A COMBUSTION MODEL WITH UNBOUNDED THERMAL CONDUCTIVITY AND REACTANT DIFFUSIVITY IN NON-SMOOTH DOMAINS

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1 Electronic Journal of Differential Equations, Vol. 2929, No. 6, pp ISSN: URL: or ftp eje.math.txstate.eu A COMBUSTION MODEL WITH UNBOUNDED THERMAL CONDUCTIVITY AND REACTANT DIFFUSIVITY IN NON-SMOOTH DOMAINS SIKIRU ADIGUN SANNI Abstract. In this article, we present a strongly couple quasilinear parabolic combustion moel with unboune thermal conuctivity an reactant iffusivity in arbitrary non-smooth omains. A priori estimates are obtaine, an the existence of a unique global strong solution is prove using a Banach fixe point theorem. 1. Introuction We consier the combustion moel u ivφ u = Qwfu, t in [,, 1.1 w ivψ w = wfu, t in [,, 1.2 u n = w = n on [,, 1.3 ux, = u x, wx, = w x, 1.4 where φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w. 1.5 Here Qwfu an wfu are the reaction kinetics etermine by a positive, uniformly boune, ifferentiable an Lipschitz continuous function fu. Further, f u is also assume to be uniformly boune an Lipschitz continuous. For our analysis, we shall take fu B, fu fũ L u ũ, 1.6 f u B, f u f ũ L u ũ. 1.7 The system represents a one step irreversible reaction, reactant prouct. wx, t is assume to be the mass fraction of the reactant, 1 wx, t, the mass fraction of the prouct, ux, t, the temperature in the reaction vessel, φx, t, u, w, the thermal conuctivity, an ψx, t, u, w, the reactant iffusivity. We assume that is an open an boune arbitrary non-smooth omain in R 3. In 2 Mathematics Subject Classification. 35B4, 35K57, 8A25. Key wors an phrases. Combustion moel; reactant iffusivity; thermal conuctivity; Banach fixe point theorem. c 29 Texas State University - San Marcos. Submitte June 25, 28. Publishe May 4, 29. 1

2 2 S. A. SANNI EJDE-29/6 theory, the reactant ecomposes at a rate which is proportional to fuwx, t, where fu is the approximate number of molecules that have sufficient energy for the reaction to begin. We remark that 1.1 an 1.2 represent an approximate set of conservation equations for a single-step irreversible reaction. For the complete set of conservation equations, we refer to Fitzgibbon an Martin [1]. We further refer the reaer to Frank-Kamenetskii for more information on chemical kinetics an combustion [2]. The literature on combustion moels is quite rich. We mention here the work of a few authors. Marion [3], Terman [4] an Wagner [5] consiere the existence of travelling wave solutions, taking = R; while their stability versus instability were analyse by Clavin [6] an Sivashinky [7]. Avrin [8, 9] investigate nontravelling wave solutions to the semilinear, constant coefficient case, obtaining existence results on both = R an = R n. With a boune C omain in R n, Fitzgibbon an Martin [1] prove global existence to the system , with boune thermal conuctivity φx, t, u an boune reactant iffusivity ψx, t, w; both assume to be C 2 [, R; R functions. Further, the reaction kinetics Qwfu an wfu were etermine by a moifie Arrhenius rate function fu. In this paper, we shall prove the existence of a unique, global strong solution to the strongly couple system , with strictly positive, unboune Lipschitz continuous with respect to u an w φ an ψ. We remark that a similar situation arises in [12], where in analysing Reynols Average Navier-Stokes Problem moel, the ey viscosities in the flui equation are assume to be nonecreasing, smooth an unboune functions of the turbulent kinetic energy. For our analysis, we assume that, for all x, t, u.w, x, t, ũ, w [, T R R, φ = h 1 x, t, u, w σ 1 >, 1.8 ψ = h 2 x, t, u, w σ 2 >, 1.9 φx, t, u, w φx, t, ũ, w A 1 u ũ w w, 1.1 ψx, t, u, w ψx, t, ũ, w A 2 u ũ w w, 1.11 φ t L 2 [,T ;L ] M, ψ t L 2 [,T ;L ] M, 1.12 φ L [,T ;H 2 ] N, ψ L [,T ;H 2 ] N, 1.13 D 3 φ L 2 [,T ;L 2 ] N 1, D 3 ψ L 2 [,T ;L 2 ] N 1, 1.14 φ t L [,T ;H 1 ] M 1, ψ t L [,T ;H 1 ] M 1, 1.15 D 2 t φ L 2 [,T ;L 2 ] M 2, D 2 t ψ L 2 [,T ;L 2 ] M All first partial erivatives of h l x, t, u, w l = 1, 2 with respect to its arguments are assume boune by λ 1, an Lipschitz continuous with respect to u an w with Lipschitz constant L 1 ; while all secon partial erivatives, except 2 h l t, are 2 assume to be boune by λ 2, Lipschitz continuous with respect to u an w with Lipschitz constant L 2. For example, hl t λ 1, 2 h l tu λ 2, 1.17 hl t h l t L 1 u ũ w w, 1.18

3 EJDE-29/6 A COMBUSTION MODEL 3 where 2 h l u 2 2 h l u 2 L 2 u ũ w w, 1.19 h = h l x, t, ũ, w. 1.2 The letter C written, with or without one or more arguments, shall represent constants which might iffer from one step to the other. We write the system in the equivalent form where u t ivφ u = ivf u g u, in [,, 1.21 w t ivψ w = ivf w g w, in [,, 1.22 u n = w =, n on [,, 1.23 ux, = u x, wx, = w x, 1.24 φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w, 1.25 φ = h 1 x,, u, w, ψ = h 2 x,, u, w, 1.26 f u = f w = t t s φs u, g u = Qwfu, 1.27 s ψs w, g w = wfu A priori estimates First, we state an prove three useful Lemmas. Lemma 2.1. Let the conitions hol. Then for θ = φ, ψ, θ θ 2 Cλ 1, L 1 u ũ 2 w w 2 u ũ 2 w w 2, 2.1 t θ θ 2 Cλ 1, L 1 u ũ 2 w w 2 t u ũ 2 t w w 2, 2.2 D 2 θ θ 2 Cλ 1, λ 2, L 1, L 2 [ 1 ũ 2 ũ 4 ũ 2 w 2 D 2 ũ 2 w 2 w 4 D 2 w 2 u ũ 2 w w 2 1 u 2 ũ 2 w 2 w 2 u ũ 2 w w 2 D 2 u ũ 2 D 2 w w 2], t θ θ 2 C λ 1, λ 2, L 1, L 2 [ 1 t ũ 2 t w 2 ũ 2 w 2 t ũ 2 ũ 2 t w 2 ũ 2 t ũ 2 w 2 t w 2 w 2 t ũ 2 t w 2 u ũ 2 w w 2 1 t u 2 t w 2 u ũ 2 w w 2 1 ũ 2 w 2 t u ũ 2 t w w 2 t u ũ 2 t w w 2]

4 4 S. A. SANNI EJDE-29/6 Proof. The estimates are prove using the chain rule to obtain appropriate ifferential coefficients of θ θ, rearranging an using the conitions For brevity, we illustrate with the proof of 2.1. θ θ 2 = hl h l hl u h l x i x i u x i ũ = hl h l hl x i x i u hl w ũ hl w x i w h l w 2 x i w x i u ũ ũ h l x i x i x i u h l ũ 2 w w w h l x i x i x i w h l w Cλ 1, L 1 u ũ 2 w w 2 u ũ 2 w w Lemma 2.2. Let R 3, be open an boune set with a C 1 bounary; an u, ũ, v, ṽ, w H 1. Then there exists ɛ >, C = C such that 1 u 2 v 2 x C u 2 H 1 ɛ v 2 L 2 ɛ v 2 H 1, uv ũṽ 2 x C v 2 H 1 ɛ u ũ 2 L 2 ɛ u ũ 2 H C ũ 2 H 1 ɛ v ṽ 2 L 2 ɛ v ṽ 2 H 1, u 2 v 2 w 2 x C u 2 H 1 v 2 H 1 w 2 H Proof. Estimates 2.6 an 2.7 are prove in [1]. We therefore rener only the proof of 2.8. By Höler an Sobolev s inequalities, we calculate 1/3 1/3 1/3 u 2 v 2 w 2 x u 6 x v 6 x w 6 x 2.9 C u 2 H 1 v 2 H 1 w 2 H 1. Lemma 2.3. Let u H 3 ; an 1.1, 1.2, 1.4 an 1.5 be satisfie. Then we have the estimate t ux, 2 H 1 twx, 2 H 1 C u 2 H 3 w 2 H 3 u 2 H 2 w H 1 where C = CQ,, B, B, N, N 1. Proof. 1. Taking 1.1, 1.2, an 1.5 at t = yiel u t ivφ u = Qw fu, in, 2.11 w ivψ w = w fu, t in, 2.12 ux, = u x, wx, = w x, 2.13 where φ = h 1 x,, u, w, ψ = h 2 x,, u, w.

5 EJDE-29/6 A COMBUSTION MODEL 5 2. Using 2.11, we estimate t u 2 x = iv φ u Qw fu 2 x = φ. u φ u Qw fu 2 x C φ 2 H 2 u 2 H 2 φ 2 u 2 C, 1 2 H 2 Q 2 fu 2 w 2 L 2 CQ,, B, N u 2 H 2 w 2 L 2, 2.14 where we have use 2.8 of Lemma 2.2 an the Sobolev s embeing φ C, 1 2 C φ H Taking the graient of 2.11, we obtain the estimate t u 2 x = ivφ u Qw fu 2 x = 2 φ. u φ. 2 u φ u φ u Q w fu w u f u 2 x C φ 2 H 3 u 2 H 2 φ 2 H 2 u 2 H 3 φ 2 u 2 C, 1 2 H 3 Q2 fu 2 w 2 H 2 Q 2 f u 2 u 2 H 2 w 2 H 1 using 2.8 C u 2 H 3 w 2 H 2 u 2 H 2 w 2 H where C = CQ,, B, B, N, N 1, an we have use the Sobolev s embeing φ C, 1 2 C φ H2. Combining 2.14 an 2.15 yiels 2.1, t u 2 H 1 C u 2 H 3 w 2 H 2 u 2 H 2 w 2 H where C = Q,, B, B, N, N 1 4. Using 2.12, we have an analogous estimate to 2.16, namely t w 2 H 1 C w 2 H 3 w 2 H 2 u 2 H 2 w 2 H where C =, B, B, N, N 1. Hence, a combination of 2.16 an 2.17 gives 2.1. Theorem 2.4. Let u o, w o H 3. Suppose there exists u, w which satisfies the system Then u, w X X, where X = L [, T ; H 2 ] L 2 [, T ; H 3 ] W 1, [, T ; H 1 ] H 1 [, T ; H 2 ] 2.18

6 6 S. A. SANNI EJDE-29/6 an we have the estimate sup [,T ] u 2 H 2 w 2 H 2 tu 2 H 1 tw 2 H 1 T T w 2 H 3 T T D 3 u 2 x t D 3 w 2 x t D 2 t u 2 x t D 2 t w 2 x t CQ,, B, B, N u 2 H 2 w 2 H 1 u 2 H 3 e CT C11 =: Λ where C = C, Q, L, L, L 1, L 2, λ 1, λ 2, M, N, N 1, M 1, M 2 an C 1 is the boun in Proof. 1. We multiply 1.21 by u an integrate by parts over ; an use 1.8 an 1.23 to obtain 1 u 2 x σ 1 u 2 x 2 t uf u x ug u x σ 1 u 2 x 1 f 2 2σ ux 2 1 u 2 x 1 g ux 2 or u 2 x u 2 x Cσ 1 u 2 x f t ux 2 gux Similarly, we may use 1.9, 1.22, 1.23 to obtain w 2 x w 2 x Cσ 2 w 2 x f t wx 2 gwx We represent the ith ifferential quotient of size h as Di h ζ = ζx he i, t ζx, t, i = 1,..., n h for h R, < h ; an efine D h ζ = D1 h ζ,..., Dnζ. h Hence, we apply D h on 1.21 to get t Dh u iv [ φ h D h u ] = iv D h f u D h φ u D h g u 2.22 Multiplying this equality by D h u, integrating over an applying Young s inequality, we obtain t D h u 2 x D h u 2 x Cσ 1 ɛ D h D h u 2 x ɛ D h u 2 x 1 D h f u D h φ u 2 x 1 g ɛ ɛ ux 2 D h u. D h uns D h u.nqwfus 2.23

7 EJDE-29/6 A COMBUSTION MODEL 7 Similarly, we have t D h w 2 x D h w 2 x Cσ 2 ɛ D h D h w 2 x ɛ D h w 2 x 1 D h f w D h φ w 2 x 1 g ɛ ɛ wx 2 D h w. D h wns D h w.nwfus Using 2.2, 2.21, 2.23 an 2.24, we euce, after taken the limit as h, chosen ɛ > sufficiently small an simplifying, that t u 2 x w 2 x D l u 2 x D l w 2 x D l u 2 x D l w 2 x C u 2 x w 2 x f u 2 x g w 2 x D l f u φ u 2 x g u 2 x f w 2 x D l f w φ w 2 x, 2.25 where D l q = lim h D h q an C = Cσ 1, σ Using 1.12, the efinitions 1.27 an 1.28, we use Höler inequality an 2.6 of Lemma 2.2 to evaluate some terms of 2.25, namely f u 2 x g u 2 x g w 2 x [ C, M, N, B, Q D l w 2 x T D l f u φ u 2 x f w 2 x D l f w φ w 2 x w 2 x 1 T T D l u 2 x D l u 2 x T ] D l w 2 x Using 2.26 in 2.25, choosing T sufficiently small an simplifying we obtain u 2 x w 2 x D l u 2 x D l w 2 x t D l u 2 x D l w 2 x C u 2 x w 2 x D l u 2 x D l w 2 x, 2.27

8 8 S. A. SANNI EJDE-29/6 where C = Cσ 1, σ 2,, M, N, B, Q. By applying the Gronwall s inequality an carrying out some simple calculations, we euce from 2.27 the estimate sup u 2 x sup w 2 x sup D l u 2 x sup D l w 2 x [,T ] [,T ] [,T ] [,T ] T T D l u 2 x t D l w 2 x t 2.28 C u 2 H 1 w 2 H 1 =: C 1 <, where C = Cσ 1, σ 2,, M, N, B, Q, T. 5. Let Y = L [, T ; L 2 ] L 2 [, T ; H 1 ]. For some h >, we infer from 2.28 that D h u, w 2 Y Y := sup D h u 2 x sup D h w 2 x [,T ] [,T ] T T D h u 2 x t D h w x t C 1 Estimate 2.29 implies sup D h u, w Y Y <. 2.3 h The space Y Y is reflexive. Therefore, there exists by the weak compactness theorem [11], a subsequence h k, an a function U, W Y Y such that D h k u, w U, W Thus, given a smooth function ζ Cc [, T, we calculate T u, w ζ x t = lim T h k T = T u, wd h k ζ x t lim D h k u, wζ x t h k U, W ζ x t Thus U, W = u, w in the weak sense, an so u, w Y Y. Hence, by the above euction, 2.28 implies sup [,T ] C T u 2 H 1 sup w 2 H 1 [,T ] u 2 H 1 w 2 H 1 =: C 1 < T D 2 u 2 x t D 2 w 2 x t where C = Cσ 1, σ 2,, M, N, B, Q, T 6. We now set out to obtain higher a priori estimates neee to complete part of the proof of Theorem 2.4. In view of estimates 2.32, we may take the limit

9 EJDE-29/6 A COMBUSTION MODEL 9 as h in 2.22 an apply D h on the ensuing equation to obtain D h u iv φ h t D 2 D h u = iv [ D h f u D h φ u φ D h u ] D h g We take the ot prouct of 2.33 with D h u to estimate D h u 2 x D 2 D h u 2 x t Cσ 1 ɛ D 2 D h u 2 1 D h f u D h φ u ɛ φ D h u 2 x 1 D h g u 2 x ɛ Similarly, we have t Cσ 1 ɛ D h w 2 x D 2 D h w 2 1 ɛ ψ D h w 2 x 1 ɛ D 2 D h w 2 x D h f w D h ψ u D h g w 2 x Combining the above inequality with 2.34, taking the limit as h, choosing ɛ > sufficiently small, an simplifying, we euce D l u 2 x D l w 2 x D 2 D l u 2 x D 2 D l w 2 x t [ Cσ 1, σ 2 D l f u D 2 φ u φ D l u 2 x D l g u 2 x ] D l f w D 2 ψ w ψ D l w 2 x D l g w 2 x 2.36 Note that this inequality is analogous to Hence, by steps similar to steps 4 an 5, we obtain sup [,T ] C T D 2 u 2 L 2 sup D 2 w 2 L 2 [,T ] D 2 u 2 L 2 D2 w 2 L 2 C 1 T D 3 u 2 x t D 3 w 2 x t. =: C 2 <, 2.37 where C = Cσ 1, σ 2,, M, N, N 1, B,, B, Q, T an C 1 is the boun in The last higher a priori estimates are obtaine in this step. Define r θx, t θx, t r :=, l θ := lim r 2.38 r r for all r > such that t r [, T. Applying l on the system yiels t lu iv φ l u = iv l f u l g u, in [,, 2.39 t lw iv ψ l w = iv l f w l g w, in [,, 2.4

10 1 S. A. SANNI EJDE-29/6 where n lu = n lw = on [,, 2.41 l ux, = l u x, l wx, = l w x, 2.42 φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w, φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w, f u = f w = t t s φs u, s ψs w, g u = Qwfu, g w = wfu. Note that the system is analogous to We therefore have an analogue to 2.25, namely l u 2 H t 1 lw 2 H 1 D 2 l u 2 x D 2 l w 2 x Cσ 1, σ 2 l u 2 H 1 lw 2 H 1 l f u 2 x l g u 2 x 2.43 l f w 2 x l g w 2 x l f u φ l u 2 x l f w ψ l w 2 x Hence, by calculations an euctions similar to that of steps 4-5, we obtain the estimates sup t u 2 H 1 sup t w 2 H 1 [,T ] T [,T ] D 2 t u 2 x t T D 2 t u 2 x t C u 2 H 3 w 2 H 3 u 2 H 2 w 2 H 1 C 1 C 2 =: C 3 <, 2.44 where C = Cσ 1, σ 2, M, N, M 1, M 2, B, B, Q,, T, C 1, an C 2 are the bouns in 2.28 an 2.37 respectively. Here we use the estimate 2.1 in Lemma Combining estimates 2.32, 2.37 an 2.44 we complete the proof of Theorem Existence of Solutions In this section, we shall prove the existence for a global unique strong solution to in a subset of the space X X, which is equippe with the norm [ u, w X X = sup u 2 H 2 w 2 H 2 tu 2 H 1 tw 2 H 1 [,T ] T T T D 3 u 2 x t D 2 t u 2 x t T D 3 w 2 x t 3.1 ] 1/2 D 2 t w 2 x t

11 EJDE-29/6 A COMBUSTION MODEL 11 Theorem 3.1. Let u, w H 3. Then, there exists a global unique strong solution to Proof. 1. The corresponing fixe point argument system to is where χ t ivφ χ = ivf u g u, in [,, 3.2 τ t ivψ τ = ivf w g w, in [,, 3.3 χ n = χ =, n on [,, 3.4 χx, = u x, τx, = w x, 3.5 φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w, 3.6 φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w. 3.7 f u = f w = t t s φs u, g u = Qwfu, 3.8 s ψs w, g w = wfu Define A : X X X X by setting A[u, w] = χ, τ. We shall prove that if T > is small enough, then A is a contraction mapping. We choose u, w, ũ, w X X, an efine A[u, w] = χ, τ, A[ũ, w] = χ, τ. For two solutions χ, τ an χ, τ of , we have t χ χ iv φ χ χ = iv f u fũ gu gũ, in [, 3.1 t τ τ iv φ τ τ = iv f w f w gw g w, in [, 3.11 χ χ = τ τ =, n n on [,, 3.12 χ χ x, =, τ τ x, =, 3.13 where φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w, 3.14 φ = h 1 x, t, ũ, w, ψ = h 2 x, t, ũ, w, 3.15 φ = h 1 x, t, u, w, ψ = h 2 x, t, u, w, 3.16 f u = fũ = f w = f w = t t t t s φs u, g u = Qwfu, 3.17 s φs ũ, gũ = Q wfũ, 3.18 s ψs w, g w = wfu, 3.19 s ψs w, g w = wfũ. 3.2

12 12 S. A. SANNI EJDE-29/6 3. Now, the system is analogous to the system Consequently, we have analogous estimates to a combination of estimates 2.25, 2.36 an From the ensuing analogues, choosing ɛ >, simplifying an integrating with respect to t over [, T ] we obtain χ χ 2 H 2 τ τ 2 H 2 t χ χ 2 H 1 t τ τ 2 H 1 D3 χ χ 2 L 2 [,T ;L 2 ] D 3 τ τ 2 L 2 [,T ;L 2 ] D2 t χ χ 2 L 2 [,T ;L 2 ] D 2 t τ τ 2 L 2 [,T ;L 2 ] T [ χ χ 2 H 2 τ τ 2 H 2 t χ χ 2 H 1 t τ τ 2 H 1 f u fũ 2 x g u gũ 2 x f w f w 2 x g w g w 2 x g u gũ 2 x g w g w 2 x f u fũ φ u ũ 2 x f w f w ψ w w 2 x t f u fũ 2 x D 2 f u fũ D 2 φ u ũ φ D 2 u ũ 2 x D 2 f w f w D 2 ψ w w ψ D 2 w w 2 x t f u fũ φ t u ũ 2 x t f w f w 2 x t f w f w ψ t w w 2 x t g u gũ 2 x ] t g w g w 2 x t 3.21 Calculations of the estimates of the necessary terms on the right sie of 3.21 are rather lengthy; an not renere here for brevity. The estimates can be obtaine by using the conitions , Sobolev s embeing an Lemmas , after some suitable rearrangement. If the estimates are substitute into 3.21, we euce, after an application of the integral form of the Gronwall s inequality, the estimates A[u, w] A[ũ, w] X X 1/2 C T T T T 1 u 2 L [,T ;H 2 ] ũ 2 L [,T ;H 2 ] 1/2 w 2 L [,T ;H 2 ] w 2 L [,T ;H 2 ] 1 ũ, w 2 X X t u 2 L [,T ;H 1 ] tw 2 L [,T ;H 1 ] u, w ũ, w X X, 3.22 where C epens on, Q an all the bouns in

13 EJDE-29/6 A COMBUSTION MODEL We efine a convex set K = { u, w u, w u, w X X, u, w X X 2 Λ }, 3.23 where X X is the set where the initial an bounary values are zero; an Λ = constant is the boun in If T > is sufficiently small, we shall show that A[K] K, A[u, w] A[ũ, w] X X 1 2 u, w ũ, w X X 3.24 for all u, w, ũ, w K. Using 2.19, we have A[u, w ] X X = χx,, τx, X X = u, w X X Λ Thus for u, w K, A[u, w] X X A[u, w ] X X A[u, w] A[u, w ] X X 1/2 Λ C T T T T 1 u 2 L [,T ;H 2 ] w 2 L [,T ;H 2 ] u 2 H 2 w 2 H 2 1 u, w 2 X X t u 2 L [,T ;H 1 ] tw 2 L [,T ;H 1 ] u, w u, w X X using 3.22, /2 Λ C T T T T 1 4Λ u 2 H 2 w 2 H 2 1 8Λ 4 Λ 2 Λ where we use 3.23, for T sufficiently small, such that 1/2 4C T T T T 1/2 1 4Λ u 2 H 2 w 2 H 2 1/2 1 8Λ < 1. Thus A[u, w] K for all u, w K. Furthermore, if T is chosen sufficiently small, such that C T T T T 1/2 1 4Λ u 2 H 2 w 2 1/2 1 H 2 1 8Λ < 2. Then 3.22 implies 1/2 A[u, w] A[ũ, w] X X 1 2 u, w ũ, w X X, 3.26 for all u, w, ũ, w K. Thus the mapping A is a strict contraction for sufficiently small T >. Hence, there exists a unique fixe point u, w K such that A[u, w] = u, w. 6. Given any T >, we select T 1 > small so that C 1/2 T1 T 1 T1 T 1 1 4Λ u 2 H 2 w 2 1/2 1 H 2 1 8Λ < 2. We can thus apply Banach s fixe point theorem to fin a unique strong solution u, w of existing on the interval [, T 1 ]. Since ux, t, wx, t K for almost everywhere t T 1, we can upon reefining T 1 if necessary, assume ux, T 1, wx, T 1 K. The argument above can then be repeate to exten our unique strong solution to the interval [T 1, 2T 1 ]. Continuing after finitely many

14 14 S. A. SANNI EJDE-29/6 steps, we construct a unique strong solution of existing on the full interval [, T ]. Acknowlegments. The author woul like to thank the anonymous referee whose comments improve the original version of this manuscript. References [1] W. E. Fitzgibbon, C. B. Martin ; The long time behavior of solutions to a quasilinear combustion moel, Nonlinear Anal. TMA, , [2] D. Frank-Kamenetskii; Diffusion an Heat Transfer in Chemical Kinetics., Plenum Press, New York [3] M. Marion; Qualitative Properties of a nonlinear system for laminar flames without ignition temperature, Nonlinear Anal. TMA, [4] D. Terman; Stability of Planar wave solutions to a combustion moel, SIAM J. Math. Anal., , [5] D. H. Wagner; Premixe laminar flame as travelling waves, in Reacting Flows: combustion an Chemical Reactors, G. S. S. Lufor, e., AMS Lectures in Applie Mathematics, 24, Provience, RI, [6] P. Clavin; Dynamical behavior of premixe fronts in laminar an turbulent flows, Prog. Energy. Comb. Sci., , [7] G. I. Sivashinsky; Instabilities, pattern formation, an turbulence in flames, Ann. rev. flui Mech., [8] J. Avrin; Qualitative theory for a moel of laminar flames with arbitrary nonnegative initial ata, J. iff. Eqns, , [9] J. Avrin; Qualitative theory of the Cauchy problem for a one-step reaction moel on boune omains, SIAM J. Math. Analysis, , [1] S. A. Sanni; A couple system of the Reynols, k-ɛ an scalar concentration equations, Ph. D. issertation, University of Leipzig, Germany 24. [11] L. c. Evans; Partial Differential Equations, American Mathematical Society, Provience, Rhoe Islan [12] J. Leerer, R. Lewanowski; A RANS moel with unboune ey viscosities, Annales e I Institu Henri Poincare C Non linear Analysis, , Sikiru Aigun Sanni Department of Mathematics, University of Uyo, Uyo, Akwa Ibom State, Nigeria aress: sikirusanni@yahoo.com