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. Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 3 of 17 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. Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 4 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 5 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 6 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 v(0, t) = 0, 0<t ≤ T v(1, t) = 0, (P3) Page 7 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 8 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 9 of 17 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. Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 10 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 11 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 12 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 13 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 14 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 15 of 17 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) Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 16 of 17 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 Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 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 References 1. Berrone, LR, Tarzia, DA, Villa, LT: Asymptotic behavior of a Non-classical heat conduction problem for a semi-infinite material. Math Methods Appl Sci. 23, 1161–1177 (2000). doi:10.1002/1099-1476(20000910)23:133.0.CO;2-Y 2. 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Addison-Wesley Publishing Company, Menlo Park, CA (1984) 14. Miller, RK: Non lineal Volterra Integral Equations. W.A. Benjamin, Inc., California (1971) 15. Friedman, A: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964) 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. Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com