The Journal of Nonlinear Science and Applications

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1 J Nonlinear Sci Appl 3 21, no 4, The Journal of Nonlinear Science and Applications GLOBAL EXISTENCE AND L ESTIMATES OF SOLUTIONS FOR A QUASILINEAR PARABOLIC SYSTEM JUN ZHOU Abstract In this paper, we study the global existence, L estimates and decay estimates of solutions for the quasilinear parabolic system u t = u m u+fu, v, v t = v n v+gu, v with zero Dirichlet boundary condition in a bounded domain R N 1 Introduction In this paper, we are concerned with the global existence, L estimates and decay estimates of solutions for the quasilinear parabolic system u t = u m u + fu, v, x, t >, v t = v n v + gu, v, x, t >, 11 ux, = u x, vx, = v x, x, ux, t = vx, t =, x, where is a bounded domain in R N N > 1 with smooth boundary and m, n > For m = n =, fu, v = u α v p, gu, v = u q v β and u x, v x, the problem 11 has been investigated extensively and the existence and nonexistence of solutions for 11 are well understood see [3, 5, 6, 13] and the references cited there We summarize some of the results Suppose that the initial data u x, v x and u, v L Then Date: Revised: 4 March 21 Corresponding author c 21 NAG 2 Mathematics Subject Classification Primary 35K55; Secondary 35K57 Key words and phrases Global existence, quasilinear parabolic system, L estimates and decay estimates 245

2 246 JUN ZHOU A1 let α > 1 or β > 1 or s = 1 α1 β pq < Problem 11 admits a global solution for small initial data and the solution for 11 must blow up in finite time for large initial data; A2 all solutions of 11 are global if α, β 1 and s The case m > for the single equation u t = u m u + fx, u, x, t >, ux, = u x, x, 12 ux, t =, x has been widely investigated in [1, 2, 4, 7, 9, 11, 12] and the references therein But the problem 11 is not considered sufficiently and there seems to be little results on global existence, L estimates and blow-up of solutions for 11 In this paper we are interested in extending the previous results A1 and A2 for m = n = to m, n > We consider problem 11 for general initial data try to be more specific here and obtain sufficient conditions for the global existence of solutions Furthermore, we obtain L and decay estimates for solutions of 11, that give the behavior of solutions as t and t Our method, very different from that on the basis of comparison principle used in [3, 5, 6, 13, 14, 15, 16], is based on a priori estimates and an improved Moser s technique as in [2, 1] In contrast with other results which results [2, 4, 7, 9, 11], our initial data u, v is neither restricted to be bounded nor nonnegative To drive the L estimates for solutions of 11, we must treat carefully the parameters m, n, p, q, α and β Definition 11 A pair of functions ux, t, vx, t is a global wea solution of 11 if ux, t, vx, t L loc,, W 1,m+1 L m+1 loc R +, W 1,m+1 L loc,, W 1,n+1 L n+1 loc R +, W 1,n+1 and the following equalities t { uϕ t + u m u ϕ fu, vϕ} dxdt = {u xϕx, ux, tϕx, t} dx, t { vϕ t + v n v ϕ gu, vϕ} dxdt = {v xϕx, ux, tϕx, t} dx are valid for any t > and ϕ C 1 R +, C 1, where R + = [, Our results read as follows Theorem 12 Suppose that H 1 The functions fu, v, gu, v C R 2 C 1 R 2, and fu, v K 1 u α v p, gu, v K 2 u q v β, u, v R 2, 13

3 where the parameters α, β, p, q satisfy GLOBAL EXISTENCE AND L ESTIMATES 247 α < 1 + m, β < 1 + n; m, n, p, q > ; 14 s = m + 1 αn + 1 β pq > H 2 u x L p, v x L q with p > max{1, q + 1 α}, q > max{1, p + 1 β} Then problem 11 admits a global wea solution ux, t, vx, t which satisfies u L R +, L p, v L R +, L q and the following estimates hold for any T > + v Ct σ, v Ct σ, t T, 15 C t 1 σ + t 1 2p+ασ + t 1 2q+βσ, t T, 16 { } N where C = CT, u p, v q, σ = min p +mn, N q +nn Theorem 13 Suppose s < Then there exist p, q > 1, d > such that if u x L p, v x L q and u p + v q < d the problem 11 admits a global wea solution ux, t, vx, t that ux, t L loc,, W 1,m+1 L m+1 loc R +, W 1,m+1 17 vx, t L loc,, W 1,n+1 L n+1 loc R +, W 1,n+1 satisfying p C1 + t 1 ϑ, v q C1 + t 1 ϑ, t, 18 where ϑ = min{m/p, n/q } To derive Theorem 12 and 13, we will use the following lemmas Lemma 14 [9] Let β, N > p 1, β + 1 q, and 1 r q β + 1Np/N p Then for u β u W 1,p, we have q C 1/β+1 1 θ r u β u θ/β+1 1,p, with θ = β + 1r 1 q 1 /N 1 p 1 + β + 1r 1 1, where C is a constant depending only on N, p and r Lemma 15 [11] Let yt be a nonnegative differentiable function on, T ] satisfying y t + At θ 1 y 1+θ t Bt yt + Ct δ with A, θ >, θ 1, B, C, 1 Then we have yt A 1/θ 2A + 2BT 1 1/θ t + 2C + BT 1 1 t 1 δ < t T This paper is organized as follows In Section 2, we apply Lemmas 14 and 15 to establish L estimates for solutions of problem 11 The proof of Theorem 13 will be given in Section 3

4 248 JUN ZHOU 2 Proof of Theorem 12 For j = 1, 2,, we choose f j u, v, g j u, v C 1 in such a way f j u, v = fu, v, g j u, v = gu, v when u 2 + v 2 j 2, f j u, v η, g j u, v η when u 2 + v 2 j 2 with some η > and f j u, v, g j u, v fu, v, gu, v uniformly in R 2 as j Let u,j, v,j C 2 and u,j u in L p, v,j v in L q as j We consider the approximate problem of 11 u t = u 2 + j 1 m/2 u + f j u, v, x, t >, v t = v 2 + j 1 n/2 v + g j u, v, x, t >, 21 ux, = u,j x, vx, = v,j x, x, ux, t = vx, t =, x, The problem 21 is a standard quasilinear parabolic system and admits a unique smooth solution u j x, t, v j x, t on [, T for each j = 1, 2,, see [7, 8] Furthermore, if T <, then lim sup u j, t + v j, t = + t T In the sequel, we will always write u, v instead of u j, v j and u p, v p for u p 1 u, v p 1 v where p > Also, let C and C i be the generic constants independent of j and p changeable from line to line Lemma 21 Let H 1 and H 2 hold If ux, t, vx, t is the solution of problem 21 Then u L R +, L p, v L R +, L q Proof Let p, q > 1 Multiplying the first equation in 21 by u p 2 u, we obtain that 1 d p dt p p + p 1m + 2 u p+m p + m f j u, v u p 2 udx 22 Notice that f j u, v u p 2 udx ηj 1 p + C 1 u α+p 1 v p dx 23 Similarly, we have 1 d q dt v q q + q 1n + 2 v q+n q + n 24 ηj 1 q + C 2 v β+q 1 u q dx, with C 1, C 2 > By Young s inequality, we obtain u γ v p + u q v ρ v pp 1 + u p2γ + u qq1 + v ρq2, 25 p 1 p 2 q 1 q 2 where γ = α + p 1, ρ = β + q 1, t = γρ pq > and p 1 = t pγ q, p 2 = t γρ p, q 1 = t qρ p, q 2 = t ργ q 26

5 GLOBAL EXISTENCE AND L ESTIMATES 249 The assumption H 2 on p, q and 14 imply that pp 1 < q + n, qq 1 < p + m Thus we have from and a Sobolev s inequality that d p p dt + v q q + C3 p m p +m p +m + q n v q +n q +n η p j 1 p + q j 1 q + C 4 u qq 1 + v pp 1 dx Using Young s inequality and letting j in 27, we conclude that and 27 d p p dt + v q q + C5 p +m p +m + v q +n q +n C 28 d p p dt + v q q + C6 p p + v q 1+ϱ q C 29 with ϱ = min{m/p, n/q } Thus 29 implies that ut L R +, L p, vt L R +, L q if u L p and v L q The proof is completed Lemma 22 Under the assumptions of Lemma 21 and for any T >, the solution ut, vt also satisfies Ct a, v Ct b, < t T, 21 + v C t 1 σ + t 1 2p+ασ + t 1 2q+βσ, < t T, 211 where the constant C depends on T, u p, v q and a = N/p m mn, b = N/q n nn, σ = min{a, b} Proof We only consider N > max{m + 2, n + 2} and the other cases can be treated in a similar way Multiplying the first equation and the second equation in 21 by u 2 u and v µ 1 v respectively, we obtain d dt + v µ µ + C1 m u +m + µ n v µ+n C 2 + µ 1 + u α+ 1 v p + u q v β+µ 1 dx By the Young s inequality, we have u γ 1 v p + u q v γ 2 v pε 1 ε 1 + u γ1ε2 ε 2 + u qη1 η v γ2η2 η 2, 213 with γ 1 = α + 1, γ 2 = β + µ 1 and pε 1 = γ 2 η 2, γ 1 ε 2 = qη 1, ε ε 1 2 = 1, η1 1 + η2 1 = 1 The direct computation shows that η 1 = τ qγ 2 p, η 2 = τ γ 2 γ 1 q, ε 1 = τ pγ 1 q, ε 2 = τ γ 1 γ 2 p,

6 25 JUN ZHOU where τ = γ 1 γ 2 pq >, µ are chosen properly so that < pε 1 < µ + n and < qη 1 < + m We tae two sequences of { } and {µ } as follows 1 = p, = = b 1 + b 12 R 1 ; 214 µ 1 = q, µ = µ = b 2 + b 22 R 1, = 2, 3, where b 1 = q + 1 α, b 12 = b 1 + m/s, b 2 = p + 1 β, b 22 = b 2 + n/s and R is chosen so that R > 1, 2 > p, µ 2 > q Notice that µ as We now derive the estimates for the integrals v pε 1 dx and u qη 1 dx If pε 1 µ and qη 1, then we have v pε 1 dx C 1 + v µ dx, u qη 1 dx C 1 + u dx 215 Without loss of generality, we suppose µ < pε 1 < µ + n, < qη 1 < + m and r = τ/γ 1 q µ >, h = τ/γ 2 p > Then from 212 and 213, we have d dt + v µ µ + 2C1 m u +m + µ n v µ+n C h +h + C2 µ 1 + v µ+r 216 where the constants C 1, C 2 are independent of and µ Furthermore, we have following by Hölder s and Sobolev s inequalities with = u +h dx θ 1 θ 2 p θ 3 N + m N m 2, θ 1 = 1 C θ 1 C 1 C m u +m + C 3 σ 1 hn, θ 2 = p m mn u +m θ 3 +m 217 hp m + 2 p m mn, hn + m θ 3 = p m mn, σ 1 = m + 1p m Nm h > hn Similarly, we can derive that v µ+r dx C 1 C2 1 µ 1 n v µ+n + C 3 µ σ 2 v µ µ, 218 with σ 2 = n + 1q n Nn r/rn > Hence it follows from that d dt + v µ µ + C1 m u +m + µ n v µ+n 219 C σ 1 + C3 µ 1 + µ σ 2 v µ µ Now we employ an improved Moser s technique as in[2, 1] Let { }, {µ }be two sequences as defined in 214 From Lemma 14, we see that C m+ 1 θ 1 u +m θ +m, 22

7 GLOBAL EXISTENCE AND L ESTIMATES 251 v µ C n+µ v 1 θ µ 1 v µ +n θ µ +n, 221 where the constant C is independent of and µ, and θ = + m m N 1 m m, m θ = µ + n n + 2 µ 1 µ N 1 n µ 1 + n n + 2µ 1 Let t = +m θ, s = µ +n µ θ Then 22 and 221 give m u +m µ n µ +n v C θ C θ +t m t 1, 222 v µ +s µ v n s µ Denote y t = + v µ µ, t Then inserting into 219 =, µ = µ, we find that y t + C 1 C θ +t m t 1 + C 1 C θ C 3 + µ + C σ Cµ σ 2+1 v µ +s µ v n s µ v µ µ We claim that there exist the bounded sequence {ξ }, {η }, {m }, {r } such that ξ t m, v µ η t r, < t T 225 Without loss of generality, we suppose that ξ, η 1 By Lemma 21, 225 holds for = if we tae m = r = and ξ = sup t p, η = sup t v q If 225 is true for 1, then we have from 224 that y t + C 3 +t ξ 1 t m m t 1 + C 3 v µ +s µ η 1 t 1 r n s 226 C + µ σ 1 + µ σ 2 v µ µ We tae σ = max{σ 1, σ 2 }, τ = min{t /, s /µ }, α = min{m t, n s } and A 1 = max{ξ 1, η 1 }, β = max{t mm 1, s nr 1 } Then we have from 226 that y t + C 3 A α 1 tβ y t+τ t C + C σ +1 Applying Lemma 15 to 227, we get y t + CA α 1 T β, < t < T227 y t B t 1+β /τ, < t < T, 228 where B = 2 C 3 A α 1 τ 1 C 3 σ β 1 τ + 2C C σ β 1 τ τ Moreover, 228 implies that B 1 1+β t τ, v µ B 1 1+β µ t µ τ, < t T 229

8 252 JUN ZHOU We tae ξ = B 1, η = B 1 µ, m = 1 + β, r = 1 + β τ τ µ By a similar argument in [2, 1], we now that {ξ }, {η } are bounded and there exist two subsequences {m l } {m } and {r l } {r } such that m l a = N p m mn, r l b = Therefore, letting l in 228, we obtain N q n nn, as l Ct a, v Ct b, < t < T, 23 This yields 21 It remains to prove the estimate 211 In order to derive 211, we use a similar argument in [1] We first choose µ > max{σ, 2p+ασ 2, 2q +βσ 2} and ht C[, C 1, such that ht = t µ, t 1; ht = 2, t 2 and ht, h t in, Then multiplying the first equation by htu and the second equation by htv in 21, and letting j, we obtain t hsgsds ht u 2 + v 2 dx t t h s u 2 + v 2 dxds + C hs u 1+α v p + u q v 1+β dxds 2 with gt = u + v, t By Young s inequality and the assumption 14, we obtain C u 1+α v p + u q v 1+β dx u τ 1 + v τ 1 dx 232 ε u + v dx + C ε C u + v + C ε for any ε > and τ 1 = α + 1β + 1 pq/β + 1 p < m + 2, τ 2 = α + 1β + 1 pq/α + 1 q < n + 2 Furthermore, we tae ε = 1/2 Then yields t hsgsds + ht v 2 2 Ct µ σ 233 Next, let ρt = t hsds, t Similarly, multiplying the first equation in 21 by ρtu t and the second equation by ρtv t, and letting j, we have from 23 and 231 that t t ρs u t 2 2+ v t 2 2ds + ρtgt C ρs u 2α v 2p + u 2q v 2β dxds + t ρ sgsds C t ρs s 2α+pσ + s 2β+qσ ds + Ct µ σ C t µ σ + t µ+2 2p+ασ + t µ+2 2q+βσ, < t T 234

9 Thus 234 implies GLOBAL EXISTENCE AND L ESTIMATES 253 gt C t 1 σ + t 1 2p+ασ + t 1 2q+βσ, < t T, 235 and 211 is proved The proof is completed Proof of Theorem 12 We notice that the estimate constant C in 23 and 235 is independent of j, we may obtain the desired solution u, v as limit of {u j, v j } or a subsequence by the standard compact argument as in [6, 8, 9, 1] The solution u, v of problem 11 also satisfies The proof is completed Remar: From the proof of Theorem 12, we see that if the assumption 13 is replaced by fu, v K u α v p, gu, v K u q v β, the conclusions in Theorem 12 still hold 3 Proof of Theorem 13 By the standard compact argument as in [2, 7, 9, 1], we only consider the estimate 18 and show that u, v L 1,m+1 loc R +, W 1,m+1 L 1,n+1 loc R +, W 1,n+1 for the solution of 21 Proof of Theorem 13 Suppose that s < holds Let p = b 1 + b 12 ε > 1, q = b 2 + b 22 ε > 1, 31 with b 1 = q+1 α, b 2 = p+1 β, b 12 = q+m+1 α/s, b 22 = p+n+1 β/s Since s <, we can tae ε > such that p max{4q, 4α, 2 + 2α}, q max{4p, 4β, 2 + 2β}, S = α + p 1β + q 1 pq > Then it follows from 25 and 27 that d q p dt + v q q + C1 u p +m + v q +n 32 C u qq 1 + v pp 1 dx, where qq 1 = S /q + β 1 p > p + m, pp 1 = S /α + p q 1 > q + n We now estimate the right-hand side of 32 Let qq 1 = p + θ, pp 1 = q + τ and θ > m, τ > n Then u qq 1 dx = p +θ p +θ C 2 θ m p u p +m, 33 Denote v pp 1 dx = v q +τ q +τ C 2 v τ n q v q +n 34 φt = p p + v q q, ft = u p +m + v q +n,

10 254 JUN ZHOU then 32 becomes φ t + C 1 ft C 2 θ m p u p +m + v τ n q v q+n C 2 u θ m p + v τ n q with α = min{θ m/p, τ n/q } > 35 implies that there is C > such that ft C3 φ α tft, 35 φ t + C ft if C 3 φ α = C 3 u p p + v q q α < C 1 36 Furthermore, we have from Sobolev embedding theorems that u p +m d 1 p +m p +m d 2 p +m p, v q +n d 2 v q +2 q, for some d 2 > Hence, ft d 2 p +m p + v q +m q d2 φ 1+ϑ, ϑ = min{m/p, n/q } Now 36 gives This implies that φ t + d 2 φ 1+ϑ, t 37 φt C1 + t 1 ϑ 38 Next, we show that u, v L 1,m+1 loc R +, W 1,m+1 L 1,n+1 loc R +, W 1,n+1 By the definition of p and q, we have from 38 that for any t, u 1+α v p dx C 1+α p v p q C 1, u q v 1+β dx C q p v 1+β q C 1 Here C 1 is a constant independent of t Thus 231 yields that t t hsgsds C ht + gsds Cht + ρt, t 39 Similarly, we have u 2α v 2p dx 2α p v 2p q C 2, Then from 234 and 39, we obtain t t t ρtgt C 3 ρsds + hsgsds C 3 It implies u 2q v 2β dx 2q p v 2β q C 2 ρsds + ht + ρt 31 gt C 4 t + t 1 + 1, t T, 311 and u, v L 1,m+1 loc R +, W 1,m+1 L 1,n+1 loc R +, W 1,n+1 This completes the proof of Theorem 12 The proof is completed Acnowledgement This wor is supported by Basic Research Foundation Project of China SWU No XDJK29C69

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