# An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab

- Natalia Nieves Salva
^{1, 2}, - Domingo Alberto Tarzia
^{1, 3}Email author and - Luis Tadeo Villa
^{1, 4}

**2011**:4

https://doi.org/10.1186/1687-2770-2011-4

© Salva et al; licensee Springer. 2011

**Received: **17 September 2010

**Accepted: **29 June 2011

**Published: **29 June 2011

## 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

## 1. Introduction

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 *u*_{
x
}(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 *u*_{
x
}(0*,t*) is given on the fixed face *x* = 0.

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 *u*_{
x
}(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^{+};

*h*is a continuously differentiable function in [0,1], which verifies the following compatibility conditions:

(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
;

*B*of ℝ × ℝ

^{+}, there exists a bounded positive function

*L*

_{ 0 }=

*L*

_{ 0 }(

*t*), which is independent on

*B*, defined for

*t*> 0, such that

(HC4) The function *F* is bounded for bounded *V* for all *t* ≥ 0;

(HD) *F*(0*,t*) = 0*, t >* 0.

*u*=

*u*(

*x,t*), which satisfies the conditions (1.1) to (1.4), can be written as below:

*x*= 0, must satisfy the following second kind Volterra integral equation

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*.

### Proof

*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

*V*

_{ o }(

*t*) satisfies hypothesis (H1). From (HC1) and the continuity of we conclude that 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:

*τ*≤

*t*≤

*K*,

*V*

_{1},

*V*

_{2}∈ B:

then the hypothesis (H6) holds.

*m*=

*m*(

*t,τ*), defined in (2.11), verifies also the complementary condition:

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

denotes the solution to (1.1) to (1.4) with null heat source (i.e. *F* ≡ 0 in such model).

*V*||] × [0,

*T*]. Now, taking into account assumptions (HA), (HB) and properties of function

*θ*, we can write

and the thesis holds.▀

## 4. Qualitative analysis of problem (P1)

### Lemma 3

- (a)

*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*∈ (0,1), ∀*t*> 0; - (d)
Under the assumptions of part (b) we have that

*u*(*x*,*t*) > 0, ∀*x*∈ [0,1], ∀*t*> 0; - (e)
Under the assumptions of part (b) we have that

*u*(*x*,*t*) ≤*u*_{1}, ∀*x*∈ [0,1], ∀*t*≥ 0.

### Proof

- (a)

*w*(

*x,*0) =

*h'*(

*x*)

*>*0 we have that the minimum of

*w*(0

*,t*) cannot be at

*x*= 0. Suppose that there exists

*t*

_{ o }

*>*0 such that

*w*(0

*,t*

_{ o }) = 0. By the Maximum Principle we know that

*w*

_{ x }(0

*,t*

_{ 0 })

*>*0. Moreover, by assumption (HD), we have that

*w*

_{ x }(0

*,t*

_{ o }) =

*F*(

*w*(0

*,t*

_{ o })

*,t*

_{ 0 }) =

*F*(0

*,t*

_{ o }) = 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*t*_{1}> 0 such that*w*(1*,t*_{1}) = 0. By the maximum principle we have that*w*_{ x }(0*,t*_{1})*<*0. In other respects, we have that*w*_{ x }(1*,t*_{1}) =*F*(*w*(0*,t*_{1})*,t*_{1}) and by assumption (HE) follows that*w*(0*,t*_{1})*<*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)

*u*(

*x*,

*t*) >

*0*, ∀

*x*∈ [0,1], ∀

*t*≥ 0.

- (e)

and the result holds.▀

### Lemma 4

### Proof

*v*(

*x*,

*t*) =

*u*(

*x*,

*t*) -

*u*

_{ 0 }(

*x*,

*t*), then

*v*(

*x,t*) is a solution to the following problem (P3):

From the maximum principle it follows that *v*(*x*,*t*) ≤ 0, ∀ *x* ∈ [0,1], ∀ *t* > 0.▀

### Lemma 5

### Proof

*u*

_{ o }(

*x*,

*t*) is a solution to the following problem (P4):

Therefore, , and by Lemma 4, and (d) and (c) 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 *V*_{
i
}= *V*_{
i
}(*t*) (*i* = 1,2) the solution to (2.6) in the minimum interval [0,T] and *u*_{
i
}= *u*_{
i
}(*x,t*) given by (2.2), respectively, for the data *f*_{
i
}*, g*_{
i
}*, h*_{
i
}and *F* (*i* = 1,2) in problem (P1). Then we obtain the following results.

### Theorem 6

### Proof

*C*

_{2}and

*C*

_{3}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

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 *u*_{
i
}= *u*_{
i
}(*x,t*)*, V*_{
i
}= *V*_{
i
}(*t*) (*i* = 1,2) be the functions given by (2.2) and (2.6) for the data *f, g, h* and *F*_{
i
}(*i* = 1,2) in problem (P1). Then, we obtain the following result:

### Theorem 7

### Proof

and the thesis holds.▀

## 6. Another related problem

The proof of their corresponding results follows a similar method to the one developed in previous Sections.

### Theorem 8

### 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*.

### Proof

It is similar to the one given for Theorem 1.▀

### Theorem 10

*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:

Let us denote by *V*_{
i
}= *V*_{
i
}(*t*) (*i* = 1,2) the solution to (6.7) and *u*_{
i
}= *u*_{
i
}(*x,t*) given by (6.5), respectively, for the data *f*_{
i
}*, g*_{
i
}*, h*_{
i
}and *F* (*i* = 1,2) in problem (P5).

### Theorem 11

### Proof

It is similar to the one given for Theorem 6.▀

Now, let *u*_{
i
}= *u*_{
i
}(*x,t*)*, V*_{
i
}= *V*_{
i
}(*t*) (*i* = 1,2) be the functions given by (6.5) and (6.7) for the data *f, g, h* and *F*_{
i
}(*i* = 1,2) in problem (P5), respectively.

### Theorem 12

### Proof

It is similar to the one given for Theorem 7.▀

### Theorem 13

### Proof

Suppose that there exists *t*_{
o
}*>* 0 such that *u*(0*,t*_{
o
}) = 0. By assumption (HE) we have that *u*_{
t
}*- u*_{
xx
}≤ 0 for all 0 < *x* < 1, 0 < *t* ≤ *t*_{
o
}. By applying the maximum principle we get *u*_{
x
}(0*,t*_{
o
}) *>* 0 which is a contradiction. Then, it implies that *u*(0*,t*) *>* 0 for all *t >* 0. Therefore, by assumption (HE), we have that *u*_{
t
}*- u*_{
xx
}≤ 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* ∈ [0,1], ∀*t* ≥ 0.

## 7. Non-classical moving boundary problems

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.

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].

### Theorem 14

*ϕ*

_{1}and

*ϕ*

_{2}must satisfy the following system of three integral equations:

Conversely, if *V, ϕ*_{1} and *ϕ*_{
2
}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 *u*_{
x
}is bounded.

### Proof

*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*,

*ϕ*

_{1}and

*ϕ*

_{ 2 }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

*x*to tend to

*s*(

*t*) and using the Lemma 14.2.3 of [13, page 218], i.e.,

*x*tends to zero in (7.15), and using the jump formulae of the fundamental solution to the heat equation [15], we obtain

and the first integral equation holds. Consequently, if *u* possesses the form (6.7), then the functions *V*, *ϕ*_{1} and *ϕ*_{
2
}must satisfy the system (7.9) to (7.11).

*V*,

*ϕ*

_{1}and

*ϕ*

_{ 2 }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

Hence, from (7.8) we have *u*_{
x
}(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.

which is an increasing function and tends to zero, when *η* tends to zero. Let us note that *H*_{
i
}(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.▀

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

*u*to the problem (P7) is given by the expression

*V*, defined by (7.8), and the unknown piecewise continuous functions

*ϕ*

_{1}and

*ϕ*

_{ 2 }are the solution to the following system of three integral equations:

Conversely, if *V, ϕ*_{1} and *ϕ*_{
2
}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

*V*, defined by (7.8), and the unknown piecewise continuous functions

*ϕ*

_{1}and

*ϕ*

_{ 2 }are solutions to the following system of three integral equations:

Conversely, if *V, ϕ*_{1} and *ϕ*_{
2
}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.

## Declarations

### 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.

## Authors’ Affiliations

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