Abstract
Nonlinear problems for the onedimensional 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 nonclassical 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 nonclassical moving boundary problems for the heat equation is also given.
2000 AMS Subject Classification: 35C15, 35K55, 45D05, 80A20, 35R35.
Keywords:
Nonclassical heat equation; Nonlinear heat conduction problems; Volterra integral equations; Moving boundary problems; Uniform heat source1. Introduction
In this article, we will consider initial and boundary value problems (IBVP), for the onedimensional nonclassical heat equation motivated by some phenomena regarding the design of thermal regulation devices that provides a heater or cooler effect [16]. In Section 2, we study the following IBVP (Problem (P1)):
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.
Nonclassical 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 semiinfinite 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 nonclassical heat equation have been studied in [911], 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 nonclassical 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 heatdiffusion 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:
(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 ;
(HC3) For each bounded set 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.
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:
where θ = θ (x,t) is the known theta function defined by
and K = K(x,t) is the fundamental solution to the heat equation defined by:
Moreover the function V = V(t), defined by
as the heat flux on the face x = 0, must satisfy the following second kind Volterra integral equation
where
with and K_{1 }(x, t; ξ, τ) defined by
Taking into account that
and θ(1,t) = θ(1,t), we can obtain a new expression for given by
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
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 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:
From (2.11), hypothesis (H3) holds. From the continuity of and (HC4) we have hypothesis (H5). From (HC3), there exists such that for 0 ≤ τ ≤ t ≤ K, V_{1},V_{2 }∈ B:
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:
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
where
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:
where M_{0 }is a positive constant which verifies the inequality
and , where we consider the bounded set [0,V] × [0,T]. Now, taking into account assumptions (HA), (HB) and properties of function θ, we can write
where
and ζ represents the Riemann's Zeta function. From (2.6), (2.7) and hypothesis (HC3) and (HD), we have:
where
Finally, in view of (3.9) and inequality (2.10), we can apply the Gronwall inequality which provides:
and then, from (3.4) we obtain for 0 < t ≤ T the following estimation:
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:
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
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 ∈ (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) Let us first observe that w(x,t), defined in (4.1), is a solution to the following auxiliary problem (P2):
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 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) Let us observe that
By assumption (HF) and part (c) we have that u(x,t) > 0, ∀ x ∈ [0,1], ∀ t ≥ 0.
(e) Let us observe that u_{t } u_{xx }< 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
and the result holds.▀
Lemma 4
Under the assumptions (HD), (HE) and (HF), we have that
Proof
Let 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
Under the same assumptions of Lemma 4, we have .
Proof
Let us observe that 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
Let us consider the problem (P1) under the assumptions (HA) to (HD), then we have:
and
Proof
From (2.6) and (2.7) we can write
Now, taking into account (HA), (HB), (HC3) and properties of function θ, we get:
where 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
Let us consider the problem (P1) under the assumptions (HA) to (HD), then we obtain the following estimation:
where
Proof
From (2.6) and (2.7) we can write
Taking into account the inequality
from (5.7) and (2.10) we obtain
where is given by (HC3), with respect to F_{2}. Using a Gronwall's inequality it follows that
Besides, in view of (5.6), (5.8) and assumption (HC3), from (2.2) we get:
and the thesis holds.▀
6. Another related problem
Now, we will consider a new nonclassical initialboundary 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):
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
where V = V(t), defined by
must satisfy the following second kind Volterra integral equation
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
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:
here C_{2 }and C_{3 }are given by (3.9) and
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
Let us consider the problem (P5) under the assumptions (HA) to (HD), then we obtain the following estimations:
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
Let us consider the problem (P5) under the assumptions (HA) to (HD), then we obtain the following estimation:
Proof
It is similar to the one given for Theorem 7.▀
We consider the following assumptions:
Theorem 13
Under the hypotheses (HG) and (HE), we have that
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. Nonclassical moving boundary problems
In this Section, we will study some initial and boundary value problems for the nonclassical heat equation in the domain
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):
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 nonclassical 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 heatdiffusion equation [12].
We will use the Neumann function, which is defined by
Theorem 14
Under the assumptions (HA) to (HD) the solution u to the problem (P6) has the expression
where the function V, defined by
and the piecewise continuous functions 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
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, 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
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.,
Letting x to tend to zero in (7.7), we obtain the third equation, i.e.,
Now let us derive u with respect to x from (7.6) and we get,
When 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).
Moreover, if the continuous functions 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.
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:
Now we will prove (8.2.40) [13]. We have for i = 1,2,3:
For the first function we have,
and by using the classical inequality
we deduce that
where . For the second function, by using (HC3), we have
By using inequality (7.23), we can get
For the third function, by using (HC3), we have
and by using inequality (7.23), we get
If we define
the hypothesis (8.2.40) [13] is satisfied. Now let us prove (8.2.41) to (8.2.42) [13]. We have
where C_{4 }and C_{5 }are positive constants. Therefore we define the function α as follows:
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.▀
Now, we can consider the following problem (P7):
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 nonclassical 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
and then the unknown function 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
Under the assumptions (HA) to (HD) the solution u to the problem (P8):
is given by:
where the unknown function 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 initialboundary value problems for the onedimensional nonclassical 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 nonclassical moving boundary problems for the heat equation is also given.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
This paper was partially sponsored by the project PIP No. 0460 of CONICET  UA (Rosario, Argentina), and Grant FA95501010023. The authors would like to thank the anonymous referee for a careful review and constructive comments.
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