A MATHEMATICAL ANALYSIS OF THERMAL EXPLOSIONS

This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory which obey the simple Arrhenius rate law and a general Newton law of heat exchange. We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. 2000 Mathematics Subject Classification. 35J65, 80A25.


Introduction and main results.
In a reacting material undergoing an exothermic reaction in which reactant consumption is neglected, heat is being produced in accordance with Arrhenius rate law and Newtonian cooling. Thermal explosions occur when the reactions produce heat too rapidly for a stable balance between heat production and heat loss to be preserved. In this paper, we are concerned with the localization of the values of a dimensionless heat evolution rate at which such critical phenomena as ignition and extinction occur. For detailed studies of thermal explosions, the reader might be referred to Aris [3,4], Bebernes-Eberly [5], Boddington-Gray-Wake [6], and Warnatz-Maas-Dibble [22].
Let D be a bounded domain of Euclidean space R N , N ≥ 2, with smooth boundary ∂D; its closure D = D ∪ ∂D is an N-dimensional, compact smooth manifold with boundary. We let be a second-order, elliptic differential operator with real coefficients such that: (1) a ij (x) ∈ C ∞ (D) with a ij (x) = a ji (x), 1 ≤ i, j ≤ N, and there exists a constant a 0 > 0 such that N i,j=1 a ij (x)ξ i ξ j ≥ a 0 |ξ| 2 , x ∈ D, ξ ∈ R N . (1.2) (2) c(x) ∈ C ∞ (D) and c(x) > 0 in D.
In this paper, we consider the following semilinear elliptic boundary value problem stimulated by a small fuel loss steady-state model in combustion theory: Here: (1) λ and ε are positive parameters.
(3) ∂/∂ν is the conormal derivative associated with the operator A a ij x n j ∂ ∂x i , (1.4) where n = (n 1 ,n 2 ,...,n N ) is the unit exterior normal to the boundary ∂D (see Figure 1.1).
The nonlinear term f (t) := exp t 1 + εt (1.5) describes the temperature dependence of reaction rate for exothermic reactions obeying the simple Arrhenius rate law in circumstances in which heat flow is purely conductive, and the parameter ε is a dimensionless inverse measure of the Arrhenius activation energy or a dimensionless ambient temperature. The equation represents heat balance with reactant consumption ignored, where the function u is a dimensionless temperature excess of a combustible material and the parameter λ, called the Frank-Kamenetskii parameter, is a dimensionless rate of heat production. On the other hand, the boundary condition represents the exchange of heat at the surface of the reactant by Newtonian cooling. Moreover the boundary condition Bu is called the isothermal condition (or Dirichlet condition) if a(x ) ≡ 0 on ∂D, and is called the adiabatic condition (or Neumann condition) if a(x ) ≡ 1 on ∂D. It should be emphasized that problem (1.3) becomes a degenerate boundary value problem from an analytical point of view. This is due to the fact that the so-called Shapiro-Lopatinskii complementary condition is violated at the points x ∈ ∂D where a(x ) = 0. In the non-degenerate case or one-dimensional case, problem (1.3) has been studied by many authors (see Brown-Ibrahim-Shivaji [7], Cohen [8], Cohen-Laetsch [9], Parter [15], Tam [21], Wiebers [23,24], and Williams-Leggett [25]).
This paper is devoted to the study of the existence of positive solutions of problem (1.3), and is an expanded and revised version of the previous paper Taira-Umezu [20]. First it follows from an application of Taira-Umezu [19, Theorem 1] that problem (1.3) has at least one positive solution u(λ) ∈ C 2 (D) for each λ > 0. Furthermore, by [18,Example 7] we know that problem (1.3) has a unique positive solution u(λ) ∈ C 2 (D) for each λ > 0 if ε ≥ 1/4. In other words, if the activation energy is so low that the parameter ε exceeds the value 1/4, then only a smooth progression of reaction rate with imposed ambient temperature can occur; such a reaction may be very rapid but it is only accelerating and lacks the discontinuous change associated with criticality and ignition. The situation may be represented schematically by The purpose of this paper is to study the case where 0 < ε < 1/4. Our main result gives sufficient conditions for problem (1.3) to have three positive solutions, which suggests that the bifurcation curve of problem (1.3) is S-shaped (see Figure 1.4).
First, to state our multiplicity theorem for problem (1.3) we introduce a function It is easy to see (see Figure 1.3) that if 0 < ε < 1/4, then the function ν(t) has a unique local maximum at t = t 1 (ε) and has a unique local minimum at t = t 2 (ε) then problem (1.3) has at least three distinct positive solutions u 1 (λ), u 2 (λ), u 3 (λ) for all λ satisfying the condition (1.14) It should be noticed that, as ε ↓ 0, the local maximum ν(t 1 (ε)) and the local minimum ν(t 2 (ε)) behave, respectively, as follows: More precisely, (1.20) By virtue of Theorems 1.1, 1.2, and 1.3, we can define two positive numbers µ I and µ E by the formulas µ I = inf µ > 0 : problem (1.3) is uniquely solvable for each λ > µ , µ E = sup µ > 0 : problem (1.3) is uniquely solvable for each 0 < λ < µ . (1.21) Then certain physical conclusions may be drawn (cf. [5,22]). If the system is in a state corresponding to a point on the lower branch and if λ is slowly increased, then the solution can be expected to change smoothly until the point µ I is reached. Rapid transition to the upper branch will then presumably occur, corresponding to ignition. A subsequent slow decrease in λ is likewise anticipated to produce a smooth decrease in burning rate until extinction occurs at the point µ E . In other words, the minimal positive solution u(λ) is continuous for λ > µ I but is not continuous at λ = µ I , while the maximal positive solution u(λ) is continuous for 0 < λ < µ E but is not continuous at λ = µ E . The situation may be represented schematically by Figures 1.5 and 1.6 (cf. Boddington-Gray-Wake [6, Figure 6]). By the maximum principle and the boundary point lemma, we can obtain from the variational formula (4.5) that the first eigenvalue λ 1 = λ 1 (a) of problem (1.16) satisfies the inequalities λ 1 (1) < λ 1 (a) < λ 1 (0). (1.22) Moreover, it follows that the unique solution φ = φ (a) of problem (1.11) satisfies the inequalities (1) in D, (1.23) so that, On the other hand, we find from formula (3.29) that the critical value β = β(a) in Theorem 1.1 satisfies the inequalities , (1.25) and further from formulas (5.54) and (5.63) that the critical value Λ = Λ(a) in Theorem 1.3 depends essentially on the first eigenvalue λ 1 = λ 1 (a). Therefore, we can conclude that the extinction phenomenon in the isothermal condition case occurs at the largest critical value µ E (0), while the extinction phenomenon in the adiabatic condition case occurs at the smallest critical value µ E (1). Similarly, we find that ignition phenomenon in the adiabatic condition case occurs at the smallest critical value µ I (1), while the ignition phenomenon in the isothermal condition case occurs at the largest critical value µ I (0). Remark 1.5. Minamoto-Yamamoto-Nakao [14] studied the case where D is the unit ball in R 3 under the Dirichlet condition a(x ) ≡ 0 on ∂D. The following numerical analysis of the critical value µ I (0) is due to them (cf. [15,24]): The rest of this paper is organized as follows. In Section 2 we collect the basic definitions and notions about the theory of positive mappings in ordered Banach spaces. This section is adapted from Amann [2]. Section 3 is devoted to the proof of Theorem 1.1. We reduce the study of problem (1.3) to the study of a nonlinear operator equation in an appropriate order Banach space just as in Taira-Umezu [20]. The methods developed here are based on a multiple positive fixed point technique formulated by Leggett-Williams [13] (see Lemma 3.2). This technique is intended to reduce the usually difficult task of establishing the existence of multiple positive solutions of problem (1.3) to the verification of a few elementary conditions on the nonlinear term f (u) and the resolvent K, just as in Wiebers [23,Theorem 4.3]. In Section 4 we make use of a variational formula (4.5) to prove Theorem 1.

Ordered Banach spaces and the fixed point index.
One of the most important tools in nonlinear functional analysis is the Leray-Schauder degree of a compact perturbation of the identity mapping of a Banach spaces into itself. In connection with nonlinear mappings in ordered Banach spaces, it is natural to consider mappings defined on open subsets of the positive cone. Since the positive cone is a retract of the Banach space, one can define a fixed point index for compact mappings on the positive cone as is shown in Amann [2, Section 11].

Ordered Banach spaces.
Let X be a nonempty set. An ordering ≤ in X is a relation in X that is reflexive, transitive and antisymmetric. A nonempty set together with an ordering is called an ordered set.
Let V be a real vector space. An ordering ≤ in V is said to be linear if the following two conditions are satisfied: If we let then it is easy to verify that the set Q has the following two conditions: (iii) If x, y ∈ Q, then αx + βy ∈ Q for all α, β ≥ 0.
(iv) If x ≠ 0, then at least one of x and −x does not belong to Q. The set Q is called the positive cone of the ordering ≤.
Let E be a Banach space with a linear ordering ≤. The Banach space E is called an ordered Banach space if the positive cone Q is closed in E. It is to be expected that the topology and the ordering of an ordered Banach space are closely related if the norm is monotone:

Retracts and retractions.
Let X be a metric space. A nonempty subset A of X is called a retract of X if there exists a continuous map r : X → A such that the restriction r | A to A is the identity map. The map r is called a retraction.
The next theorem, due to Dugundji [11,12], gives a sufficient condition in order that a subset of a Banach space be a retract.
Theorem 2.1. Every nonempty closed convex subset of a Banach space E is a retract of E.

The fixed point index.
Let E and F be Banach spaces, and let A be a nonempty subset of E. A map f : A → F is said to be compact if it is continuous and the image f (A) is relatively compact in F . Theorem 2.1 tells us that the positive cone Q is a retract of the Banach space E. Therefore one can define a fixed point index for compact mappings defined on the positive cone; more precisely, the next theorem asserts that one can define a fixed point index for compact maps on closed subsets of a retract of E.
then define an integer i(f ,U,X) satisfying the following four conditions: The integer i(f ,U,X) is called the fixed point index of f over U with respect to X. In fact, the integer i(f ,U,X) is defined by the formula    [20].
To do this, we consider the following linearized problem: for any given function g ∈ L p (D), find a function u in D such that Bu = 0 on ∂D.  as follows: for any g ∈ L p (D), the function u = Kg ∈ W 2,p (D) is the unique solution of problem (3.1). Then, by the Ascoli-Arzelà theorem we find that the operator K, considered as is compact. Indeed it follows from an application of Sobolev's imbedding theorem that Then the space C(D) is an ordered Banach space with the linear ordering , and with the positive cone For u, v ∈ C(D), the notation u v means that u−v ∈ P \{0}. Then it follows from an application of the maximum principle (cf. [16]) that the resolvent K is strictly positive, that is, Kg is positive everywhere in D if g 0 (see [18,Lemma 2.7]). Moreover it is easy to verify that a function u( Assume that there exist constants 0 < δ < τ and σ > 0 such that the set is nonempty, where • A denotes the interior of a subset A of Q, and that Then the mapping G has at least three distinct fixed points. Proof. Let i(G,U,Q) denote the fixed point index of the mapping G(·) over an open subset U with respect to the positive cone Q as is stated in Theorem 2.2.
We letG Then we have, by condition (3.8), This implies that By the continuity of η we find that the set W is open, so that the index i(G,W ,Q) is well defined. Moreover, by condition (3.9) one can choose a point w 0 ∈ W . We notice that if w ∈ ∂W , then it follows that either w = τ or η(w) = σ .
(i) First, if w = τ, we let Then we have, by condition (3.8), This implies that (3.27) Therefore, by the solution property (vi) of the index we can find three distinct fixed points u 1 ,u 2 ,u 3 of G(·) such that Let Ꮾ be the set of all subdomains Ω of D with smooth boundary such that dist(Ω,∂D) > 0, and let where χ Ω denotes the characteristic function of a set Ω. It is easy to see that the constant β is positive, since the resolvent K of problem (3.1) is strictly positive. Since lim t→∞ ν(t) = lim t→∞ t/f (t) = ∞, one can find a constant t 1 (ε) such that (see Figure 3.2) (3.30) It should be noticed that and that .  To do this, it suffices to verify that conditions of Lemma 3.2 are fulfilled for all λ satisfying condition (1.14).
Then it is easy to see that η is a continuous and concave functional of P . If u ∈ P (t 1 (ε)), then we have This verifies condition (3.11) for the functional η.

Proof of Theorem 1.2. We let
If u 1 = u 1 (λ) and u 2 = u 2 (λ) are two positive solutions of problem (1.3), then we have, by the mean value theorem, where We will prove Theorem 1.2 by using a variant of variational method. To do this, we introduce an unbounded linear operator U from the Hilbert space L 2 (D) into itself as follows: (a) The domain of definition D(U) of U is the space Then it follows from [18,Theorem 2.6] that the operator U is a positive and selfadjoint operator in L 2 (D), and has a compact resolvent. Hence we obtain that the first eigenvalue λ 1 of U is characterized by the following variational formula: Thus it follows from formulas (4.2) and (4.5) that However, it is so easy to see that Hence, combining formula (4.7) with inequality (4.6) we obtain that The proof of Theorem 1.2 is complete.  Let φ(x) = K1(x) be the unique solution of problem (1.11). Then it follows from [18, Lemma 2.7] that the function φ(x) belongs to C ∞ (D) and satisfies the conditions By using the function φ(x), we can introduce a subspace of C(D) as follows: The space C φ (D) is given a norm by the formula If we let then it is easy to verify that the space C φ (D) is an ordered Banach space having the positive cone P φ with nonempty interior. For u, v ∈ C φ (D), the notation u v means that u − v is an interior point of P φ . It follows from [18, Proposition 2.8] that K maps C φ (D) compactly into itself, and that K is strongly positive, that is, Kg 0 for all g ∈ P φ \{0}.
It is easy to see that a function u(x) is a solution of problem (1.3) if and only if it satisfies the nonlinear operator equation However we know from [18, Theorem 0] that the first eigenvalue λ 1 of U is positive and simple, with positive eigenfunction ϕ 1 (x): Without loss of generality, one may assume that We let Here we remark that t 1 (ε) → 1 as ε ↓ 0, so that the constant γ is positive. Proof. (i) Let c be a parameter satisfying 0 < c < 1. Then However, since we have (see Figure 5.1) it follows that First we obtain from formula (5.8) that Secondly we have, for all λ > λ 1 /γ, However, one can find a constant ε 0 ∈ (0, 1/4] such that, for all 0 < ε ≤ ε 0 , Hence it follows that Therefore, combining inequalities (5.12), (5.13), and (5.16) we obtain that, for all λ > λ 1 /γ and 0 < ε ≤ ε 0 , The proof of Proposition 5.1 is complete.

End of proof of Theorem 1.3. (I) First we introduce a function
The next lemma summarizes some elementary properties of the function F(t).
Lemma 5.3. Let 0 < ε < 1/4. Then the function F(t) has the properties  Proof. First, since t 2 (ε) < 2ε −2 , it follows from Lemma 5.3 that We define two functions Moreover, we define two sets Then M ⊂ L for all u 4ε −2 ϕ 1 , and so By using Friedrichs' mollifiers, we can construct a function v(x) ∈ C ∞ (D) such that v 0 and that On the other hand, by Lemma 5.3 we remark that it follows that If α is a constant greater than 4, we define a set Then we have, for all u αε −2 ϕ 1 , However we have, as ε ↓ 0, Therefore the desired inequality (5.28) follows from inequality (5.43) if we take the positive integer k so large that (5.47) However Proposition 5.4 tells us that there exists an element v ∈ • P φ such that Now let Ꮽ be an arbitrary compact subset of αε −2 ϕ 1 + P φ . Then, by combining inequalities (5.47) and (5.48) one can find a constant s 0 > 1 such that Therefore, by using inequality (5.51) m-times we obtain that Proof. Assume to the contrary that there exists a fixed point u of λK(f (·)) with u u. Then we can choose a constants > 1 such that Let φ(x) = K1(x) be the unique positive solution of the linear eigenvalue problem (1.11). Then it is easy to see that, for all λ > 0, the functions λφ(x) and λe 1/ε φ(x) are a subsolution and a supersolution of problem (1.3), respectively. Indeed, since the function is increasing for all t ≥ 0, and satisfies the condition it follows that A(λφ) = λ < λf (λφ) in D, B(λφ) = 0 on ∂D, (6.4) and that A λe 1/ε φ = λe 1/ε > λf λe 1/ε φ in D, B λe 1/ε φ = 0 on ∂D. (6.5) Hence, by applying the method of super-subsolutions (see [18,Theorem 2]) to our situation one can find a solution v(λ) ∈ C 2 (D) of problem (1.3) such that On the other hand, it follows from Theorem 3.1 that the resolvent K maps L p (D) continuously into W 2,p (D).
(II) Secondly we prove assertion (1.20). The proof is carried out in the same way as in the proof of assertion (1.19).
Let u(λ) be the unique positive solution for λ sufficiently large as in Theorem 1.3 Since we have, for all x ∈ D, φ(x) > 0, (6.13) it follows from assertion (6.7) that, for all x ∈ D, u(λ)(x) → ∞ as λ ↑ ∞. (6.14) Therefore, just as in step (I) we obtain assertion (6.3) that The proof of Theorem 1.4 is complete.