Salva et al. Boundary Value Problems 2011, 2011:4
http://www.boundaryvalueproblems.com/content/2011/1/4
RESEARCH
Open Access
An initial-boundary value problem for the
one-dimensional non-classical heat equation
in a slab
Natalia Nieves Salva1,2, Domingo Alberto Tarzia1,3* and Luis Tadeo Villa1,4
* Correspondence: DTarzia@austral.
edu.ar
1
CONICET, Rosario, Argentina
Full list of author information is
available at the end of the article
Abstract
Nonlinear problems for the one-dimensional heat equation in a bounded and
homogeneous medium with temperature data on the boundaries x = 0 and x = 1, and
a uniform spatial heat source depending on the heat flux (or the temperature) on the
boundary x = 0 are studied. Existence and uniqueness for the solution to non-classical
heat conduction problems, under suitable assumptions on the data, are obtained.
Comparisons results and asymptotic behavior for the solution for particular choices of
the heat source, initial, and boundary data are also obtained. A generalization for
non-classical moving boundary problems for the heat equation is also given.
2000 AMS Subject Classification: 35C15, 35K55, 45D05, 80A20, 35R35.
Keywords: Non-classical heat equation, Nonlinear heat conduction problems, Volterra integral equations, Moving boundary problems, Uniform heat source
1. Introduction
In this article, we will consider initial and boundary value problems (IBVP), for the
one-dimensional non-classical heat equation motivated by some phenomena regarding
the design of thermal regulation devices that provides a heater or cooler effect [1-6]. In
Section 2, we study the following IBVP (Problem (P1)):
ut − uxx = −F(ux (0, t), t), 0 < x < 1,
u(0, t) = f (t),
(P1)
t>0
u(1, t) = g(t),
u(x, 0) = h(x),
t>0
(1:1)
(1:2)
t>0
0 ≤ x ≤ 1,
(1:3)
(1:4)
where the unknown function u = u(x,t) denotes the temperature profile for an homogeneous medium occupying the spatial region 0 < x <1, the boundary data f and g are
real functions defined on ℝ+, the initial temperature h(x) is a real function defined on
[0,1], and F is a given function of two real variables, which can be related to the evolution of the heat flux ux(0,t) (or of the temperature u(0,t)) on the fixed face x = 0. In
Sections 6 and 7 the source term F is related to the evolution of the temperature u(0,t)
when a heat flux ux(0,t) is given on the fixed face x = 0.
© 2011 Salva et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
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Page 2 of 17
Non-classical problems like (1.1) to (1.4) are motivated by the modelling of a system
of temperature regulation in isotropic media and the source term in (1.1) describes a
cooling or heating effect depending on the properties of F which are related to the
evolution of the heat ux(0,t). It is called the thermostat problem.
A heat conduction problem of the type (1.1) to (1.4) for a semi-infinite material was
analyzed in [5,6], where results on existence, uniqueness and asymptotic behavior for
the solution were obtained. In other frameworks, a class of heat conduction problems
characterized by a uniform heat source given as a multivalued function from ℝ into
itself was studied in [3] with results regarding existence, uniqueness and asymptotic
behavior for the solution. Other references on the subject are [2,4,7,8]. Recently, free
boundary problems (Stefan problems) for the non-classical heat equation have been
studied in [9-11], where some explicit solutions are also given.
Section 2 is devoted to prove the existence and the uniqueness of the solution to an
equivalent Volterra integral formulation for problems (1.1) to (1.4). In Section 3, 4 and
5, boundedness, comparisons results and asymptotic behavior regarding particular initial
and boundary data are obtained. In Section 6, a similar problem to (P1) is presented: the
heat source F depends on the temperature on the fixed face x = 0 when a heat flux
boundary condition is imposed on x = 0, and we obtain the existence of a solution
through a system of three second kind Volterra integral equations. In Section 7, we
solve a more general problem for a non-classical heat equation with a moving boundary
x = s(t) on the right side which generalizes the boundary constant case and it can be useful for the study of free boundary problems for the classical heat-diffusion equation [12].
2. Existence and uniquenes of problem (P1)
For data h = h(x), g = g(t), f = f(t) and F in problems (1.1) to (1.4) we shall consider the
following assumptions:
(HA) g and f are continuously differentiable functions on ℝ+;
(HB) h is a continuously differentiable function in [0,1], which verifies the following
compatibility conditions:
h(0) = f (0),
h(1) = g(0);
(2:1)
(HC) The function F = F(V,t) verifies the following conditions:
(HC1) The function F is defined and continuous in the domain ℝ × ℝ+;
(HC2) For each M >0 and for |V| ≤ M, the function F is uniformly Hölder continuous in variable t for each compact subset of R+0;
(HC3) For each bounded set B of ℝ × ℝ +, there exists a bounded positive function
L0 = L0(t), which is independent on B, defined for t > 0, such that
F(V2 , t) − F(V1 , t) |≤ LO (t) |V2 − V1 , ∀(V2 , t), (V1 , t) ∈ B;
(HC4) The function F is bounded for bounded V for all t ≥ 0;
(HD) F(0,t) = 0, t >0.
Under these assumptions, from Th. 20.3.3 of [13] an integral representation for the
function u = u(x,t), which satisfies the conditions (1.1) to (1.4), can be written as below:
t
t
θx (x, t − τ )f (τ )dτ + 2
θx (x − 1, t − τ )g(τ )dτ
θ (x − ξ , t) − θ (x + ξ , t) h(ξ )dξ −2
0
0
0
t 1
−
θ (x − ξ , t − τ ) − θ (x + ξ , t − τ ) dξ F(V(τ ), τ )dτ
u(x, t) =
1
0
0
(2:2)
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where θ = θ (x,t) is the known theta function defined by
θ (x, t) = K(x, t) +
∞
j=1
[K(x + 2j, t) + K(x − 2j, t)]
(2:3)
and K = K(x,t) is the fundamental solution to the heat equation defined by:
x2
−
1
K(x, t) = √ e 4t ,
2 πt
(2:4)
t > 0.
Moreover the function V = V(t), defined by
V(t) = ux (0, t),
t > 0,
(2:5)
as the heat flux on the face x = 0, must satisfy the following second kind Volterra
integral equation
t
(2:6)
V(t) = V0 (t) −
K(t − τ )F(V(τ ), τ )dτ
0
where
Vo (t) =
1
(θξ (−ξ , t) − θξ (ξ , t))h(ξ )dξ − 2
0
=2
1
t
θ (0, t − τ )ḟ (τ )dτ + 2
0
θ (ξ , t)h′ (ξ )dξ − 2
0
t
t
θ (−1, t − τ )ġ(τ )dτ
0
θ (0, t − τ )ḟ (τ )dτ + 2
0
t
θ (−1, t − τ )ġ(τ )dτ ,
(2:7)
t > 0,
0
with K = K(t) and K1 (x, t; ξ, τ) defined by
1
K(t) =
K1 (0, t; ξ , 0)dξ , t > 0,
(2:8)
0
K1 (x, t; ξ , τ ) = θx (x − ξ , t − τ ) − θx (x + ξ , t − τ ), t > τ .
(2:9)
Taking into account that
1
K1 (x, t; ξ , 0) dξ =
0
=
x
1+x
1
θx (x − ξ , t) dξ −
0
1
θx (x + ξ , t) dξ
0
x−1
θx (y, t) dy −
θx (y, t) dy = 2θ (x, t) − θ (x − 1, t) − θ (x + 1, t)
x
and θ(-1,t) = θ(1,t), we can obtain a new expression for K(t) given by
K(t) = 2 θ (0, t) − θ (1, t) ,
t > 0.
(2:10)
Then, problem (2.2), (2.5) to (2.7) provides an integral formulation for the problem
(1.1) to (1.4).
Theorem 1
Under the assumptions (HA) to (HC), there exists a unique solution to the problem
(P1). Moreover, there exists a maximal time T > 0, such that the unique solution to
(1.1) to (1.4) can be extended to the interval 0 ≤ t ≤ T.
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Proof
In order to prove the existence and uniqueness of problem (P1) on the interval [0,T],
we will verify the hypotheses (H1), (H2), (H3), (H5) and (H6) of the Theorem 1.2 of
[[14], p. 91]. From (HA) and (HB) we conclude that Vo(t) satisfies hypothesis (H1).
From (HC1) and the continuity of K we conclude that K̄(t − τ )F(V(τ ), τ ) satisfies
hypothesis (H2). If B is a bounded subset of D, then by (HC4) we have |F(V(τ),τ)| <M
and, therefore, there exists m = m(t,τ) such that:
K̄(t − τ )F(V(τ ), τ ) < M K̄(t − τ ) < M
1
π (t − τ )
+ 2erf
3
√
2 t−τ
1
<M
π (t − τ )
+ 2 = m(t, τ )
(2:11)
From (2.11), hypothesis (H3) holds. From the continuity of K and (HC4) we have
hypothesis (H5). From (HC3), there exists k(t, τ ) = Lo (t)K̄(t − τ ) such that for 0 ≤ τ ≤
t ≤ K, V1,V2 Î B:
K̄(t − τ )F(V1 , τ ) − K̄(t − τ )F(V2 , τ ) = K̄(t−τ ) F(V1 , τ ) − F(V2 , τ ) ≤ k(t, τ ) |V1 − V2 |
then the hypothesis (H6) holds.
In order to extend the solution to a maximal interval we can apply the Theorem 2.3
[[14], p. 97]. Taking into account that function m = m(t,τ), defined in (2.11), verifies
also the complementary condition:
lim+
t→0
T+t
m(T + t, τ ) dτ = 0
T
(2:12)
then the required hypothesis (2.3) of [[14], p. 97] is fulfilled and the thesis holds.▀
3. Boundedness of the solution to problem (P1)
We obtain the following result.
Theorem 2
Under assumptions (HA) to (HD), the solution u to problem (P1) in [0,1] × [0,T],
given by Theorem 1, is bounded in terms of the initial and boundary data h, f and g.
Proof
The integral representation of the solution u to problem (P1) can be written as
t 1
u(x, t) = u0 (x, t) −
θ (x − ξ , t − τ ) − θ (x + ξ , t − τ ) F(V(τ ), τ )dξ dτ ,(3:1)
0
0
where
u0 (x, t) =
1
0
θ (x − ξ , t) − θ (x + ξ , t) h(ξ )dξ −2
t
0
θx (x, t − τ )f (τ )dτ +2
t
θx (x − 1, t − τ )g(τ )dτ ,
(3:2)
0
denotes the solution to (1.1) to (1.4) with null heat source (i.e. F ≡ 0 in such model).
From the continuity of function θ and hypothesis (HC3) and (HD), we have:
u(x, t) ≤ u0 (x, t) +
t
1
θ (x − ξ , t − τ ) − θ (x + ξ , t − τ ) F(V(τ ), τ ) dξ dτ
0 0
t
t
F(V(τ ), τ ) dτ ≤ u0 (x, t) + C0
V(τ ) dτ ,
≤ u0 (x, t) + M0
0
0
(3:3)
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where M0 is a positive constant which verifies the inequality
1
0
θ (x − ξ , t − τ ) − θ (x + ξ , t − τ ) dξ ≤ M0 , 0 < τ < t ≤ T,
C0 = M0 L0
and L0
T
(3:4)
0 ≤ x ≤ 1,
T,
(3:5)
= max L0 (t), where we consider the bounded set [0,||V||] × [0,T]. Now,
0≤t≤T
taking into account assumptions (HA), (HB) and properties of function θ, we can write
|u0 (x, t) |≤ M0 h ∞ + C1 f T + gT , 0 < τ < t ≤ T,
(3:6)
where
C1 = 1 +
16 ζ (3) 3/2
√ T
3 π
(3:7)
and ζ represents the Riemann’s Zeta function. From (2.6), (2.7) and hypothesis (HC3)
and (HD), we have:
t
K(t − τ ) V(τ ) dτ
K(t − τ ) F(V (τ ) , τ ) dτ ≤ V0 (t) + Co
M0 0
0
C t
′
o
≤ C2 h ∞ + C3 ḟ + ġ T +
K(t − τ ) V(τ ) dτ
T
Mo 0
V(t) ≤ Vo(t) +
where
t
(3:8)
1
T
C2 = √ + 1, C3 = 2
+ T.
π
πt
(3:9)
Finally, in view of (3.9) and inequality (2.10), we can apply the Gronwall inequality
which provides:
V(t) ≤ C2 h′ + C3
ḟ + ġT exp
∞
T
C0
M0
t
0
K(t − τ ) dτ ,
0 < t ≤ T,
(3:10)
and then, from (3.4) we obtain for 0 <t ≤ T the following estimation:
⎫
2C0 √ ⎪
√ τ⎬
′
C2 h ∞ + C3 ḟ + ġT e M0 π
dτ
⎪
T
0 ⎪
⎩
⎭
√
2C0 T
√
≤ M0 h ∞ + C1 f T + gT + C0 C3 e M0 π h′ ∞ + T ḟ + ġT
u(x, t) ≤ M0 h
f + g + C0
∞ + C1
T
T
t
⎧
⎪
⎨
(3:11)
T
and the thesis holds.▀
4. Qualitative analysis of problem (P1)
In this section, we shall consider problem (1.1) to (1.4) with the following assumptions:
(HE)
V F(V, t) > 0,
(HF)
f (t) ≡ 0
∀ V = 0,
∀t > 0,
∀ t > 0;
g(t) ≡ u10 > 0
∀t > 0,
h′ (x) > 0
∀ x ∈ [0, 1],
h(1) ≤ u10 .
Lemma 3
(a) Under the hypothesis (HD) and (HF), we have that w(0,t) > 0, ∀ t > 0, where w(x,t)
is defined by
w(x, t) = ux (x, t)
(4:1)
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and u(x,t) is the solution to problem (P1);
(b) Under the assumptions (HD), (HE) and (HF) we have that w(1,t) > 0, ∀ t > 0;
(c) Under the assumptions of part (b) we have that w(x,t) > 0, ∀ x Î (0,1), ∀ t > 0;
(d) Under the assumptions of part (b) we have that u(x,t) > 0, ∀ x Î [0,1], ∀ t > 0;
(e) Under the assumptions of part (b) we have that u(x,t) ≤ u1, ∀ x Î [0,1], ∀ t ≥ 0.
Proof
(a) Let us first observe that w(x,t), defined in (4.1), is a solution to the following auxiliary problem (P2):
wt − wxx = 0,
(x, t) ∈ ≡ {(x, t) : 0 < x < 1, 0 < t ≤ T}
wx (0, t) = F(w(0, t), t),
(P2)
0<t≤T
wx (1, t) = F(w(0, t), t),
w(x, 0) = h′ (x),
0<t ≤ T
0≤x≤1
(4:2)
(4:3)
(4:4)
(4:5)
As w(x,0) = h’(x) >0 we have that the minimum of w(0,t) cannot be at x = 0. Suppose that there exists to >0 such that w(0,to) = 0. By the Maximum Principle we know
that wx (0,t0) >0. Moreover, by assumption (HD), we have that wx (0,to) = F(w(0,to),t0)
= F(0,to) = 0, which is a contradiction. Therefore we have w(0,t) > 0, ∀ t > 0.
(b) As w(1,0) >0, we have that the minimum of w(1,t) cannot be at x = 0. Suppose
that there exists t1 > 0 such that w(1,t1) = 0. By the maximum principle we have that
wx (0,t1) <0. In other respects, we have that wx (1,t1) = F(w(0,t1),t1) and by assumption
(HE) follows that w(0,t1) <0, which is a contradiction. Therefore, we have w(1,t) > 0, ∀
t > 0.
(c) It is sufficient to use part (a), (b), h’(x) >0 and the maximum principle.
(d) Let us observe that
u(x, t) = u(0, t) +
x
w(ξ , t)dξ .
(4:6)
0
By assumption (HF) and part (c) we have that u(x,t) >0, ∀ x Î [0,1], ∀ t ≥ 0.
(e) Let us observe that ut - uxx <0, which follows from (HE) and part (c). According
to the Maximum Principle, the maximum of u(x,t) must be on the parabolic boundary,
from which we obtain that
u(x, t) ≤ Max h(1) , u1o = u1o ,
(4:7)
and the result holds.▀
Lemma 4
Under the assumptions (HD), (HE) and (HF), we have that
0 ≤ u(x, t) ≤ uo (x, t),
∀ x ∈ [0, 1], ∀t > 0.
(4:8)
Proof
Let v(x,t) = u(x,t) - u0(x,t), then v(x,t) is a solution to the following problem (P3):
vt − vxx < 0,
(x, t) ∈ ≡ {(x, t) : 0 < x < 1, 0 < t ≤ T}
(4:9)
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v(0, t) = 0,
0<t ≤ T
v(1, t) = 0,
(P3)
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v(x, 0) = 0,
(4:10)
0<t ≤ T
(4:11)
0 ≤ x ≤ 1
(4:12)
From the maximum principle it follows that v(x,t) ≤ 0, ∀ x Î [0,1], ∀ t > 0.▀
Lemma 5
Under
the
same
assumptions
lim u(x, t) ≤ u10 x ≤ u10 , ∀x ∈ [0, 1].
of
Lemma
4,
we
have
t→+∞
Proof
Let us observe that uo (x,t) is a solution to the following problem (P4):
uo t − uo xx = 0,
uo (0, t) = 0,
(P4)
(x, t) ∈ ≡ {(x, t) : 0 < x < 1, 0 < t ≤ T}
(4:13)
0<t ≤ T
uo (1, t) = u10 ,
uo (x, 0) = h(x),
(4:14)
0<t≤T
(4:15)
0 ≤ x ≤ 1.
(4:16)
lim u0 (x, t) = u10 x ≤ u10 , ∀x ∈ (0, 1), and by Lemma 4, and (d) and (c)
Therefore, t→+∞
of Lemma 3, the thesis holds.▀
5. Local comparison results
Now we will consider the continuous dependence of the functions V = V(t) and u = u
(x,t) given by (2.2) and (2.6), respectively, upon the data f, g, h and F. Let us denote by
Vi = Vi(t) (i = 1,2) the solution to (2.6) in the minimum interval [0,T] and ui = ui(x,t)
given by (2.2), respectively, for the data fi, gi, hi and F (i = 1,2) in problem (P1). Then
we obtain the following results.
Theorem 6
Let us consider the problem (P1) under the assumptions (HA) to (HD), then we have:
⎛
V2 (t) − V1 (t) ≤ C2 h2 ′ − h1 ′ + C3
⎝
ḟ2 − f 1 + ġ2 − ġ1 t exp L0
∞
t
t
t
0
⎞
K̄(t − τ )dτ ⎠ (5:1)
and
Proof
u2 (x, t) − u1 (x, t) ≤ M0 h2 − h1 ∞ + C1 f2 − f1 + g2 − g1 +
t
t
√
2C0 t
h2 ′ − h1 ′ + t
.
√
+C0 C3 exp
ḟ
−
ḟ
+
−
ġ
ġ
2
1
2
1 t
∞
t
M0 π
(5:2)
From (2.6) and (2.7) we can write
V2 (t) − V1 (t) = 2
1
θ (ξ , t) h2 ′ (ξ ) − h1 ′ (ξ ) dξ − 2
0
+2
t
0
θ (−1, t − τ ) ġ2 (τ ) − ġ1 (τ ) dτ +
t
0
θ (0, t − τ ) ḟ2 (τ ) − ḟ1 (τ ) dτ +
t
0
K(t − τ ) F(V1 (τ ), τ ) − F(V2 (τ ), τ ) dτ .
(5:3)
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Now, taking into account (HA), (HB), (HC3) and properties of function θ, we get:
t
V2 (t) − V1 (t) ≤ C2 h2 ′ − h1 ′ +C3
ḟ2 − ḟ1 + ġ2 − ġ1 t + L0 t K̄(t − τ ) |V2 − V1 | dτ , 0 < τ < t ≤ T,
∞
(5:4)
t
0
where C 2 and C3 are given by (3.10). Then, (5.1) follows from (5.4) by using the
Gronwall’s inequality. To obtain (5.2) we note that from (2.2) we can write
u2 (x, t) − u1 (x, t) =
1
θ (x − ξ , t) − θ (x + ξ , t) h2 (ξ ) − h1 (ξ ) dξ − 2
0
+2
t
t
θx (x, t − τ ) f2 (τ ) − f1 (τ ) dτ
0
θx (x − 1, t − τ ) g2 (τ ) − g1 (τ ) dτ +
0
t 1
0
θ (x − ξ , t − τ ) − θ (x + ξ , t − τ ) F(V1 (τ ), τ ) − F(V2 (τ ), τ ) dξ dτ .
0
Now, taking into account assumptions (HA), (HB) and (HC), and using the same
constants as in (3.5) and (3.7) it follows (5.2).▀
Now, let ui = ui(x,t), Vi = Vi(t) (i = 1,2) be the functions given by (2.2) and (2.6) for
the data f, g, h and Fi (i = 1,2) in problem (P1). Then, we obtain the following result:
Theorem 7
Let us consider the problem (P1) under the assumptions (HA) to (HD), then we obtain
the following estimation:
⎡
√ ⎤
2 t
L02 ∞ √ ⎥
⎢ 2L √
π⎥
u2 (x, t) − u1 (x, t) ≤ M0 F2 − F1 t,M ⎢t + √02 ∞ t e
(5:5)
⎣
⎦
π
where
F1 − F2
t,M
= sup F1 (z(τ ), τ ) − F2 (z(τ ), τ ) .
(5:6)
z t ≤M
0<τ ≤t
Proof
From (2.6) and (2.7) we can write
t
V2 (t) − V1 (t) =
K(t − τ ) F1 (V1 (τ ), τ ) − F2 (V2 (τ ), τ ) dτ .
(5:7)
0
Taking into account the inequality
F2 (V2 (τ ), τ ) − F1 (V1 (τ ), τ ) ≤ F2 (V2 (τ ), τ ) − F2 (V1 (τ ), τ )+F2 (V1 (τ ), τ ) − F1 (V1 (τ ), τ )
(5:8)
from (5.7) and (2.10) we obtain
V2 (t) − V1 (t) ≤ √2 F2 − F1
π
t,M
√ t
t + K(t − τ )L02 (τ ) V2 (τ ) − V1 (τ ) dτ .
(5:9)
0
where L02 (t) is given by (HC3), with respect to F2. Using a Gronwall’s inequality it
follows that
V2 (t) − V1 (t) ≤ √2 F2 − F1
π
t,M
√
⎛
t exp ⎝
t
0
⎞
K(t − τ )L02 (τ )dτ ⎠ , 0 < t ≤ T.
(5:10)
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Besides, in view of (5.6), (5.8) and assumption (HC3), from (2.2) we get:
u2 (x, t) − u1 (x, t) ≤ Mo F2 − F1
t,M t
+ Mo
and the thesis holds.▀
t
0
L02 (τ ) V2 (t) − V1 (t) dτ ,
(5:11)
6. Another related problem
Now, we will consider a new non-classical initial-boundary value problem (P5) for the
heat equation in the slab [0,1], which is related to the previous problem (P1), i.e. (6.1)
to (6.4):
(x, t) ∈ ≡ {(x, t) : 0 < x < 1, t > 0}
ut − uxx = −F(u(0, t), t),
ux (0, t) = f (t),
t>0
ux (1, t) = g(t),
(P5)
u(x, 0) = h(x),
(6:1)
(6:2)
t>0
(6:3)
0 ≤ x ≤ 1.
(6:4)
The proof of their corresponding results follows a similar method to the one developed in previous Sections.
Theorem 8
Under the assumptions (HA) to (HD), the solution u to the problem (P5) has the
expression
t
t
θ (x − 1, t − τ )g(τ )dτ
θ (x, t − τ )f (τ )dτ + 2
θ (x − ξ , t) + θ (x + ξ , t) h(ξ )dξ −2
0
0
0
t 1
θ (x − ξ , t − τ ) + θ (x + ξ , t − τ ) dξ F(V(τ ), τ )dτ
−
u(x, t) =
1
(6:5)
0
0
where V = V(t), defined by
V(t) = u(0, t), t > 0
(6:6)
must satisfy the following second kind Volterra integral equation
V(t) = 2
1
θ (ξ , t)h(ξ )dξ − 2
θ (0, t − τ )f (τ )dτ + 2
0
0
−2
t
t 1
0
t
0
θ (−1, t − τ )g(τ )dτ
(6:7)
θ (ξ , t − τ )dξ F(V(τ ), τ )dτ .
0
Proof
We follow the Theorem 1.▀
Theorem 9
Under the assumptions (HA) to (HD), there exists a unique solution to the problem
(P5). Moreover, there exists a maximal time T > 0, such that the unique solution to
(1.1) to (1.4) can be extended to the interval 0 ≤ t ≤ T.
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Proof
It is similar to the one given for Theorem 1.▀
Theorem 10
Under the assumptions (HA) to (HD), the solution u to problem (P5) in [0,1]×[0,T]
given by Theorem 9, it is bounded in terms of the initial and boundary data h, f and g,
in the following way:
u(x, t) ≤ M1 h
∞ +C3
f + g + M1 L0
T
T
TT
C2 h
∞
+ C3 f T + gT exp(C3 L0
T)
here C2 and C3 are given by (3.9) and
1
θ (x − ξ , t − τ ) + θ (x + ξ , t − τ ) dξ ≤ M1 , 0 < τ < t ≤ T, 0 ≤ x ≤ 1.
(6:8)
(6:9)
0
Let us denote by Vi = Vi(t) (i = 1,2) the solution to (6.7) and ui = ui(x,t) given by
(6.5), respectively, for the data fi, gi, hi and F (i = 1,2) in problem (P5).
Theorem 11
Let us consider the problem (P5) under the assumptions (HA) to (HD), then we obtain
the following estimations:
V2 (t) − V1 (t) ≤ C2 h2 − h1
∞
+ C3 f2 − f1 t + g2 − g1 t exp(C3 L0 t ),
u2 (x, t) − u1 (x, t) ≤ M1 h2 − h1 ∞ + C3 f2 − f1 + g2 − g1 +
t
t
+ M1 L0 t C3 h2 − h1 ∞ + t f2 − f1 t + g2 − g1 t exp(C3 L0 t ).
(6:10)
(6:11)
Proof
It is similar to the one given for Theorem 6.▀
Now, let ui = ui(x,t), Vi = Vi(t) (i = 1,2) be the functions given by (6.5) and (6.7) for
the data f, g, h and Fi (i = 1,2) in problem (P5), respectively.
Theorem 12
Let us consider the problem (P5) under the assumptions (HA) to (HD), then we obtain
the following estimation:
u2 (x, t) − u1 (x, t) ≤ M1 F2 − F1 t,M t 1 + L0 C3 exp(C3 L0 ) .
(6:12)
2 t
2 t
Proof
It is similar to the one given for Theorem 7.▀
We consider the following assumptions:
(HG)
f (t) ≡ 0
∀t > 0,
g(t) ≡ 0
∀t > 0,
h(x) > 0 ∀ x ∈ [0, 1]
(6:13)
Theorem 13
Under the hypotheses (HG) and (HE), we have that
0 < u(x, t) < h
∞,
∀x ∈ [0, 1], ∀ t ≥ 0.
(6:14)
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Proof
Suppose that there exists to >0 such that u(0,to) = 0. By assumption (HE) we have that
ut - uxx ≤ 0 for all 0 <x < 1, 0 <t ≤ to. By applying the maximum principle we get ux(0,
to) >0 which is a contradiction. Then, it implies that u(0,t) >0 for all t >0. Therefore,
by assumption (HE), we have that ut - uxx ≤ 0 for all (x,t) in Ω, and by the Maximum
Principle, the minimum of u must be at t = 0, which implies, by assumption (HG),
that u(x,t) > 0, ∀x Î [0,1], ∀t ≥ 0.
7. Non-classical moving boundary problems
In this Section, we will study some initial and boundary value problems for the nonclassical heat equation in the domain
s ≡ (x, t) : 0 < x < s(t),
t>0
(7:1)
ut − uxx = −F(u(0, t), t),
0 < x < s(t), t > 0
(7:2)
where s = s(t) is a continuous function of t over the interval t > 0 and s(0) = 1. The
IBVP are reduced to equivalent systems of integral equations in order to get the existence of a solution.
We consider the following problem (P6):
ux (0, t) = f (t),
(P6)
t>0
u(s(t), t) = g(t),
u(x, 0) = h(x),
(7:3)
t>0
(7:4)
0 ≤ x ≤ 1.
(7:5)
The function F is now related to the evolution of the temperature instead of the heat
flux at x = 0. The problem (P6) can be considered a non-classical moving boundary
problem for the heat equation as a generalization of the moving boundary problem for
the classical heat equation [13] which can be useful in the study of free boundary problems for the heat-diffusion equation [12].
We will use the Neumann function, which is defined by
N(x, t; ξ , τ ) = K(x − ξ , t − τ ) + K(x + ξ , t − τ ).
(7:6)
Theorem 14
Under the assumptions (HA) to (HD) the solution u to the problem (P6) has the
expression
u(x, t) =
+∞
−∞
−
K(x − ξ , t)h(ξ )dξ − 2
t
K(x, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
0
t
0
Kx (x − s(τ ), t − τ ) ϕ2 (τ )dτ
(7:7)
N(x, ξ , t, τ )F(V(τ ), τ )dξ dτ
0
where the function V, defined by
V(t) = u(0, t),
(7:8)
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and the piecewise continuous functions j1 and j2 must satisfy the following system
of three integral equations:
ϕ1 (t) = f (t) −
+∞
Kx (−ξ , t)h(ξ )dξ − 2
−∞
ϕ2 (t) = g(t) −
+∞
K(s(t) − ξ , t)h(ξ )dξ + 2
t 1
0
K(−ξ , t)h(ξ )dξ − 2
K(s(t), t − τ ) ϕ1 (τ )dτ − 2
0
t
Kx (s(t) − s(τ ), t − τ ) ϕ2 (τ )dτ +
0
(7:10)
N(s(t), ξ , t, τ )F(V(τ ), τ )dξ dτ ,
t
K(0, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
(7:9)
0
−∞
−2
t
0
+
+∞
Kxx (−s(τ ), t − τ ) ϕ2 (τ )dτ ,
0
−∞
V(t) =
t
t
Kx (−s(τ ), t − τ ) ϕ2 (τ )dτ +
0
(7:11)
K(ξ , t − τ )dξ F(V(τ ), τ )dτ .
0
Conversely, if V, j1 and j2 are solutions to the integral system (7.9)-(7.11), and u has
the expression (7.7), then u is a solution to the problem (P6). Moreover, V(t) = u(0,t)
and the solution u is unique among the class of solutions for which ux is bounded.
Proof
We first make a smooth extension of h outside of 0≤x≤1, so that the extended h is
bounded and has compact support. The solution u is now assumed to have the form (7.7),
where V, j1 and j2 are unknown continuous functions that they are to be determined.
Note that the initial condition (7.5) is satisfied. From the differential equation we obtain
ut (x, t) − uxx (x, t) = −F(V(t), t)
(7:12)
and therefore by (7.8) the differential equation is satisfied. The system of integral
equations is derived from the boundary conditions. The second equation is obtained
allowing x to tend to s(t) and using the Lemma 14.2.3 of [13, page 218], i.e.,
g(t) =
+∞
t
K(s(t) − ξ , t)h(ξ )dξ − 2
−∞
K(s(t), t − τ ) ϕ1 (τ )dτ + ϕ2 (t) + 2
0
−
t 1
0
t
Kx (s(t) − s(τ ), t − τ ) ϕ2 (τ )dτ
0
(7:13)
N(s(t), ξ , t, τ )F(V(τ ), τ )dξ dτ .
0
Letting x to tend to zero in (7.7), we obtain the third equation, i.e.,
V(t) =
+∞
K(−ξ , t)h(ξ )dξ − 2
−∞
−
K(0, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
0
t
t
Kx (−s(τ ), t − τ ) ϕ2 (τ )dτ
0
(7:14)
2K(ξ , t − τ )dξ F(V(τ ), τ )dτ .
0
Now let us derive u with respect to x from (7.6) and we get,
ux (x, t) =
+∞
Kx (x − ξ , t)h(ξ )dξ − 2
−∞
−
Kx (x, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
0
t
0
Nx (x, ξ , t, τ )F(V(τ ), τ )dξ dτ .
t
0
Kxx (x − s(τ ), t − τ ) ϕ2 (τ )dτ
(7:15)
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When x tends to zero in (7.15), and using the jump formulae of the fundamental
solution to the heat equation [15], we obtain
f (t) =
+∞
Kx (−ξ , t)h(ξ )dξ + ϕ1 (t) + 2
−∞
t
Kxx (−s(τ ), t − τ ) ϕ2 (τ )dτ
(7:16)
0
and the first integral equation holds. Consequently, if u possesses the form (6.7),
then the functions V, j1 and j2 must satisfy the system (7.9) to (7.11).
Moreover, if the continuous functions V, j1 and j2 verify the system (7.9) to (7.11)
for all 0 ≤ t ≤ T, then we can consider the expression (7.7) for u, which satisfies the
initial condition (7.5). Allowing x to tend to zero in (7.15), and using (7.10) we obtain
(7.8), and therefore the differential equation is satisfied. From Lemma 4.2.3 of [13,
page 50] we see that
lim ux (x, t) =
x↓0
+∞
−∞
t
Kx (−ξ , t)h(ξ )dξ + ϕ1 (t) + 2 Kxx (−s(τ ), t − τ )ϕ2 (τ )dτ .
(7:17)
0
Hence, from (7.8) we have ux (0,t) = f(t). Likewise, u assumes the value g as x tends
to s(t), and therefore the equivalence between (7.3) to (7.6) and (7.9) to (7.11) holds.
Finally, in order to prove the uniqueness and existence of solution to the system of
integral equations (7.9) to (7.11), we will verify hypothesis (8.2.40) to (8.2.44) of the
Corollary 8.2.1 of [13, p. 91]. First we define the following functions:
H1 (t, τ , V(τ ), ϕ1 (τ ), ϕ2 (τ )) = −2Kxx (−s(τ ), t − τ ) ϕ2 (τ ),
1
H2 (t, τ , V(τ ), ϕ1 (τ ), ϕ2 (τ )) = 2K(s(t), t−τ ) ϕ1 (τ )−2Kx (s(t)−s(τ ), t−τ ) ϕ2 (τ )+ N(s(t), ξ , t, τ )dξ F(V(τ ), τ ),
(7:18)
(7:19)
0
H3 (t, τ , V(τ ), ϕ1 (τ ), ϕ2 (τ )) = −2K(0, t−τ ) ϕ1 (τ )+2Kx (−s(τ ), t−τ ) ϕ2 (τ )−2
1
K(ξ , t − τ )dξ F(V(τ ), τ ).
(7:20)
0
Now we will prove (8.2.40) [13]. We have for i = 1,2,3:
Hi (t, τ , V, ϕ1 , ϕ2 ) − Hi (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) ≤ L(t, τ ) V − Ṽ + |ϕ1 − ϕ̃1 | + |ϕ2 − ϕ̃2 | . (7:21)
For the first function we have,
H1 (t, τ , V, ϕ1 , ϕ2 ) − H1 (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) ≤ 2 Kxx (−s(τ ), t − τ ) |ϕ2 − ϕ̃2 |
(7:22)
and by using the classical inequality
exp(
x2
)
α(t − τ )
(t − τ )n/2
≤
n α n/2
,
2ex2
α, x > 0, t > τ , n ∈ N,
(7:23)
we deduce that
1
6 3/2
10 5/2
|ϕ2 − ϕ̃2 |
2
+
H1 (t, τ , V, ϕ1 , ϕ2 ) − H1 (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) ≤ √
3
e
e
4 π (s(τ ))
1
6 3/2
10 5/2
+
≤ √ 3 2
V − Ṽ + |ϕ1 − ϕ̃1 | + |ϕ2 − ϕ̃2 | ,
e
e
4 πD
(7:24)
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min s(τ ). For the second function, by using (HC3), we have
where D = 0<τ
<T
H2 (t, τ , V, ϕ1 , ϕ2 ) − H2 (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) ≤ 2K(s(t), t − τ ) |ϕ1 − ϕ̃1 | + 2 Kx (s(t) − s(τ ), t − τ ) |ϕ2 − ϕ̃2 |
+
1
0
N(s(t), ξ , t, τ )dξ L0 (τ ) V − Ṽ .
(7:25)
By using inequality (7.23), we can get
′
s
L0 (τ )
2
T
|ϕ2 − ϕ̃2 | +
|ϕ1 − ϕ̃1 | +
H2 (t, τ , V, ϕ1 , ϕ2 ) − H2 (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) ≤ √
V − Ṽ
2π eD
2 π (t − τ )
π (t − τ )
′
s + 2 L0 T
2
T
≤ √
+
+ V − Ṽ + |ϕ1 − ϕ̃1 | + |ϕ2 − ϕ̃2 | .
2π eD
2 π (t − τ )
(7:26)
For the third function, by using (HC3), we have
H3 (t, τ , V, ϕ1 , ϕ2 ) − H3 (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) = 2K(0, t − τ ) |ϕ1 − ϕ̃1 | + 2 Kx (−s(τ ), t − τ ) |ϕ2 − ϕ̃2 |
+2
1
0
K(ξ , t − τ )dξ L0 (τ ) V − Ṽ
(7:27)
and by using inequality (7.23), we get
|ϕ1 − ϕ̃1 |
1
L0 (τ )
6 3/2
|ϕ2 − ϕ̃2 | +
+ √
H3 (t, τ , V, ϕ1 , ϕ2 ) − H3 (t, τ , Ṽ, ϕ̃1 , ϕ̃2 ) ≤
V − Ṽ
2 e
π (t − τ )
π (t − τ )
2 π s(τ )
1
6 3/2
1 + L0 (τ )
≤
+ √
V − Ṽ + |ϕ1 − ϕ̃1 | + |ϕ2 − ϕ̃2 | .
π (t − τ ) 2 π D2 e
(7:28)
If we define
1
L(t, τ ) = √ 3
4 πD
2(1 + D)
6
e
3/2
+
10
e
5/2
+
8D
e
1/2
1
+ s′ T + 1 + L0
2
T
1
π (t − τ )
(7:29)
the hypothesis (8.2.40) [13] is satisfied. Now let us prove (8.2.41) to (8.2.42) [13]. We
have
t2
t1
√
L(t2 , τ )dτ ≤ C4 (t2 − t1 ) + C5 t2 − t1
(7:30)
where C4 and C5 are positive constants. Therefore we define the function a as follows:
√
α(η) = C4 η + C5 η
(7:31)
which is an increasing function and tends to zero, when h tends to zero. Let us note
that Hi (t,τ,0,0,0) = 0 for all i = 1, 2,3, and therefore hypothesis (8.2.43) and (8.2.44)
[13] are satisfied.▀
Now, we can consider the following problem (P7):
ut − uxx = −F(u(0, t), t),
(P7)
ux (0, t) = f (t),
u(1, t) = g(t),
u(x, 0) = h(x),
0 < x < 1, t > 0
t>0
t>0
0 ≤ x ≤ 1.
(7:32)
(7:33)
(7:34)
(7:35)
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In this case, the function F depends on the evolution of the temperature of the temperature u(0,t) on the fixed face x = 0 while a heat flux condition is given by (7.33).
This non-classical problem (P7) can be consider as a complementary problem to the
previous problem (P1) given by (1.1) to (1.4) in which the source term F depends on
the heat flux on the fixed face x = 0 while a temperature boundary condition (1.2) is
given on the face x = 0.
Corollary 15
Under the same assumptions of Theorem 9, the solution u to the problem (P7) is given
by the expression
+∞
u(x, t) =
−∞
−
K(x − ξ , t)h(ξ )dξ − 2
t
K(x, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
0
t
Kx (x − 1, t − τ ) ϕ2 (τ )dτ
0
(7:36)
N(x, ξ , t, τ )F(V(τ ), τ )dξ dτ ,
0
and then the unknown function V, defined by (7.8), and the unknown piecewise continuous functions j1 and j2 are the solution to the following system of three integral
equations:
ϕ1 (t) = f (t) −
+∞
Kx (−ξ , t)h(ξ )dξ − 2
−∞
ϕ2 (t) = g(t)−
+∞
V(t) =
K(1 − ξ , t)h(ξ )dξ +2
+2
0
t
K(1, t − τ ) ϕ1 (τ )dτ +
0
K(ξ , t)h(ξ )dξ − 2
−∞
t
K(0, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
Kxx (−1, t − τ ) ϕ2 (τ )dτ ,
(7:37)
0
−∞
+∞
t
t 1
0
t
N(1, ξ , t, τ )F(V(τ ), τ )dξ dτ ,
(7:38)
0
Kx (−1, t − τ ) ϕ2 (τ )dτ +
0
(7:39)
K(ξ , t − τ )dξ F(V(τ ), τ )dτ .
0
Conversely, if V, j1 and j2 are solutions to the integral system (7.37) to (7.39), and
we define u by the expression (7.36), then u is a solution to the problem (P7). Moreover, we have V(t) = u(0,t).
Theorem 16
Under the assumptions (HA) to (HD) the solution u to the problem (P8):
ut − uxx = −F(u(0, t), t)(x, t) ∈ s
(P8)
ux (0, t) = f (t),
ux (s(t), t) = g(t),
u(x, 0) = h(x),
0<t ≤ T
0<t ≤ T
0 ≤ x ≤ 1
(7:40)
(7:41)
(7:42)
(7:43)
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is given by:
u(x, t) =
+∞
−∞
−
K(x − ξ , t)h(ξ )dξ − 2
t
K(x, t − τ ) ϕ1 (τ )dτ + 2
0
t 1
0
t
K(x − s(τ ), t − τ ) ϕ2 (τ )dτ
0
(7:44)
N(x, ξ , t, τ )F(V(τ ), τ )dξ dτ
0
where the unknown function V, defined by (7.8), and the unknown piecewise continuous functions j1 and j2 are solutions to the following system of three integral equations:
ϕ1 (t) = f (t) −
+∞
Kx (ξ , t)h(ξ )dξ − 2
−∞
ϕ2 (t) = g(t) −
−2
t
+∞
Kx (s(t) − ξ , t)h(ξ )dξ + 2
−∞
K(ξ , t)h(ξ )dξ − 2
t
(7:46)
Nx (s(t), ξ , t, τ )F(V(τ ), τ )dξ dτ ,
0
K(0, t − τ ) ϕ1 (τ )dτ + 2
0
−2
t 1
0
(7:45)
K x (s(t), t − τ ) ϕ1 (τ )dτ +
t 1
0
−∞
t
0
Kx (s(t) − s(τ ), t − τ ) ϕ2 (τ )dτ +
+∞
Kx (−s(τ ), t − τ ) ϕ2 (τ )dτ ,
0
0
V(t) =
t
t
0
K(−s(τ ), t − τ ) ϕ2 (τ )dτ +
(7:47)
K(ξ , t − τ )dξ F(V(τ ), τ )dτ .
0
Conversely, if V, j1 and j2 are solutions to the integral system (7.45) to (7.47), and u
has the form (7.44), then u is a solution to the problem (P8). Moreover, we have V(t) =
u(0,t).
Proof
It is similar to the one given for Theorem 14.▀
Conclusions
In this article, we have proposed and obtained the existence and uniqueness of several
initial-boundary value problems for the one-dimensional non-classical heat equation in
the slab [0,1] with a heat source depending on the heat flux (or the temperature) on
the boundary x = 0. Moreover, a generalization for non-classical moving boundary problems for the heat equation is also given.
Acknowledgements
This paper was partially sponsored by the project PIP No. 0460 of CONICET - UA (Rosario, Argentina), and Grant
FA9550-10-1-0023. The authors would like to thank the anonymous referee for a careful review and constructive
comments.
Author details
CONICET, Rosario, Argentina 2TEMADI, Centro Atómico Bariloche, Av. Bustillo 9500, 8400 Bariloche, Argentina 3Depto.
de Matemática, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina 4Facultad de Ingeniería, Universidad
Nacional de Salta, Buenos Aires 144, 4400 Salta, Argentina
1
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Page 17 of 17
Authors’ contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and
approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 17 September 2010 Accepted: 29 June 2011 Published: 29 June 2011
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doi:10.1186/1687-2770-2011-4
Cite this article as: Salva et al.: An initial-boundary value problem for the one-dimensional non-classical heat
equation in a slab. Boundary Value Problems 2011 2011:4.
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