –a
a +
|∇u|+ a
gu f a +f a
P–Pt≥ inD×(,T). (.)
With (.), we have
∂P
∂n= e
u∂ut ∂n + e
uu t
∂u
∂n–βf ∂u
∂n= e
u ∂u ∂n t
= on∂D×(,T). (.)
It follows from (.) that
max
D
P(x, ) =max
D
eu
gu(x,u)
∇ ·a(u)b(x)∇u
+f(u)
–βf(u)
=max
D
f(u)
eu
f(u)gu(x,u) ∇ ·
a(u)b(x)∇u
+f(u)
–β
Combining (.)-(.) and applying the maximum principle, we know that the maximum ofPinD×[,T) is zero; that is,
P(x,t)≤ inD×[,T). (.)
From (.), we get
eu
f(u)ut≤β. (.)
For each fixedx∈D, by integrating (.) over [,t], we have
β
t
eu
f(u)utdt= β
u(x,t)
u(x)
es
f(s)ds≤t. (.)
It follows from (.) and (.) thatu(x,t) must be a global solution. Furthermore, by (.), we have
u(x,t)–u(x)
=
u(x,t)
m es
f(s)ds–
u(x)
m es
f(s)ds=
u(x,t)
u(x)
es
f(s)ds≤βt.
Hence,
u(x,t)≤–βt+u(x)
.
The proof is complete.
4 Applications
Wheng(x,u)≡g(u),a(u)≡, andb(x)≡ org(x,u)≡g(u) andb(x)≡, the conclusions of Theorems . and . are valid. In this sense, our results extend and supplement the results of [, ].
In what follows, we present several examples to demonstrate the applications of the ob-tained results.
Example . Letu(x,t) be a solution of the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ut=u+|∇u|++|x|
i=xi∂∂xui+ eu
+|x| inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) = – ( –|x|) inD,
whereD={x= (x,x,x)||x|=x+x+x< }is the unit ball ofR. The above problem
may be turned into the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(eu( +|x|))t=∇ ·(eu( +|x|)∇u) + eu inD×(,T), ∂u
∂n= on∂D×(,T),
Now
a(u) = eu, b(x) = +|x|, f(u) = eu,
g(x,u) = eu +|x|, u
(x) = –
–|x|. By setting
s=|x|,
we have ≤s≤ and
α=min
D
eu
f(u)gu(x,u) ∇ ·
a(u)b(x)∇u
+f(u)
=min
D
+ |
x|– |x|– |x|+ |x|+ exp[ – ( –|x|)]
( +|x|)exp[ – ( –|x|)]
= min ≤s≤
+ s– s– s+ s+ exp[ – ( –s)] ( +s)exp[ – ( –s)]
= ..
It is easy to check that (.)-(.) hold. By Theorem .,u(x,t) must blow up in a finite timeTand
T≤
α
+∞
M es
f(s)ds= .
+∞
esds= .
as well as
u(x,t)≤–α(T–t)=ln
.(T–t).
Example . Letu(x,t) be a solution of the following problem:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ut=u+ |∇u|++|x|
i=xi∂∂xu
i+ e– u
(+|x|) inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) = + ( –|x|) inD,
whereD={x= (x,x,x)||x|=x+x+x< }is the unit ball ofR. The above problem
can be transformed into the following problem:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
(eu( +|x|))t=∇ ·(eu( +|x|)∇u) + eu inD×(,T), ∂u
∂n= on∂D×(,T),
u(x, ) = + ( –|x|) inD.
Now we have
a(u) = eu, b(x) = +|x|, f(u) = eu,
g(x,u) = eu +|x|, u(x) = +
In order to determine the constantβ, we assume
s=|x|,
then ≤s≤ and
β=max
D
eu
f(u)gu(x,u) ∇ ·
a(u)b(x)∇u
+f(u)
=max
D
exp
+
–|x|
×
– + |x|– |x|– |x|+ |x|+exp
– –
–|x|
+|x|
= max ≤s≤
exp
[ +
( –s)
][– + s– s– s+ s+exp(– –
( –s)
)]
+s
= ..
Again, it is easy to check that (.)-(.) hold. By Theorem .,u(x,t) must be a global solution and
u(x,t)≤–βt+u(x)
= ln
.t+exp
u(x)
= ln
.t+exp
+
–|x|
.
Competing interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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), the Research Project Supported by Shanxi Scholarship Council of China (Nos. 2011-011 and 2012-011), and the Higher School ‘131’ Leading Talent Project of Shanxi Province.
Received: 8 April 2014 Accepted: 25 June 2014 References
1. Quittner, P, Souplet, P: Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States. Birkhäuser Advanced Texts. Birkhäuser, Basel (2007)
2. Galaktionov, VA, Vázquez, JL: The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst.8, 399-433 (2002)
3. Deng, K, Levine, HA: The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl.243, 85-126 (2000)
4. Bandle, C, Brunner, H: Blow-up in diffusion equations: a survey. J. Comput. Appl. Math.97, 3-22 (1998) 5. Samarskii, AA, Galaktionov, VA, Kurdyumov, SP, Mikhailov, AP: Blow-Up in Problems for Quasilinear Parabolic
Equations. Nauka, Moscow (1987) (in Russian); English translation, Walter de Gruyter, Berlin, 1995 6. Levine, HA: The role of critical exponents in blow-up theorems. SIAM Rev.32, 262-288 (1990)
7. Ding, JT: Blow-up solutions and global existence for quasilinear parabolic problems with Robin boundary conditions. Abstr. Appl. Anal.2014, Article ID 324857 (2014)
10. Friedman, A: Partial Differential Equation of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
11. Qu, CY, Bai, XL, Zheng, SN: Blow-up versus extinction in a nonlocalp-Laplace equation with Neumann boundary conditions. J. Math. Anal. Appl.412, 326-333 (2014)
12. Li, FS, Li, JL: Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions. J. Math. Anal. Appl.385, 1005-1014 (2012)
13. Gao, WJ, Han, YZ: Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl. Math. Lett.
24, 784-788 (2011)
14. Song, JC: Lower bounds for the blow-up time in a non-local reaction-diffusion problem. Appl. Math. Lett.24, 793-796 (2011)
15. Ding, JT, Li, SJ: Blow-up and global solutions for nonlinear reaction-diffusion equations with Neumann boundary conditions. Nonlinear Anal. TMA68, 507-514 (2008)
16. Jazar, M, Kiwan, R: Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire25, 215-218 (2008)
17. Soufi, AE, Jazar, M, Monneau, R: A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire24, 17-39 (2007) 18. Payne, LE, Schaefer, PW: Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl.
Anal.85, 1301-1311 (2006)
19. Ishige, K, Yagisita, H: Blow-up problems for a semilinear heat equations with large diffusion. J. Differ. Equ.212, 114-128 (2005)
20. Ishige, K, Mizoguchi, N: Blow-up behavior for semilinear heat equations with boundary conditions. Differ. Integral Equ.16, 663-690 (2003)
21. Mizoguchi, N: Blow-up rate of solutions for a semilinear heat equation with Neumann boundary condition. J. Differ. Equ.193, 212-238 (2003)
22. Mizoguchi, N, Yanagida, E: Blow-up of solutions with sign changes for a smilinear diffusion equation. J. Math. Anal. Appl.204, 283-290 (1996)
23. Deng, K: Blow-up behavior of the heat equations with Neumann boundary conditions. J. Math. Anal. Appl.188, 641-650 (1994)
24. Chen, XY, Matano, H: Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations. J. Differ. Equ.78, 160-190 (1989)
25. Lair, AV, Oxley, ME: A necessary and sufficient condition for global existence for degenerate parabolic boundary value problem. J. Math. Anal. Appl.221, 338-348 (1998)
26. Zhang, HL: Blow-up solutions and global solutions for nonlinear parabolic problems. Nonlinear Anal. TMA69, 4567-4574 (2008)
27. Gao, XY, Ding, JT, Guo, BZ: Blow-up and global solutions for quasilinear parabolic equations with Neumann boundary conditions. Appl. Anal.88, 183-191 (2009)
doi:10.1186/s13661-014-0168-5
