1. Art and Diseño

TEMA Tend. Mat. Apl. Comput., 6, No. 2 (2005), 273-284.
c Uma Publica¸c˜
ao da Sociedade Brasileira de Matem´
atica Aplicada e Computacional.
Existence and Uniqueness of Solutions of a
Nonlinear Heat Equation
M.A. RINCON1, Instituto de Matem´
atica, Universidade Federal do Rio de Janeiro,
Rio de Janeiro, RJ, Brazil
J. L´IMACO2, Instituto de Matem´
atica,
Niteroi, RJ, Brazil
Universidade Federal Fluminense,
I-SHIH LIU3, Instituto de Matem´
atica, Universidade Federal do Rio de Janeiro,
Rio de Janeiro, RJ, Brazil.
Abstract. A nonlinear partial differential equation of the following form is considered:
u′ − div a(u)∇u + b(u) |∇u|2 = 0,
which arises from the heat conduction problems with strong temperature-dependent
material parameters, such as mass density, specific heat and heat conductivity.
Existence, uniqueness and asymptotic behavior of initial boundary value problems
under appropriate assumptions on the material parameters are established.
1.
Introduction
Metallic materials present a complex behavior during heat treatment processes involving phase changes. In a certain temperature range, change of temperature
induces a phase transformation of metallic structure, which alters physical properties of the material. Indeed, measurements of specific heat and conductivity show
a strong temperature dependence during processes such as quenching of steel.
In this paper, we are interested in a nonlinear heat equation with temperaturedependent material parameters, in contrast to the usual linear heat equation with
constant coefficients.
Let θ(x, t) be the temperature field, then we can write the conservation of energy
in the following form:
ρ ε′ + div q = 0,
(1.1)
where prime denotes the time derivative.
1 [email protected]
2 [email protected]
3 [email protected]
274
Rincon, L´ımaco and Liu
The mass density ρ = ρ(θ) > 0 may depend on temperature due to possible
change of material structure, while the heat flux q is assumed to be given by the
Fourier law with temperature-dependent heat conductivity,
q = −κ∇θ,
κ = κ(θ) > 0.
(1.2)
The internal energy density ε = ε(θ) generally depends on the temperature, and
the specific heat c is assumed to be positive defined by c(θ) = ∂ε/∂θ > 0, which is
not necessarily a constant.
We can reformulate the equation (1.1) in terms of the energy ε instead of the
temperature θ. Rewriting Fourier law as
q = −κ(θ)∇θ = −α(ε)∇ε,
(1.3)
and observing that ∇ε = ∂ε/∂θ ∇θ = c(θ)∇θ, we have c(θ)α(ε) = κ(θ), and hence
α = α(ε) > 0.
Now let u be defined as u = ε(θ), then the equation (1.1) becomes
ρ(u) u′ − div(α(u)∇u) = 0.
Since ρ(u) > 0, dividing the equation by ρ, we obtain
u′ − div
Since c =
∇
α(u)∇u ρ(u)
1 + ∇
· α(u)∇u = 0.
ρ(u)
(1.4)
du
∂ε
=
,
∂θ
dθ
1 dρ(u)
1 dρ dθ
1 dρ 1
1 =−
∇u = −
∇u = −
∇u.
2
2
ρ(u)
ρ(u)
du
ρ(u) dθ du
ρ(u)2 dθ c
Substituting into equation (1.4) we obtain
u′ − div
α(u)∇u ρ(u)
−
1 dρ 1
∇u · α(u)∇u = 0,
2
ρ(u) dθ c
which is equivalent to
u′ − div a(u)∇u + b(u) |∇u|2 = 0,
(1.5)
α(u)c(u)
k(u)
α(u) dρ
α(u)
=
=
> 0 and b(u) = −
> 0.
ρ(u)
ρ(u)c(u)
ρ(u)c(u)
c ρ(u)2 dθ
The positiveness of a(u) and b(u) is the consequence of thermodynamic considerations (see [8]), and reasonable physical experiences: the specific heat c > 0, the
thermal conductivity κ > 0, the mass density ρ > 0, and the thermal expansion
dρ/dθ < 0. In this paper we shall formulate the problem based on the nonlinear
heat equation (1.5).
where a(u) =
Existence and Uniqueness of Solutions of a Heat Equation
1.1.
275
Formulation of the problem
Let Ω be a bounded open set of IR, with regular boundary. We represent by Q =
Ω × (0, T ) for T > 0, a cylindrical domain, whose lateral boundary we re´present
by Σ = Γ × (0, T ). We shall consider the following non-linear problem:

 u′ − div a(u)∇u + b(u) |∇u|2 = 0 in Q,
(1.6)

u = 0 on Σ,
u(x, 0) = u0 (x) in Ω.
Mathematical models of semi-linear and nonlinear parabolic equations under
Dirichlet or Neumann boundary conditions has been considered in several papers,
among them, let us mention ([1], [2], [3]) and ([4], [5], [9]), respectively.
Feireisl et al [6] prove that with non-negative initial data, the function a(u) ≡ 1
and g(u, ∇u) ≤ h(u)(1 + |∇u|2 ), instead of the non-linear term b(u) |∇u|2 in (1.6),
there exists an admissible solution positive in some interval [0, Tmax ) and if Tmax <
∞ then lim ku(t, .)k∞ = ∞.
t→Tmax
For Problem (1.6) we will prove global existence, uniqueness and asymptotic
behavior for the one-dimensional case.
2.
Existence and Uniqueness
Let ((·, ·)), k · k and (·, ·), | · | be respectively the scalar product and the norms in
H01 (Ω) and L2 (Ω). Thus, when we write |u| = |u(t)|, kuk = ku(t)k its means the
L2 (Ω), H01 (Ω) norm of u(x, t) respectively. To prove the existence and uniqueness
of solutions for the one-dimensional case, we need the following hypotheses:
H1: a(u) and b(u) belongs to C 1 (IR) and there are positive constants a0 , a1
such that, a0 ≤ a(u) ≤ a1 and b(u)u ≥ 0.
da db H2: There is a positive constant M > 0 such that max (s); (s) ≤ M
s∈IR
du
du
H3: u0 ∈ H01 (Ω) ∩ H 2 (Ω) such that |∆u0 | < ε, for some constant ε > 0.
Remark. From hypothesis (H1), we have that b(u)u ≥ 0, ∀u ∈ IR. Thus from
hypothesis (H2), |b(u)| ≤ M |u|.
Theorem 2.1. Under the hypotheses (H1), (H2) and (H3), there exist a positive
constant ε0 such that, if 0 < ε ≤ ε0 then the problem (1.6) admits a unique solution
u : Q → IR, satisfying the following conditions:
i. u ∈ L2 (0, T ; H01 (Ω) ∩ H 2 (Ω)),
u′ ∈ L2 (0, T ; H01 (Ω))
ii. u′ − div(a(u)∇u) + b(u) |∇u|2 = 0,
in L2 (Q),
iii. u(0) = u0 .
Remark. The positive constant ε0 will be determined later (see (2.14), (2.15),
(2.20) and (3.11)).
276
Rincon, L´ımaco and Liu
Proof. To prove the theorem, we employ Galerkin method with the Hilbertian basis
from H01 (Ω), given by the eigenvectors (wj ) of the spectral problem: ((wj , v)) =
λj (wj , v) for all v ∈ V and j = 1, 2, · · ·. We represent by Vm the subspace of
V generated by vectors {w1 , w2 , ..., wm }. We propose the following approximate
problem: Determine um ∈ Vm , so that

 (u′m , v) + a(um )∇um , ∇v + b(um ) |∇um |2 , v = 0 ∀ v ∈ Vm ,
(2.1)

um (0) = u0m → u0 in H01 (Ω) ∩ H 2 (Ω).
⊔
⊓
Existence
The system of ordinary differential equations (2.1) has a local solution in the interval
(0, Tm ). To extend the local solution to the interval (0, T ) independent of m the
following a priori estimates are needed.
Estimate I: Taking v = um (t) in the equation (2.1) and integrating over (0, T ),
we obtain
Z T
Z TZ
1
1
2
2
|um | + a0
(2.2)
kum k +
b(um )um |∇um |2 < |u0 |2 ,
2
2
0
0
Ω
where we have used hypothesis (H1). Taking a
ˆ0 = min{a0 , 12 } > 0, we obtain
|um |2 +
Z
T
kum k2 +
0
Z
0
T
Z
b(um )um |∇um |2 <
Ω
1
|u0 |2 .
2ˆ
a0
Thus, applying the Gronwall’s inequality in (2.3) yields
(um ) is bounded in L∞ 0, T ; L2 (Ω) ∩ L2 0, T ; H01 (Ω) .
(2.3)
(2.4)
Estimate II: Taking v = u′m in the equation (2.1)1 and integrating over (0, T ),
we obtain
Z
Z
Z
d
da
2|u′m |2 +
a(um )|∇um |2 =
(um )u′m |∇um |2 + 2
b(um )u′m |∇um |2 .
dt Ω
Ω du
Ω
(2.5)
On the other hand, from hypothesis (H2), we have the following inequality,
Z
1
1
da
(um )u′m |∇um |2 ≤ M C0 ku′m k kum k2 ,
(2.6)
2 Ω du
2
and since | · |L∞ (Ω) ≤ C0 k · k and |b(um )| ≤ M |um |, we obtain
Z
b(um )u′m |∇um |2 ≤ M C02 ku′m kkum k3 ,
Ω
where C0 = C0 (Ω) is a constant depending on Ω.
(2.7)
277
Existence and Uniqueness of Solutions of a Heat Equation
Substituting (2.6) and (2.7) into
Z
1 d
|u′m |2 +
a(um )|∇um |2 ≤
2 dt Ω
the left hand side of (2.5) we get
1
M C0 ku′m k kum k2 + M C02 ku′m kkum k3
2
(2.8)
1
1
≤ (M C0 )2 ku′m k2 kum k2 + kum k2 + (M C02 )2 ku′m k2 kum k4 .
4
2
Now taking the derivative of the equation (2.1)1 with respect to t and making
v = u′m , we have
Z
Z
Z
db
1 d ′ 2
da
a(um )|∇u′m |2 = −
|um | +
(um )|u′m |2 |∇um |2 −
(um ) u′m |∇um |2
2 dt
Ω du
Ω
Ω du
Z
1
−2
b(um )∇um ∇u′m u′m ≤ M C02 (3 + M )ku′m k2 kum k2 + kum k2 .
2
Ω
(2.9)
From inequalities (2.8) and (2.9), we have
Z
a0
d 1 ′ 2 1
a(um )|∇um |2 + |u′m | + ku′m k2 +
|um | +
dt 2
2 Ω
2
(2.10)
na
o
0
′ 2
2
4
2
kum k
− α0 kum k − α1 kum k ≤ kum k ,
2
where we have defined α0 = (M C02 )2 and α1 = M C0 41 M C0 + 3C0 + 1 .
Now, under the condition that the following inequality,
α0 kum k4 + α1 kum k2 <
a0
4
∀ t ≥ 0,
(2.11)
be valid, the coefficients of the term ku′m k in the relation (2.10) is positive and we
can integrate it with respect to t,
Z t
Z t
Z
′ 2
2
′ 2
ku′m k2 ≤ C.
(2.12)
|um | + a0
a(um )|∇um | +
|um | +
Ω
0
0
Therefore, applying the Gronwall’s inequality in (2.12), we obtain the following
estimate:
(u′m ) is bounded in L∞ 0, T ; L2 (Ω) ∩ L2 0, T ; H01 (Ω) .
(2.13)
Now we want to prove that the inequality (2.11) is valid if the initial data are
sufficiently small. Suppose that
2
a0
1
α1 α0
2
2
(2.14)
S0 + a1 ku0 k + |u0 |
+
S0 + a1 ku0 k2 + ku0 k2 <
2
a0
a0
a0
4
and
α0 ku0 k4 + α1 ku0 k2 <
a0
,
4
(2.15)
2
where we have denoted S0 = M (ku0 k + |∆u0 |2 + ku0 k|∆u0 |2 ) + a1 |∆u0 | .
278
Rincon, L´ımaco and Liu
We shall prove by contradiction. Suppose that (2.11) is false, then there
such that
a0
if 0 < t < t∗
α0 kum (t)k4 + α1 kum (t)k2 <
4
and
a0
α0 kum (t∗ )k4 + α1 kum (t∗ )k2 = .
4
∗
Integrating (2.10) from 0 to t , we obtain
Z
Z
1 ′ ∗ 2
a(um (0))|∇um (0)|2
a(um )|∇um (t∗ )|2 ≤
|um (t )| +
2
Ω
Ω
Z t∗
1 ′
1
+ |um (0)|2 +
kum k2 ≤ a1 ku0 k2 + |u0 |2
2
a0
0
2
+ M ( ku0 k + |∆u0 |2 + ku0 k|∆u0 |2 ) + a1 |∆u0 |2 ,
and consequently, kum (t∗ )k2 ≤
we obtain
∗
4
α0 kum (t )k
is a t∗
(2.16)
(2.17)
(2.18)
a1
1
1
S0 + ku0 k2 + 2 |u0 |2 . Using (2.14) and (2.15)
a0
a0
a0
2
α0
1
2
2
+α1 kum (t )k ≤ 2 S0 + a1 ku0 k + |u0 |
a0
a0
1
a1
1
a0
+α1
S0 + ku0 k2 + ku0 k2 < ,
a0
a0
a0
4
∗
2
(2.19)
hence, comparing with (2.17), we have a contradiction.
Estimate III: Taking v = −∆um (t) in the equation (2.1)1 , and using hypothesis
H1, we obtain
Z
d
a(um )|∆um |2 ≤ a1 kum k |∆um | + b1 |∆um |2 kum k,
kum k2 +
dt
Ω
where we have used the following inequality,
Z
Z
2
b(um )|∇um | |∆um | ≤ b1 |∇um |L∞
|∇um | |∆um | ≤ b1 |∇um |H 1 (Ω) kum k |∆um |
Ω
Ω
≤ b1 |∆um | kum k |∆um | = b1 kum k |∆um |2 .
a0 a0
,
]. Note
4α1 4α1
that, from (2.11), we have |um (t)| ≤ kum (t)k < a0 /4α1 and b1 = b1 + M .
In this expression we have denoted b1 = sup |b(s)|, for all s ∈ [−
Using hypothesis (H1), we obtain
a0
d
kum k2 + |∆um |2 ≤ a1 kum k |∆um | + b1 |∆um |2 kum k,
dt
2
a
d
0
or equivalently,
kum k2 +
− b1 kum k |∆um |2 ≤ a1 kum k |∆um |.
dt
2
279
Existence and Uniqueness of Solutions of a Heat Equation
Note that the constant C from (2.12) is given by C = S0 + a1 ku0 k2 +
Considering
1
a
1
0
(S0 + a1 ku0 k2 +
|u0 |2 )1/2 <
a0
2a0
4
a
a0
0
and from (2.12) we obtain
≤
− b1 kum k , it implies that
4
2
b1
1
|u0 |2 .
2a0
(2.20)
d
a0
a0
kum k2 + |∆um |2 ≤ a1 kum k |∆um | ≤ |∆um |2 + Ckum k2 .
(2.21)
dt
4
8
Z T
2
b
|∆um |2 ≤ C.
Now, integrating from 0 to t, we obtain the estimate: kum k +
0
Hence, we have
(um )
is bounded in L∞ 0, T ; H01 (Ω) ∩ L2 0, T ; H01 (Ω) ∩ H 2 (Ω) .
(2.22)
Limit of the approximate solutions
From the estimates (2.4), (2.13) and (2.22), we can take the limit of the nonlinear
system (2.1). In fact, there exists a subsequence of (um )m∈N , which we denote as
the original sequence, such that
u′m −→ u′ weak
in L2 0, T ; H01 (Ω) ,
(2.23)
um −→ u weak
in L2 0, T ; H01 (Ω) ∩ H 2 (Ω) .
Thus, by compact injection of H01 (Ω×(0, T )) into L2 (Ω×(0, T )) it follows by compactness arguments of Aubin-Lions [7], we can extract a subsequence of (um )m∈N ,
still represented by (um )m∈N such that
um −→ u
strong in L2 0, T ; H01 (Ω) , um −→ u a.e. in Q,
(2.24)
∇um −→ ∇u strong in L2 (Q), ∇um −→ ∇u a.e. in Q.
Let us analyze the nonlinear terms from the approximate system (2.1). From
the first term, we know that
Z
Z
2
|a(um )∇um | ≤ a1
|∇um |2 ≤ a1 C,
(2.25)
Ω
Ω
and since that um → u a.e. in Q and that a(x, .) is continuous, we get
a(um ) −→ a(u) and ∇um −→ ∇u a.e. in Q. Hence, we also have
|a(um )∇um |2 −→ |a(u)∇u|2
a.e. in Q.
(2.26)
From (2.25) and (2.26), and Lions Lemma, we obtain
a(um )∇um −→ a(u)∇u
weak in L2 (Q).
(2.27)
280
Rincon, L´ımaco and Liu
From the second term, we know that
Z
0
≤
T
Z
Ω
b21 C0
|b(um )|∇um |2 |2 ≤ b21
Z
Z
T
0
|∇um |2L∞ (Ω)
T
2
kum k
0
|∇um |2H 1
≤
b21 C0 kum k2
Z
Z
|∇um |2
Ω
(2.28)
T
2
|∆um | ≤ C,
0
where C0 has been defined in (2.7).
By the same argument that leads to (2.26), we get
|b(um )|∇um |2 |2 −→ |b(u)|∇u|2 |2
a.e. in
Q
(2.29)
weak in L2 (Q).
(2.30)
Hence, from (2.28) and (2.29), we obtain the convergence
b(um )|∇um |2 −→ b(u)|∇u|2
Taking into account (2.23), (2.27) and (2.30) into (2.1)1 , there exists a function u =
u(x, t) defined over Ω × [0, T [ with value in IR satisfying (1.6). Moreover, from the
convergence results obtained, we have that um (0) = u0m → u0 in H01 (Ω) ∩ H 2 (Ω),
and the initial condition is well defined.
Hence, we conclude that equation (1.6) holds in the sense of L2 (0, T ; L2 (Ω)).
Uniqueness
Let w(x, t) = u(x, t) − v(x, t), where u(x, t) and v(x, t) are solutions of Problem
(1.6). Then we have
 ′
w − div a(u)∇w − div(a(u) − a(v))∇v




+ b(u) |∇u|2 − |∇v|2 + (b(u) − b(v))|∇v|2 = 0 in Q,
(2.31)




w = 0 on Σ,
w(x, 0) = 0 in Ω.
Multiplying by w(t), integrating over Ω, we obtain
Z
Z
1 d
da
| (b
a(u)|∇w|2 ≤
|w|2 +
u)| |w| |∇v| |∇w|
2 dt
Ω du
Ω
Z
Z
∂b
+
|b(u)| |∇u| + |∇v| |∇w| |w| +
|∇v|2 | | (u)| |w|2
∂u
Ω
Ω
Z Z
|∇w| |w| |∇v| + C0
|∇u| + |∇v| |∇w| |w|
≤M
Ω
ZΩ
+M
|∇v|2 |w|2 ≤ M |∇v|L∞ (Ω) kwk |w| + C0 |∇u|L∞ (Ω) + |∇v|L∞ (Ω) kwk |w|
Ω
+M |∇v|2L∞ (Ω) |w|2 ≤
where we have used
a0
kwk2 + C(∆u|2 + |∆v|2 )|w|2 ,
2
(2.32)
281
Existence and Uniqueness of Solutions of a Heat Equation
(a) The generalized mean-value theorem, i.e.,
da
da
u)(u − v) ≤ | (b
u)| |w|,
u≤u
b ≤ v,
|a(u) − a(v)| = (b
du
du
(b) b(u) (|∇u|2 − |∇v|2 ) ≤ |b(u)| |∇w| |∇u| + |∇v| ,
(c) |∇v|L∞ (Ω) ≤ k∇vkH 1 (Ω) ≤ kvkH 2 (Ω) ≤ |∆v|L2 (Ω) .
The last inequality is valid only for the one-dimensional case.
Integrating (2.32) from 0 to t, we obtain
1 2 a0
|w| +
2
2
Z
t
1
kwk ≤ |w(0)|2 + C
2
2
0
Z
t
(|∆u|2 + |∆v|2 )|w|2 ,
0
where C denotes a different positive constant. Since w(0) = u0 − v0 = 0, using the
Z t
Gronwall’s inequality, we obtain |w|2 +
kwk2 = 0, which implies the uniqueness,
0
w(x, t) = u(x, t) − v(x, t) = 0 and the theorem is proved. ⊔
⊓
3.
Asymptotic Behavior
In the following we shall prove that the solution u(x, t) of Problem (1.6) decays
exponentially when time t → ∞, using the same procedure developed in Lions [7]
and Prodi [10].
Theorem 3.1. Let u(x, t) be the solution of Problem (1.6). Then there exist positive
constants sb0 and C = C{ku0 k, |∆u0 |} such that
s0 t
kuk2 + |u′ |2 ≤ C exp−b
.
(3.1)
Proof. To prove the theorem, complementary estimates are needed.
⊔
⊓
Estimate I′ : Consider the approximate system (2.1). Using the same argument
as Estimate I, i.e, taking v = um (t), we have
Z
Z
d
2
2
|um | +
a(um )|∇um | +
b(um )um |∇um |2 = 0.
(3.2)
dt
Ω
Ω
Integrating (3.3) from (0, T ), we obtain
1
|um |2 + a0
2
Z
0
T
2
kum k +
Z
0
T
Z
b(um )um |∇um |2 <
Ω
1
|u0 |2 .
2
(3.3)
From H1 hypothesis, (3.2) and (3.3) we conclude
a0 kum k2 ≤ 2|u′m | |um | ≤ 2|u′m | |u0 |.
(3.4)
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Rincon, L´ımaco and Liu
Estimate II′ : Taking derivative of the system (2.1) with respect to t and taking
v = u′m , we obtain
Z
Z
d ′ 2
′ 2
′ 2
|u | + a0 kum k ≤ C
|um | |∇um | +
|u′m ||∇um |2
dt m
Ω
Ω
(3.5)
Z
′
′
′ 2
′ 2
2
|∇um ||∇um ||um | ≤ C1 kum k kum k + kum k kum k .
+C
Ω
Hence,
a
d ′ 2 a0 ′ 2
0
|um | + kum k + ku′m k2
− C1 kum k − C1 kum k2 ≤ 0.
dt
2
2
(3.6)
Using (3.4) then we can write the inequality (3.6) in the form,
a
d ′ 2 a0 ′ 2
0
|um | + kum k +ku′m k2
−C1
dt
2
2
1/2
2|u0 | ′ |um | ≤ 0. (3.7)
a0
Let v = u′m (0) in (2.1). Then, |u′m (0)|2 ≤ Cku0 k + C2 |∆u0 | + |∆u0 )|2 |u′m (0)|
and we have
2
.
(3.8)
|u′m (0)|2 ≤ C ku0 k + |∆u0 | + |∆u0 )|2 + |∆u0 |3
2|u0 |
a0
|u′m |1/2 −C1
We define the operator
J(u0 ) =
2|u0 |
a0
1/2 C ku0 k + |∆u0 | + |∆u0 |2 + |∆u0 |3
2|u0 |
C ku0 k + |∆u0 | + |∆u0 |2 + |∆u0 |3 .
a0
1/2
Then we can show that
1/2
2|u0 | ′
2|u0 |
|u′m (0)|1/2 +
|um (0)| ≤ J(u0 ).
a0
a0
(3.9)
(3.10)
Consider C1 a positive constant and u0 small enough, that is,
C1 J(u0 ) <
a0
.
4
(3.11)
Then the following inequality holds,
C1
2|u0 |
a0
1/2
|u′m |1/2 + C1
a0
2|u0 | ′
|um | <
a0
4
∀ t ≥ 0.
(3.12)
Indeed, we can proof it by contradiction. Suppose that there is a t∗ such that
C1
2|u | 1/2
2|u0 | ′ ∗
a0
0
|u′m (t∗ )|1/2 + C1
|um (t )| = .
a0
a0
4
(3.13)
Existence and Uniqueness of Solutions of a Heat Equation
Integrating (3.7) from 0 to t∗ , we obtain
(3.12), we conclude that
C1
283
|u′ (t∗ )|2 ≤ |u′ (0)|2 . From (3.11) and
2|u | 1/2
2|u0 | ′ ∗
0
|u′m (t∗ )|1/2 + C1
|um (t )|
a0
a0
2|u | 1/2
2|u0 | ′
a0
0
≤ C1
|u′m (0)|1/2 + C1
|um (0)| ≤ C1 J(u0 ) < .
a0
a0
4
(3.14)
Therefore, we have a contradiction by (3.13).
From (3.7), (3.12) and using the Poincar´e inequality, we obtain
d ′ 2
|u | + s0 |u′m |2 ≤ 0, where s0 = (a0 c0 )/2 and c0 is a positive constant such
dt m
o
dn
that k . kH01 (Ω) ≥ c0 | · |L2 (Ω) . Consequently, we have
exps0 t |u′m |2 ≤ 0 and
dt
hence
2
|u′m |2 ≤ |u′m (0)|2 exp−s0 t ≤ C ku0 k + |∆u0 | + |∆u0 |2 exp−s0 t
(3.15)
−s0 t
e
≤ C ku0 k, |∆u0 | exp
.
We also have from (3.4) that
kum k2 ≤
2
2 |u0 | |u′m | ≤
C ku0 k, |∆u0 | |u0 | exp−s0 t/2 .
a0
a0
(3.16)
Defining sb0 = s0 /2 then the result follows from (3.15), (3.16) inequality and of the
Banach-Steinhaus theorem. ⊔
⊓
Resumo A equa¸c˜
ao diferencial parcial n˜
ao linear,
u′ − div a(u)∇u + b(u) |∇u|2 = 0,
representa o problema de condu¸c˜
ao do calor em que os parˆ
ametros, densidade de
massa, calor espec´ıfico e condutividade t´ermica s˜
ao fortemente dependentes da temperatura. Nesse trabalho, provamos a existˆencia, unicidade e o comportamento
assint´
otico para o problema de valor inicial com apropriadas hip´
oteses sobre os
parˆ
ametros do material.
References
[1] N. Alaa and M. Iguername, Weak periodic solution of some quasilinear
parabolic equations with data measures, Journal of Inequalities in Pure and
Applied Mathematics, 3 (2002), 1-14.
[2] H. Amann, Periodic solutions of semilinear parabolic equations, Nonlinear
Analysis, (1978), 1-29.
[3] L. Boccardo, F. Murat, and J.P. Puel, Existence results for some quasilinear
parabolic equations, Nonlinear Analysis, 13 (1989), 373-392.
284
Rincon, L´ımaco and Liu
[4] L. Boccardo and T. Gallouet, Nonlinear elliptic and parabolic equations involving measure data, Journal of Functional Analysis, 87 (1989), 149-168.
[5] M. Chipot and F.B. Weissler, Some blow up results for a nonlinear parabolic
equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.
[6] E. Feireisl, H. Petzeltov´a, and F. Simondon, Admissible solutions for a class of
nonlinear parabolic problems with non-negative data, Proceedings of the Royal
Society of Edinburgh, Section A - Mathematics, 131, No. 5 (2001), 857-883.
[7] J.L. Lions, “Quelques m´ethodes de r´esolution des probl`emes aux limites non
lin´eares”, First edition, Dunod-Gauthier Villars, Paris, 1969.
[8] I-S. Liu, “Continuum Mechanics”, Spring-Verlag, Berlin-Heidelberg, 2002.
[9] M. Nakao, On boundedness periodicity and almost periodicity of solutions of
some nonlinear parabolic equations, J. Differential Equations, 19 (1975), 371385.
[10] G. Prodi, Un Teorema di unicita per le equazioni di Navier-Stokes, Annali di
Mat., 48 (1959), 173-182.