Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 113, pp. 1–5.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
LOWER BOUNDS FOR THE BLOW-UP TIME OF NONLINEAR
PARABOLIC PROBLEMS WITH ROBIN BOUNDARY
CONDITIONS
KHADIJEH BAGHAEI, MAHMOUD HESAARAKI
Abstract. In this article, we find a lower bound for the blow-up time of solutions to some nonlinear parabolic equations under Robin boundary conditions
in bounded domains of Rn .
1. Introduction
In this article, we consider the nonlinear initial-boundary value problem
(b(u))t = ∇ · (g(u)∇u) + f (u),
x ∈ Ω, t > 0
∂u
+ γu = 0, x ∈ ∂Ω, t > 0,
∂ν
u(x, 0) = u0 (x) ≥ 0, x ∈ Ω
(1.1)
where Ω ⊆ Rn , n ≥ 3, is a bounded domain with smooth boundary, ν is the outward
normal vector to ∂Ω, γ is a positive constant and u0 (x) ∈ C 1 (Ω) is the initial value.
We assume that f is a nonnegative C(R+ ) function and the nonnegative functions
g and b satisfy
g ∈ C 1 (R+ ),
2
+
b ∈ C (R ),
g(s) ≥ gm > 0,
0
0 < b (s) ≤
b0M ,
g 0 (s) ≤ 0,
00
b (s) ≤ 0,
∀s > 0,
∀s > 0,
(1.2)
where gm and b0M are positive constants.
The reader is referred to [1, 3, 4, 5, 6, 8] for results on bounds for blow-up time
in nonlinear parabolic problems. Ding [2] studied problem (1.1) under assumptions
(1.2) and derived conditions on the data which imply blow-up or the global existence
of solutions. In addition, Ding obtained a lower bound for the blow-up time when
Ω ⊆ R3 is a bounded convex domain. Here we obtain a lower bound for the blow-up
time for (1.1) in general bounded domains Ω ⊆ Rn , n ≥ 3.
2. A lower bound for the blow-up time
In this section we find a lower bound for the blow-up time T in an appropriate
measure. The idea of the proof of the following theorem comes from [1].
2000 Mathematics Subject Classification. 35K55, 35B44.
Key words and phrases. Parabolic equation; Robin boundary condition; blow-up; lower bound.
c
2014
Texas State University - San Marcos.
Submitted June 6, 2013. Published April 16, 2014.
1
2
K. BAGHAEI, M. HESSARAKI
EJDE-2014/113
Theorem 2.1. Let Ω be a bounded domain in Rn , n ≥ 3, and let the functions
f, g, b satisfy
Z s b0 (y) p+1
0 < f (s) ≤ cg(s)
dy
, s > 0,
(2.1)
0 g(y)
for some constants c > 0 and p ≥ 1. If u(x, t) is a nonnegative classical solution to
problem (1.1), which becomes unbounded in the measure
Z Z u(x,t) 0
b (y) 2k
dy
Φ(t) =
dx,
g(y)
Ω
0
where k is a parameter restricted by the condition
k > max p(n − 2), 1 ,
then T is bounded from below by
Z +∞
Φ(0)
dξ
k1 + k2 ξ
3(n−2)
3n−8
2n−3
(2.2)
,
(2.3)
+ k3 ξ 2(n−2)
where k1 , k2 and k3 are positive constants which will be determined later in the
proof.
Proof. To simplify our computations we define
Z s 0
b (y)
v(s) =
dy,
0 g(y)
s > 0.
(2.4)
Hence,
dΦ
d
=
dt
dt
Z
v
ZΩ
2k
Z
dx = 2k
Ω
v 2k−1
b0 (u)
ut dx
g(u)
(b(u))t
dx
= 2k
v 2k−1
g(u)
ZΩ
i
1 h
∇ · (g(u)∇u) + f (u) dx
= 2k
v 2k−1
g(u)
Ω
Z
Z
g 0 (u)
= −2k(2k − 1)
v 2k−2 v 0 (u)|∇u|2 dx + 2k
v 2k−1
|∇u|2 dx
g(u)
Ω
Ω
Z
Z
f (u)
− 2kγ
v 2k−1 u ds + 2k
v 2k−1
dx
g(u)
∂Ω
Ω
Z
Z
b0 (u)
f (u)
≤ −2k(2k − 1)
v 2k−2
|∇u|2 dx + 2k
v 2k−1
dx,
g(u)
g(u)
Ω
Ω
where in the above inequality we used u ≥ 0 and g 0 (u) ≤ 0 from (1.2). From (2.4),
we have
g(u) 2
|∇v|2 .
(2.5)
|∇u|2 = 0
b (u)
By (1.2), (2.5), and (2.1) we have
Z
Z
dΦ
f (u)
2k−2 g(u)
2
≤ −2k(2k − 1)
v
|∇v| dx + 2k
v 2k−1
dx
0 (u)
dt
b
g(u)
Ω
Ω
Z
Z
2(2k − 1)gm
≤−
|∇v k |2 dx + 2kc
v 2k+p dx.
kb0M
Ω
Ω
(2.6)
EJDE-2014/113
LOWER BOUNDS FOR THE BLOW-UP TIME
3
From (2.2), Hölder, and Young inequalities, we infer
Z
Z k(2n−3) m2
v 2k+p dx ≤ |Ω|m1
v n−2 dx
Ω
Ω
Z
k(2n−3)
≤ m1 |Ω| + m2
v n−2 dx,
(2.7)
Ω
where
m1 =
k(2n − 3) − (n − 2)(2k + p)
,
k(2n − 3)
m2 =
(n − 2)(2k + p)
.
k(2n − 3)
From (2.7) and the Cauchy-Schwartz inequality we have:
Z
Z
1/2 Z 2k(n−1) 1/2
k(2n−3)
2k
n−2
dx ≤
v
v dx
v n−2 dx
Ω
Ω
Ω
Z
34 Z
1/4
2n
≤
v 2k dx
(v k ) n−2 dx
.
Ω
(2.8)
Ω
Applying the Sobolev inequality (see [7]) to the last term in (2.8), for n > 3, we
obtain
n
kv k k 2(n−2)
2n
L n−2 (Ω)
n
n
2(n−2)
≤ (cs ) 2(n−2) kv k kW
1,2 (Ω)
n
n
n
k 2(n−2)
≤ (cs ) 2(n−2) k∇v k kL2(n−2)
2 (Ω) + kv kL2 (Ω)
(2.9)
In the case, n = 3, we have
n
kv k k 2(n−2)
2n
L n−2 (Ω)
n
n
2(n−2)
≤ (cs ) 2(n−2) kv k kW
1,2 (Ω)
n
n
4−n
n
k 2(n−2)
.
≤ 2 2(n−2) (cs ) 2(n−2) k∇v k kL2(n−2)
2 (Ω) + kv kL2 (Ω)
(2.10)
Here, cs is the best constant in the Sobolev inequality.
By inserting (2.9) in (2.8) for n > 3 and (2.10) in (2.8) for n = 3, we have
Z
k(2n−3)
v n−2 dx
Ω
Z
43 n
n
k 2(n−2)
≤ c0
v 2k dx
k∇v k kL2(n−2)
(2.11)
2 (Ω) + kv kL2 (Ω)
Ω
Z
Z
Z
2n−3
n
34 4(n−2)
2(n−2)
= c0
v 2k dx
|∇v k |2 dx
+ c0
v 2k dx
,
Ω
Ω
Ω
where
 4−n
n
2 2(n−2) (cs ) 2(n−2)
, for n = 3,
c0 =
n
(c ) 2(n−2)
,
for n > 3.
s
Now, using Young’s inequality we obtain
Z
k(2n−3)
v n−2 dx
Ω
(2.12)
4(n−2)
c 3n−8 (3n − 8) 3(n−2)
n
≤ 0
Φ 3n−8 +
n
4(n − 2)
4(n − 2) 3n−8
Z
k 2
|∇v | dx + c0 Φ
Ω
2n−3
2(n−2)
,
4
K. BAGHAEI, M. HESSARAKI
EJDE-2014/113
where is a positive constant to be determined later. Substituting (2.12) into (2.7)
yields
Z
Z
n (3n − 8)
4(n−2)
3(n−2)
n
3n−8
3n−8 +
2kc
v 2k+p dx ≤ 2kcm2
c
|∇v k |2 dx
Φ
n
0
3n−8
4(n
−
2)
4(n
−
2)
Ω
Ω
o
2n−3
2(n−2)
+ c0 Φ
+ 2kcm1 |Ω|.
By inserting the last inequality in (2.6), we have
Z
2n−3
3(n−2)
nkcm2 dΦ 2(2k − 1)gm
≤ −
+
|∇v k |2 dx + k1 + k2 Φ 3n−8 + k3 Φ 2(n−2) ,
0
dt
kbM
2(n − 2) Ω
where
4(n−2)
k1 = 2kcm1 |Ω|,
2kcm2 (c0 ) 3n−8 (3n − 8)
k2 =
,
n
4(n − 2) 3n−8
For
=
k3 = 2kcc0 m2 .
4(n − 2)(2k − 1)gm
,
nk 2 cm2 b0M
the above inequality becomes
2n−3
3(n−2)
dΦ
≤ k1 + k2 Φ 3n−8 + k3 Φ 2(n−2) .
dt
Thus,
dΦ
≤ dt.
2n−3
3(n−2)
k1 + k2 Φ 3n−8 + k3 Φ 2(n−2)
We integrate from 0 to t to obtain
Z Φ(t)
dξ
≤ t,
2n−3
3(n−2)
Φ(0) k1 + k2 ξ 3n−8 + k3 ξ 2(n−2)
where
Z Z u0 (x) 0
b (y) 2k
dx.
Φ(0) =
dy
g(y)
Ω
0
Passing to the limit as t → T − , we conclude that
Z +∞
dξ
≤ T.
2n−3
3(n−2)
Φ(0) k1 + k2 ξ 3n−8 + k3 ξ 2(n−2)
The proof is complete.
(2.13)
References
[1] A. Bao, X. Song; Bounds for the blow-up time of the solutions to quasi-linear parabolic problems, Z. Angew. Math. Phys. (2013), DOI: 10.1007/s00033-013-0325-1.
[2] J. Ding; Global and blow-up solutions for nonlinear parabolic equations with Robin boundary
conditions, Comput. Math. Appl. 65 (2013), 1808-1822.
[3] C. Enache; Lower bounds for blow-up time in some non-linear parabolic problems under Neumann boundary conditions, Glasg. Math. J. 53 (2011), 569-575.
[4] L. E. Payne, G. A. Philippin, P. W. Schaefer; Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. 69 (2008), 3495-3502.
[5] L. E. Payne, G. A. Philippin, P. W. Schaefer; Bounds for blow-up time in nonlinear parabolic
problems, J. Math. Anal. Appl., 338 (2008), 438-447.
[6] L. E. Payne, J. C. Song, P. W. Schaefer; Lower bounds for blow-up time in a nonlinear
parabolic problems, J. Math. Anal. Appl., 354 (2009), 394-396.
[7] G. Talenti; Best constants in Sobolev inequality, Ann. Math. Pura. Appl., 110 (1976), 353-372.
EJDE-2014/113
LOWER BOUNDS FOR THE BLOW-UP TIME
5
[8] H. L. Zhang; Blow-up solutions and global solutions for nonlinear parabolic problems, Nonlinear Anal., 69 (2008), 4567-4575.
Khadijeh Baghaei
Department of mathematics, Iran University of Science and Technology, Tehran, Iran
E-mail address: khbaghaei@iust.ac.ir
Mahmoud Hesaaraki
Department of mathematics, Sharif University of Technology, Tehran, Iran
E-mail address: hesaraki@sina.sharif.edu