Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition
We investigate the long-time behavior of the reaction-diffusion equation, which has a nonlinearity of polynomial growth of any order, with Robin boundary condition. Sufficient conditions are obtained for the solutions of the problem to be bounded or approaching infinity at a finite time.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507726341210112 |
|---|---|
| author | Öztürk, E. Озтюрк, Е. |
| author_facet | Öztürk, E. Озтюрк, Е. |
| author_sort | Öztürk, E. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:29:34Z |
| description | We investigate the long-time behavior of the reaction-diffusion equation, which has a nonlinearity of polynomial growth of any order, with Robin boundary condition. Sufficient conditions are obtained for the solutions of the problem to be bounded or approaching infinity at a finite time. |
| first_indexed | 2026-03-24T02:13:54Z |
| format | Article |
| fulltext |
UDC 517.9
E. Öztürk, K. Kalh (Hacettepe Univ., Ankara, Turkey)
EXISTENCE AND NONEXISTENCE OF SOLUTIONS
OF REACTION-DIFFUSION EQUATION
WITH ROBIN BOUNDARY CONDITION
IСНУВАННЯ ТА НЕIСНУВАННЯ РОЗВ’ЯЗКIВ РЕАКЦIЙНО-ДИФУЗIЙНОГО
РIВНЯННЯ З ГРАНИЧНИМИ УМОВАМИ РОБЕНА
We investigate the long-time behavior of the reaction-diffusion equation, which has a nonlinearity of polynomial growth
of any order, with Robin boundary condition. Sufficient conditions are obtained for the solutions of the problem to be
bounded or approaching infinity at a finite time.
Дослiджено довготривалу поведiнку реакцiйно-дифузiйного рiвняння з граничними умовами Робена, яке мiстить
нелiнiйнiсть полiномiального росту будь-якого порядку. Отримано достатнi умови для того, щоб розв’язки граничної
задачi були обмеженими, або прямували до нескiнченностi на скiнченному промiжку часу.
1. Introduction. We consider the following reaction-diffusion equation with Robin boundary
condition:
ut - \Delta u+ a(x, t)| u| \rho u - b(x, t)| u| \nu u = h(x, t), (x, t) \in QT = \Omega \times (0, T ), (1.1)\biggl(
\partial u
\partial \eta
+ k(x\prime , t)u
\biggr) \bigm| \bigm| \bigm| \bigm|
\partial \Omega
= \varphi (x\prime , t), (x\prime , t) \in \Sigma T = \partial \Omega \times [0, T ), (1.2)
u(x, 0) = u0(x), x \in \Omega , (1.3)
where \Omega \subset \BbbR n, n \geq 3, is a bounded domain with sufficiently smooth boundary \partial \Omega ; \rho , \nu > 0
are given some constants; T is a positive number; \Delta is the n-dimensional Laplace operator; a :
QT \rightarrow \BbbR 1
+, b : QT \rightarrow \BbbR 1
+ and k : \Sigma T \rightarrow \BbbR 1 are given functions; h and \varphi are given generalized
functions.
\partial u
\partial \eta
denotes the normal derivative of the function u in direction of the outer normal vector
\eta . Here u(x, t) is an unknown function which can represent temperature, population density, or in
general the quantity of a substance.
Existence and nonexistence of solutions of nonlinear parabolic problems extensively investigated
during the past few decades. We refer the reader to the survey paper of Galaktionov and Vazquez
[5], Levine [7] and books of Quittner, Souplet [17] and Samarskii et al. [19]. There are many studies
on blow-up of solutions of semilinear parabolic equation without time-dependent coefficients under
homogeneous Dirichlet or Neumann boundary condition (see [1, 2, 4, 11, 12]). Also, some nonlin-
ear initial value parabolic problems with time-dependent coefficients under homogeneous Dirichlet
boundary condition and homogeneous Neumann boundary condition were investigated in [16, 14]
respectively.
One of the first papers on this subject is due to Hale, Rocha in [6], with homogeneous Robin
boundary condition in a bounded domain \Omega \subset \BbbR n, n \leq 3, and where it is shown the existence of
attractor.
c\bigcirc E. ÖZTÜRK, K. KALH, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3 423
424 E. ÖZTÜRK, K. KALH
In [3], blow-up phenomena was investigated for the nonlinear parabolic equation under homoge-
neous Robin boundary condition without time dependent coefficients.
In [8], Marras and Piro considered the semilinear parabolic equation ut = \Delta u + f(u) under
the Robin boundary conditions by taking constant coefficient on the bounded domain of \BbbR 2. They
determined sufficient conditions on the geometry and data to preclude the blow-up of the solution
and to obtain an exponential decay bound for the solution and its gradient.
In [18], Rault investigated the Fujita phenomenon for nonlinear parabolic problems \partial tu = \Delta u+up
in an exterior domain of \BbbR n under the homogeneous Robin boundary conditions with time dependent
coefficient in the superlinear case for positive solutions.
In [13], Payne and Schaefer studied the semilinear parabolic equation ut = \Delta u + f(u) under
a Robin boundary condition without time dependent coefficient where f satisfies the constraint
uf(u) \geq 2(1 + \alpha )F (u), F (u) =
\int u
0
f(s)ds, \alpha > 0. They determined sufficient conditions which
ensure that blow-up does occur or does not occur.
After that in [15], Payne and Philippin dealt with the time dependent semilinear parabolic equation
ut = \Delta u + b(t)f(u) under a Robin boundary condition. They determined upper and lower bounds
for the blow-up time on a region \Omega \subset \BbbR 3.
In [9], a lower bound for the blow-up time was derived for a nonlinear parabolic problem with a
gradient term; ut = \Delta u + k1(t)u
p - k2(t)| \nabla u| q, p, q > 1 under the Robin boundary conditions by
taking constant coefficient.
In a recent paper [10], we considered problem (1.1) – (1.3) and showed the existence of generalized
solution by using a general result in [20] and the existence of global attractor for the autonomous
case.
The purpose of this paper is to investigate the asymptotic behavior of solutions and to give
some condition for blow-up of solutions of the problem (1.1) – (1.3) in finite time. In this study,
we investigate Robin type boundary-value problem for reaction-diffusion equation by taking Yamabe
type polynom as the nonlinear part of the equation. Also, differently from articles above, we consider
the problem in nonstationary case, i.e., the coefficients in the equation and boundary condition depend
on time. Moreover, here the asymptotic behavior is studied in the space where the solution exists.
The plan of this paper is as follows: In the next section we give some results on the existence and
uniqueness of the solution of the problem (1.1) – (1.3). In Section 3, we give some conditions under
which the solutions of the problem (1.1) – (1.3) is bounded in L2(\Omega ) for all t \geq 0. We also obtain
additional condition under which the solutions tend to zero as t \rightarrow \infty . In Section 4, we investigated
the effect of exponents \rho , \nu and the data on the behavior of the solutions, some sufficient conditions
are obtained for solutions of the problem tending to infinity at a finite time.
2. Preliminaries. First, we give the definition of the generalized solution and then recall the
existence and uniqueness theorems since we will investigate the behavior of the solution which exists.
For more details, we refer to [10].
We shall assume h \in L2(0, T ; (W
1
2 (\Omega ))
\ast ) + L\rho +2
\rho +1
(QT ), \varphi \in L2(0, T ;W
- 1/2
2 (\partial \Omega )) and define
the following class of functions u : QT \rightarrow \BbbR 1:
P0 \equiv L2(0, T ;W
1
2 (\Omega )) \cap L\rho +2(QT ) \cap W 1
2 (0, T ; (W
1
2 (\Omega ))
\ast ) \cap \{ u : u(x, 0) = u0(x)\} .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE AND NONEXISTENCE OF SOLUTIONS OF REACTION-DIFFUSION EQUATION . . . 425
We will understand the solution of the considered problem in the following sense:
Definition 2.1. A function u \in P0 is called the generalized solution of problem (1.1) – (1.3) if it
satisfies the equality;
-
T\int
0
\int
\Omega
u
\partial v
\partial t
dxdt+
\int
\Omega
u(x, T )v(x, T )dx+
T\int
0
\int
\Omega
DuDv dx dt+
+
T\int
0
\int
\Omega
(a(x, t) | u| \rho u - b(x, t) | u| \nu u) vdxdt -
-
\int
\Omega
u(x, 0)v(x, 0)dx+
T\int
0
\int
\partial \Omega
k(x\prime , t)uvdx\prime dt =
=
T\int
0
\int
\Omega
hvdxdt+
T\int
0
\int
\partial \Omega
\varphi vdx\prime dt,
for all v \in W 1
2 (0, T ; (W
1
2 (\Omega ))
\ast ) \cap L2(0, T ;W
1
2 (\Omega )) \cap L\rho +2(QT ).
Theorem 2.1. Assume that the following conditions are satisfied with 0 < \nu \leq \rho :
(i) a and b are positive functions,
a \in L\infty (\BbbR +;L\infty (\Omega )), b \in
\left\{ L \rho +2
\rho - \nu
(\BbbR +;L \rho +2
\rho - \nu
(\Omega )), if \nu < \rho ,
L\infty (\BbbR +;L\infty (\Omega )), if \nu = \rho .
If \nu < \rho , then there exists a number a0 > 0 such that a(x, t) \geq a0 for almost every (x, t) \in
\in \Omega \times \BbbR +.
If \nu = \rho , then there exists a number b0 > 0 such that a(x, t) - b(x, t) \geq b0 for almost every
(x, t) \in \Omega \times \BbbR +.
(ii) k \in L\infty (\BbbR +;Ln - 1 (\partial \Omega )) and there exists a number k0 \geq 0 such that k(x\prime , t) \geq - k0 for
almost every (x\prime , t) \in \partial \Omega \times \BbbR +,
k0 <
\left\{
\mathrm{m}\mathrm{i}\mathrm{n} \{ a\prime , \theta 1\}
c23
, if 0 < \nu < \rho ,
\mathrm{m}\mathrm{i}\mathrm{n} \{ b\prime , \theta 1\}
c23
, if \nu = \rho .
Then problem (1.1) – (1.3) is solvable in P0 for any (h, \varphi ) \in [L2(0, T ; (W
1
2 (\Omega ))
\ast )+L\rho +2
\rho +1
(QT )]\times
\times L2(0, T ;W
- 1/2
2 (\partial \Omega )) and u0 \in W 1
2 (\Omega ) \cap L\rho +2(\Omega ) (here \theta 1, a
\prime and b\prime are positive numbers such
that a\prime < a0, b
\prime < b0, \theta 1 < 1 and c3 comes from Sobolev’s embedding inequality \| u\| L2(\partial \Omega ) \leq
\leq c3\| u\| W 1
2 (\Omega )).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
426 E. ÖZTÜRK, K. KALH
Theorem 2.2. Assume that the conditions of Theorem 2.1 are satisfied. If there exists a positive
number b1 such that b(x, t) \leq b1 < a0 for almost every (x, t) \in QT when 0 < \nu < \rho , then the
solution is unique. Moreover, if u and v are solutions of problem (1.1) – (1.3), with initial data u0
and v0, respectively, then
| u(x, t) - v(x, t)| 2L2(\Omega ) \leq \| u0 - v0\| 2L2(\Omega )e
2(b1(\rho +1)+1)t as \nu < \rho ,
| u(x, t) - v(x, t)| 2L2(\Omega ) \leq \| u0 - v0\| 2L2(\Omega )e
2t as \nu = \rho .
3. Asymptotic behavior of solutions in homogeneous case. In this section, we show that under
some conditions the solutions of problem (1.1) – (1.3) is bounded in L2(\Omega ) for all t \geq 0 in the case
of h(x, t) = 0, \varphi (x\prime , t) = 0. We also describe the asymptotic behavior of these solutions.
The main result of this section is the following theorem.
Theorem 3.1. Let condition (i) of Theorem 2.2 be fulfilled. Assume that function k belongs to
L\infty (\BbbR +;Ln - 1 (\partial \Omega )) and there exist positive numbers b1 and k0 such that b(x, t) \leq b1, k(x
\prime , t) \geq k0
hold for almost every (x, t) \in \Omega \times \BbbR + and (x\prime , t) \in \partial \Omega \times \BbbR +. Then we have the following inequalities
for the solution of problem (1.1) – (1.3) for all t \geq 0:
| u| 2L2(\Omega )(t) \leq
2\Biggl[
K2
K1
+ e -
\rho
2K1t
\Biggl( \biggl(
1
2
\int
\Omega
u20dx
\biggr) - \rho /2
- K2
K1
\Biggr) \Biggr] 2/\rho , (3.1)
where K1 = K1(\~c, c2, k0, b1, \rho , \nu ) > 0, K2 = K2(a0, c5, \rho ) > 0 as 0 < \nu < \rho and K1 =
= K1(\~c, c2, k0) < 0, K2 = K2(b0, c5, \rho ) > 0 as \nu = \rho . (Here \~c, c2, c5 come from inequalities
\| u\| L2(\Omega ) \leq c2\| u\| W 1
2 (\Omega ), \| u\| 2W 1
2 (\Omega )
\leq \~c(\| Du\| 2L2(\Omega ) + \| u\| 2L2(\partial \Omega )), \| u\| L2(\Omega ) \leq c5\| u\| L\rho +2(\Omega )).
Proof. Conditions of Theorem 2.1 provide that problem (1.1) – (1.3) has a solution in P0. Let
define the auxiliary function:
E(t) =
1
2
\int
\Omega
u2dx,
where u(x, t) is solution of problem (1.1) – (1.3) and compute E\prime (t) =
\int
\Omega
uutdx:
E\prime (t) =
\int
\Omega
u\Delta udx -
\int
\Omega
a(x, t)| u| \rho +2dx+
\int
\Omega
b(x, t)| u| \nu +2dx,
after applying the integrating by parts, and using the conditions of theorem 3.1, we get
E\prime (t) \leq - 1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} \| u\| 2L2(\Omega ) -
\int
\Omega
a(x, t)| u| \rho +2dx+
\int
\Omega
b(x, t)| u| \nu +2dx.
Here first consider the case 0 < \nu < \rho . By using the assumptions of Theorem 2.1 we have
E\prime (t) \leq - 1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} \| u\| 2L2(\Omega ) - a0| u| \rho +2
L\rho +2(\Omega ) + b1
\int
\Omega
| u| \nu +2dx.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE AND NONEXISTENCE OF SOLUTIONS OF REACTION-DIFFUSION EQUATION . . . 427
We separate the end term of the right-hand side such that the following:
E\prime (t) \leq - 1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} \| u\| 2L2(\Omega ) - a0| u| \rho +2
L\rho +2(\Omega ) + b1
\int
\Omega
| u|
2
\rho (\rho - \nu )| u|
\nu
\rho (\rho +2)
dx.
Applying Hцlder and Young inequalities for the last term, we deduce that
E\prime (t) \leq
\biggl(
- 1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} + c(\varepsilon )b
\rho /(\rho - \nu )
1
\biggr)
\| u\| 2L2(\Omega ) - (a0 - \varepsilon )
1
c\rho +2
5
| u| \rho +2
L2(\Omega ), (3.2)
where \varepsilon < \mathrm{m}\mathrm{i}\mathrm{n}
\left\{ a0,
\nu
\rho
\Biggl(
b
\rho /(\rho - \nu )
1 (\rho - \nu )\~cc22
\rho \mathrm{m}\mathrm{i}\mathrm{n}\{ 1, k0\}
\Biggr) (\rho - \nu )/\nu
\right\} , thus we obtain
E\prime (t) \leq 2
\biggl(
- 1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} + c(\varepsilon )b
\rho /(\rho - \nu )
1
\biggr)
E(t) - (a0 - \varepsilon )
\biggl(
2
c25
\biggr) (\rho +2)/2
(E(t))(\rho +2)/2,
for convenience we denote coefficients by K1, K2:
K1 = 2
\biggl(
- 1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} + c(\varepsilon )b
\rho /(\rho - \nu )
1
\biggr)
,
K2 = (a0 - \varepsilon )
\biggl(
2
c25
\biggr) (\rho +2)/2
,
then we have
E\prime (t) \leq K1E(t) - K2(E(t))(\rho +2)/2,
where K1 > 0, K2 > 0 by depending on choosing of \varepsilon . Now we solve the following inequality:
E\prime (t) \leq K1E(t) - K2(E(t))(\rho +2)/2
with
E(0) =
1
2
\int
\Omega
u20dx,
making use of the substitution v = (E(t)) - \rho /2, then we obtain
v\prime +
\rho
2
K1v \geq K2
\rho
2
,
that is
d
dt
(ve
\rho
2K1t) \geq K2e
\rho
2K1t \rho
2
,
by integrating we have
ve
\rho
2K1t - v(0) \geq K2
K1
e
\rho
2K1t - K2
K1
,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
428 E. ÖZTÜRK, K. KALH
that is
v \geq
\biggl[
K2
K1
e
\rho
2K1t - K2
K1
+ v(0)
\biggr]
e -
\rho
2K1t,
it follows that
| u| 2L2(\Omega )(t) \leq
2\Biggl[
K2
K1
+ e -
\rho
2K1t
\Biggl( \biggl(
1
2
\int
\Omega
u20dx
\biggr) - \rho /2
- K2
K1
\Biggr) \Biggr] 2/\rho
which completes the first part of the proof.
Now consider the case \nu = \rho . By making use of same arguments as in the case 0 < \nu < \rho , we
obtain a differential inequality of the following form:
E\prime (t) \leq K1E(t) - K2(E(t))(\rho +2)/2
with
K1 = - 2
1
\~cc22
\mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\} , K2 = b0
\biggl(
2
c25
\biggr) (\rho +2)/2
,
if we solve this inequality then we arrive at the desired result for the case \nu = \rho .
Theorem 3.1 is proved.
Now, we investigate the decay to zero of the solutions under some extra conditions.
Corollary 3.1. Assume that the following inequality is satisfied with 0 < \nu < \rho :
\nu
\rho
\Biggl(
b
\rho /(\rho - \nu )
1 (\rho - \nu )\~cc22
\rho \mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\}
\Biggr) (\rho - \nu )/\nu
< a0. (3.3)
Then
\| u\| L2(\Omega )(t) \leq \| u0\| L2(\Omega ) \forall t \geq 0, (3.4)
and
u(x, t) \rightarrow 0 as t \rightarrow \infty , (3.5)
under the assumptions of Theorem 3.1.
By using the assumption (3.3) we can apply the Young inequality to (3.2) with
\nu
\rho
\Biggl(
b
\rho /(\rho - \nu )
1 (\rho - \nu )\~cc22
\rho \mathrm{m}\mathrm{i}\mathrm{n} \{ 1, k0\}
\Biggr) (\rho - \nu )/\nu
< \varepsilon < a0,
then we have negative constant K1. Considering this in (3.1), we obtain (3.4), (3.5) immediately.
Corollary 3.2. If u0 = 0, then the solution is zero regardless of the sign of K1 under the
assumptions of Theorem 3.1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE AND NONEXISTENCE OF SOLUTIONS OF REACTION-DIFFUSION EQUATION . . . 429
In light of inequality (3.1) we have the following:
| u| 2L2(\Omega ) \leq
2| u0| 2L2(\Omega )\biggl[
K2
K1
| u0| \rho L2(\Omega )(1 - e -
\rho
2K1t) + 2\rho /2e -
\rho
2K1t
\biggr] 2/\rho .
Corollary 3.3. If \nu = \rho , then \| u\| L2(\Omega )(t) \leq \| u0\| L2(\Omega ) for all t \geq 0 and u(x, t) \rightarrow 0 as t \rightarrow \infty
under the assumptions of Theorem 3.1.
By using the inequality (3.1), we give the following results on the existence of invariant set and
the asymptotic behavior of the solution.
Corollary 3.4. Under the assumptions of Theorem 3.1,
\| u\| 2L2(\Omega )(t) \leq 2
\biggl(
K1
K2
\biggr) 2/\rho
when \| u0\| 2L2(\Omega ) \leq 2
\biggl(
K1
K2
\biggr) 2/\rho
for all t \geq 0 and
\| u\| 2L2(\Omega )(t) \leq 2
\biggl(
K1
K2
\biggr) 2/\rho
as t \rightarrow \infty .
4. On blow-up. In this section we will give sufficient conditions which ensure that the solution
of problem (1.1) – (1.3) blows-up at some finite time t\ast in the case of h(x, t) = 0, \varphi (x\prime , t) = 0, for
the sufficiently smooth solution;\left\{
ut - \Delta u+ a(x, t)| u| \rho u - b(x, t)| u| \nu u = 0, (x, t) \in \Omega \times (0, t\ast ),
\biggl(
\partial u
\partial \eta
+ k(x\prime , t)u
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm|
\partial \Omega
= 0, (x\prime , t) \in \partial \Omega \times (0, t\ast ),
u(x, 0) = u0(x), x \in \Omega ,
where \Omega \subset \BbbR n, n \geq 2, is a bounded domain with smooth boundary \partial \Omega .
Theorem 4.1. Assume that the following conditions are satisfied for every (x\prime , t) \in \partial \Omega \times (0, t\ast ],
(x, t) \in \Omega \times (0, t\ast ]:
(i) a(x, t) \geq 0, b(x, t) > 0, k(x\prime , t) > 0,
\partial
\partial t
a(x, t) \leq 0,
\partial
\partial t
b(x, t) \geq 0,
\partial
\partial t
k(x\prime , t) \leq 0;
(ii) \nu \geq \rho > 0 and let inequality a(x, t) < b(x, t) hold when \nu = \rho ;
(iii) u0 > 0 satisfies given inequality
2
\int
\Omega
b(x, 0)u\nu +2
0 dx - 2(\nu + 2)
\rho + 2
\int
\Omega
a(x, 0)u\rho +2
0 dx >
> (\nu + 2)
\left( \int
\Omega
(\nabla u0)
2dx+
\int
\partial \Omega
k(x\prime , 0)u20dx
\prime
\right) .
If u is a positive solution of problem (1.1) – (1.3), then u blows-up in L2(\Omega ) in finite time t\ast .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
430 E. ÖZTÜRK, K. KALH
Proof. Let define the auxiliary function G(t) =
\int
\Omega
u2dx where u(x, t) is solution of the problem
and compute G\prime (t):
G\prime (t) = - 2
\int
\Omega
(\nabla u)2dx - 2
\int
\partial \Omega
k(x\prime , t)u2dx\prime +
+2
\int
\Omega
b(x, t)u\nu +2dx - 2
\int
\Omega
a(x, t)u\rho +2dx,
G\prime (t) \geq - 2
\Bigl(
1 +
\nu
2
\Bigr) \left( \int
\Omega
(\nabla u)2dx+
\int
\partial \Omega
k(x\prime , t)u2dx\prime
\right) +
+2
\int
\Omega
b(x, t)u\nu +2dx - 2
\biggl(
\nu + 2
\rho + 2
\biggr) \int
\Omega
a(x, t)u\rho +2dx.
We denote right side of this inequality by H(t) and compute H \prime (t), then we get
H \prime (t) = 2(\nu + 2)
\int
\Omega
(ut)
2dx - 2
\Bigl(
1 +
\nu
2
\Bigr) \int
\partial \Omega
\partial k
\partial t
(x\prime , t)u2dx\prime -
- 2
\biggl(
\nu + 2
\rho + 2
\biggr) \int
\Omega
\partial a
\partial t
(x, t)u\rho +2dx+
+2
\int
\Omega
\partial b
\partial t
(x, t)u\nu +2dx.
In the last inequality, by using condition (i), we obtain
H \prime (t) \geq 2(\nu + 2)
\int
\Omega
(ut)
2dx.
Since H \prime (t) \geq 0 and H(0) > 0 (condition (iii)), it follows that H(t) > 0 for t \geq 0.
By using G(t), (3.1) and Schwarz inequality, we have
(G\prime (t))2 =
\left( 2
\int
\Omega
uutdx
\right) 2
\leq 4
\int
\Omega
u2dx
\int
\Omega
u2tdx \leq 2
\nu + 2
G(t)H \prime (t)
and since G\prime (t) \geq H(t), we get
2
\nu + 2
G(t)H \prime (t) \geq G\prime (t)H(t).
It follows that
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
EXISTENCE AND NONEXISTENCE OF SOLUTIONS OF REACTION-DIFFUSION EQUATION . . . 431
H \prime (t)
H(t)
\geq
\biggl(
2 + \nu
2
\biggr)
G\prime (t)
G(t)
.
Integrating the above inequality from 0 to t, we obtain that
H(t)
H(0)
\geq
\biggl[
G(t)
G(0)
\biggr] \Bigl( 2+\nu
2
\Bigr)
. (4.1)
By using G\prime (t) \geq H(t) in (4.1), we have,
G\prime (t)
[G(t)]
\Bigl(
2+\nu
2
\Bigr) \geq M, M =
H(0)
[G(0)]
\Bigl(
2+\nu
2
\Bigr) .
We integrate above inequality, we get
1
[G(0)]\nu /2
- 1
[G(t)]\nu /2
\geq \nu
2
Mt
or
1
[G(t)]\nu /2
\leq 1
[G(0)]\nu /2
- \nu
2
Mt. (4.2)
Since inequality (4.2) does not exist for all time t \geq 0, we say that u blows-up at some finite time t\ast
and t\ast is bounded above by
t\ast \leq 2G(0)
\nu H(0)
.
Theorem 4.1 is proved.
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Received 08.12.14
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
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| id | umjimathkievua-article-1849 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:54Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/81/49601c6dc0c1eb0d6795dbc360ab8c81.pdf |
| spelling | umjimathkievua-article-18492019-12-05T09:29:34Z Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition Існування та неiснування розв’язкiв реакцiйно-дифузiйного рiвняння з граничними умовами Робена Öztürk, E. Озтюрк, Е. We investigate the long-time behavior of the reaction-diffusion equation, which has a nonlinearity of polynomial growth of any order, with Robin boundary condition. Sufficient conditions are obtained for the solutions of the problem to be bounded or approaching infinity at a finite time. Дослiджено довготривалу поведiнку реакцiйно-дифузiйного рiвняння з граничними умовами Робена, яке мiстить нелiнiйнiсть полiномiального росту будь-якого порядку. Отримано достатнi умови для того, щоб розв’язки граничної задачi були обмеженими, або прямували до нескiнченностi на скiнченному промiжку часу. Institute of Mathematics, NAS of Ukraine 2016-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1849 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 3 (2016); 423-432 Український математичний журнал; Том 68 № 3 (2016); 423-432 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1849/831 Copyright (c) 2016 Öztürk E. |
| spellingShingle | Öztürk, E. Озтюрк, Е. Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition |
| title | Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition |
| title_alt | Існування та неiснування розв’язкiв реакцiйно-дифузiйного рiвняння з граничними умовами Робена |
| title_full | Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition |
| title_fullStr | Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition |
| title_full_unstemmed | Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition |
| title_short | Existence and nonexistence of solutions of reaction-diffusion equation with Robin boundary condition |
| title_sort | existence and nonexistence of solutions of reaction-diffusion equation with robin boundary condition |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1849 |
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