In this paper we prove the existence and uniqueness of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators by combining compactness and monotonicity methods.
HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2019-0025 Natural Science, 2019, Volume 64, Issue 6, pp 3-11 This paper is available online at http://stdb.hnue.edu.vn EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A CLASS OF QUASILINEAR DEGENERATE PARABOLIC EQUATIONS Tran Thi Quynh Chi and Le Thi Thuy Faculty of Mathematics, Electric Power University Abstract In this paper we prove the existence and uniqueness of weak solutions to a class of quasilinear degenerate parabolic equations involving weighted p-Laplacian operators by combining compactness and monotonicity methods Keywords: Quasilinear degenerate parabolic equation, weighted p-Laplacian operator, weak solution, compactness method, monotonicity method Introduction In this paper we consider the following parabolic problem: p−2 ut − div(a(x)|∇u| ∇u) + f (u) = g(x), x ∈ Ω, t > 0, u(x, t) = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0 (x), x ∈ Ω, (1.1) where Ω is a bounded domain in RN (N ≥ 2) with smooth boundary ∂Ω, ≤ p ≤ N, u0 ∈ L2 (Ω) given, the coefficient a(·), the nonlinearity f and the external force g satisfy the following conditions: (H1) The function a : Ω → R satisfies the following assumptions: a ∈ L1loc (Ω) and a(x) = for x ∈ Σ, and a(x) > for x ∈ Ω \ Σ, where Σ is a closed subset of Ω with meas(Σ) = Furthermore, we assume that N Ω [a(x)] α dx < ∞ for some α ∈ (0, p); (1.2) (H2) f : R → R is a C -function satisfying C1 |u|q − C0 ≤ f (u)u ≤ C2 |u|q + C0 , f ′ (u) ≥ −ℓ, for some q ≥ 2, (1.3) (1.4) where C0 , C1, C2 , ℓ are positive constants; Received March 11, 2019 Revised June 5, 2019 Accepted June 12, 2019 Contact Tran Thi Quynh Chi, e-mail address: chittq@epu.edu.vn Tran Thi Quynh Chi and Le Thi Thuy q pN , q − (N + 1)p − N + α (H3) g ∈ Ls (Ω), where s ≥ The degeneracy of problem (1.1) is considered in the sense that the measurable, nonnegative diffusion coefficient a(x) is allowed to vanish somewhere The physical motivation of the assumption (H1) is related to the modeling of reaction diffusion processes in composite materials, occupying a bounded domain Ω, in which at some points they behave as perfect insulator Following [1, p 79], when at some points the medium is perfectly insulating, it is natural to assume that a(x) vanishes at these points As mentioned in [2], the assumption (H1) implies that the degenerate set may consist of an infinite many number of points, which is different from the weight of Caldiroli-Musina type in [3, 4] that is only allowed to have at most a finite number of zeroes A typical example of the weight a is dist(x, ∂Ω) Problem (1.1) contains some important classes of parabolic equations, such as the semilinear heat equation (when a = 1, p = 2), semilinear degenerate parabolic equations (when p = 2), the p-Laplacian equations (when a = 1, p = 2), etc It is noticed that the existence and long-time behavior of solutions to (1.1) when p = 2, the semilinear case, have been studied recently by Li et al in [2] We also refer the interested reader to [4-11] for related results on degenerate parabolic equations Preliminary results To study problem (1.1), we introduce the weighted Sobolev space W01,p (Ω, a), defined as the closure of C0∞ (Ω) in the norm u W01,p (Ω,a) p := p a(x)|∇u| dx , Ω ′ and denote by W −1,p (Ω, a) its dual space We now prove some embedding results, which are generalizations of the corresponding results in the case p = of Li et al [2] Proposition 2.1 Assume that Ω is a bounded domain in RN , N ≥ 2, and a(·) satisfies (H1) Then the following embeddings hold: (i) W01,p (Ω, a) ֒→ W01,β (Ω) continuously if ≤ β ≤ NpN ; +α (ii) W01,p (Ω, a) ֒→ Lr (Ω) continuously if ≤ r ≤ p∗α , where p∗α = pN N −p+α (iii) W01,p (Ω, a) ֒→ Lr (Ω) compactly if ≤ r < p∗α Proof Applying the Hăolder inequality, we have pN pN N N+α |∇u| N+α dx |∇u| N+α dx = N [a(x)] Ω Ω [a(x)] N+α ≤ N Ω α N+α [a(x)] α p dx a(x)|∇u| dx Ω N α Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations Using the assumption (H1), we complete the proof of (i) The conclusions (ii) and (iii) follow from (i) and the well-known embedding results for the classical Sobolev spaces Putting Lp,a u = −div(a(x)|∇u|p−2∇u), u ∈ W01,p (Ω, a) The following proposition, its proof is straightforward, gives some important properties of the operator Lp,a Proposition 2.2 The operator Lp,a maps W01,p (Ω, a) into its dual W −1,p (Ω, a) Moreover, (i) Lp,a is hemicontinuous, i.e., for all u, v, w ∈ W01,p (Ω, a), the map λ → Lp,a (u+ λv), w is continuous from R to R; (ii) Lp,a is strongly monotone when p ≥ 2, i.e., ′ Lp,a u − Lp,a v, u − v ≥ δ u − v p W01,p (Ω,a) for all u, v ∈ W01,p (Ω, a) Existence and uniqueness of global weak solutions Denote ΩT = Ω × (0, T ), V = Lp (0, T ; W01,p(Ω, a)) ∩ Lq (0, T ; Lq (Ω)), ′ ′ ′ ′ V ∗ = Lp (0, T ; W −1,p (Ω, a)) + Lq (0, T ; Lq (Ω)) Definition 3.1 A function u is called a weak solution of problem (1.1) on the interval (0, T ) if du ∈ V ∗, dt = u0 a.e in Ω, u ∈ V, u|t=0 and ΩT ∂u η + a(x)|∇u|p−2∇u∇η + f (u)η − gη dxdt = 0, ∂t (3.1) for all test functions η ∈ V du It is known (see e.g [4]) that if u ∈ V and ∈ V ∗ , then u ∈ C([0, T ]; L2 (Ω)) dt This makes the initial condition in problem (1.1) meaningful Lemma 3.1 Let {un } be a bounded sequence in Lp (0, T ; W01,p(Ω, a)) such that {u′n } is bounded in V ∗ If (H1) and (H3) hold, then {un } converges almost everywhere in ΩT up to a subsequence Tran Thi Quynh Chi and Le Thi Thuy Proof By Proposition 2.1, one can take a number r ∈ [2, p∗α ) such that W01,p (Ω, a) ֒→֒→ Lr (Ω) (3.2) Since r ′ ≤ 2, we have ′ Lp (Ω) ∩ Lq (Ω) ֒→ Lr (Ω), and therefore, ′ ′ Lr (Ω) ֒→ Lp (Ω) + Lq (Ω) (3.3) Using Proposition 2.1 once again and noticing that p ≤ p∗α since α ∈ (0, p), we see that W01,p (Ω, a) ֒→ Lp (Ω) This and (3.3) follow that ′ ′ Lr (Ω) ֒→ W −1,p (Ω, a) + Lq (Ω) Now with (3.2), we have an evolution triple W01,p (Ω, a) ֒→֒→ Lr (Ω) ֒→ W −1,p (Ω, a) + Lq (Ω) ′ ′ The assumption of {u′n } in V ∗ implies that ′ ′ {u′n } is also bounded in Ls (0, T ; W −1,p (Ω, a) + Lq (Ω)), where s = min{p′ , q ′} Thanks to the well-known Aubin-Lions compactness lemma (see [12, p 58]), {un } is precompact in Lp (0, T ; Lr (Ω)) and therefore in Lt (0, T ; Lt (Ω)), t = min(p, r), so it has an a.e convergent subsequence The following lemma is a direct consequence of Young’s inequality and the ∗ pN embedding W01,p (Ω, a) ֒→ Lpα (Ω), where p∗α = N −p+α , which is frequently used later Lemma 3.2 Let condition (H3) hold and u ∈ W01,p (Ω, a) ∩ Lq (Ω) Then for any ε > 0, we have gudx ≤ Ω ε u ε u p + C(ε) g sLs(Ω) W01,p (Ω,a) q s Lq (Ω) + C(ε) g Ls (Ω) if s ≥ if s ≥ pN , (N +1)p−N +α q q−1 The following theorem is the main result of the paper Theorem 3.1 Under assumptions (H1) − (H3), for each u0 ∈ L2 (Ω) and T > given, problem (1.1) has a unique weak solution on (0, T ) Moreover, the mapping u0 → u(t) is continuous on L2 (Ω) Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations Proof (i) Existence Consider the approximating solution un (t) in the form n unk (t)ek , un (t) = k=1 1,p q where {ej }∞ j=1 is a basis of W0 (Ω, a) ∩ L (Ω), which is orthogonal in L (Ω) We get un from solving the problem dun , ek + Lp,a un , ek + f (un), ek = g, ek , dt (u (0), e ) = (u , e ), k = 1, , n n k k By the Peano theorem, we obtain the local existence of un We now establish some a priori estimates for un Since 1d un (t) dt L2 (Ω) a(x)|∇un |p dx + + f (un )un dx = Ω Ω gun dx Ω Using (1.3) and Lemma 3.2, we have d un dt L2 (Ω) a(x)|∇un |p dx + +C Ω |un |q dx ≤ C( g Integrating from to t, ≤ t ≤ T and using the fact that un (0) obtain t un (t) L2 (Ω) L2 (Ω) ≤ u0 L2 (Ω) , we t a(x)|∇un |p dxdt + C +C ≤ Ls (Ω) , |Ω|) Ω Ω u0 L2 (Ω) |un |q dxdt + T C( g Ω Ls (Ω) , |Ω|) It follows that • {un } is bounded in L∞ (0, T ; L2 (Ω)); • {un } is bounded in Lp (0, T ; W01,p(Ω, a)); • {un } is bounded in Lq (0, T ; Lq ()) The Hăolder inequality yields T | T a(x)|∇un |p−2 ∇un ∇vdxdt| Lp,a un , v dt| = | 0 T Ω ≤ a(x) ≤ un p−1 p |∇un |p−1)(a(x) p |∇v| dxdt Ω p p′ Lp (0,T ;W01,p (Ω,a)) v Lp (0,T ;W01,p (Ω,a)) , Tran Thi Quynh Chi and Le Thi Thuy for any v ∈ Lp (0, T ; W01,p(Ω, a)) Using the boundedness of {un } in ′ ′ 1,p p L (0, T ; W0 (Ω, a)), we infer that {Lp,a un } is bounded in Lp (0, T ; W −1,p (Ω, a)) From (1.3), we have |f (u)| ≤ C(|u|p−1 + 1) Hence, since {un } is bounded in Lq (0, T ; Lq (Ω)), one can check that {f (un )} is bounded ′ ′ in Lq (0, T ; Lq (Ω)) Rewriting (1.1) in V ∗ as u′n = g − Lp,a un − f (un ) (3.4) and using the above estimates, we deduce that {u′n } is bounded in V ∗ From the above estimates, we can assume that • u′n ⇀ u′ in V ∗ ; ′ ′ • Lp,a un ⇀ ψ in Lp (0, T ; W −1,p (Ω, a)); ′ • f (un ) ⇀ χ in Lq (ΩT ) By Lemma 3.1, un → u a.e in ΩT , so f (un ) → f (u) a.e in ΩT since f (·) is continuous Thus, χ = f (u) thanks to Lemma 1.3 in [12] Now taking (3.4) into account, we obtain the following equation in V ∗ , u′ = g − ψ − f (u) (3.5) We now show that ψ = Lp,a u We have for every v ∈ Lp (0, T ; W01,p (Ω, a)), T Xn := Lp,a un − Lp,a v, un − v ≥ 0 Noticing that T T a(x)|∇un |p dxdt Lp,a un , un dt = 0 Ω T (gun − f (un )un − u′n un )dxdt = Ω T = (gun − f (un )un )dxdt + Ω un (0) 2 L2 (Ω) − un (T ) (3.6) Therefore, T Xn = (gun − f (un )un )dxdt + Ω T − un (0) 2 L2 (Ω) − un (T ) T Lp,aun , v dt − L2 (Ω) Lp,a v, un − v dt L2 (Ω) Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations It follows from the formulation of un (0) that un (0) → u0 in L2 (Ω) Moreover, by the lower semi-continuity of L2 (Ω) we obtain u(T ) L2 (Ω) ≤ lim inf un (T ) n→∞ L2 (Ω) (3.7) Meanwhile, by the Lebesgue dominated theorem, one can check that T T (gu − f (u)u)dxdt = lim (gun − f (un )un )dxdt n→∞ Ω Ω This fact and (3.6), (3.7) imply that T lim sup Xn ≤ n→∞ (gu − f (u)u)dxdt + Ω T − u(0) 2 L2 (Ω) − u(T ) 2 L2 (Ω) (3.8) T ψ, v dt − Lp,av, u − v dt In view of (3.5), we have T (gu − f (u)u)dxdt + Ω u(0) 2 L2 (Ω) − u(T ) T L2 (Ω) = ψ, u dt This and (3.8) deduce that T ψ − Lp,a v, u − v dt ≥ (3.9) Putting v = u − λw, w ∈ Lp (0, T ; W01,p(Ω, a)), λ > Since (3.9) we have T ψ − Lp,a (u − λw), w dt ≥ λ Then T ψ − Lp,a (u − λw), w dt ≥ 0 Taking the limit λ → and noticing that Lp,a is hemicontinuous, we obtain T ψ − Lp,a u, w dt ≥ 0, for all w ∈ Lp (0, T ; W01,p(Ω, a)) Thus, ψ = Lp,a u We now prove u(0) = u0 Choosing some test function ϕ ∈ C ([0, T ]; W01,p (Ω, a)∩ q L (Ω)) with ϕ(T ) = and integrating by parts in t in the approximate equations, we have T T ′ − un , ϕ dt + Lp,a un , ϕ dt + (f (un )ϕ − gϕ)dxdt = (un (0), ϕ(0)) ΩT Tran Thi Quynh Chi and Le Thi Thuy Taking limits as n → ∞, we obtain T T ′ − u, ϕ dt + Lp,a u, ϕ dt + (f (u)ϕ − gϕ)dxdt = (u0 , ϕ(0)), (3.10) ΩT since un (0) → u0 On the other hand, for the ”limiting equation”, we have T T − u, ϕ′ dt + Lp,au, ϕ dt + (f (u)ϕ − gϕ)dxdt = (u(0), ϕ(0)) (3.11) ΩT Comparing (3.10) and (3.11), we get u(0) = u0 (ii) Uniqueness and continuous dependence Let u, v be two weak solutions of problem (1.1) with initial data u0 , v0 in L2 (Ω) Then w := u − v satisfies dw + (Lp,a u − Lp,a v) + (f (u) − f (v)) = 0, dt w(0) = u − v 0 Hence 1d w dt L2 (Ω) + Lp,a u − Lp,a v, u − v + (f (u) − f (v))(u − v)dx = Ω Using (1.4) and the monotonicity of the operator Lp,a , we have d w dt L2 (Ω) ≤ 2ℓ w L2 (Ω) Applying the Gronwall inequality, we obtain w(t) L2 (Ω) ≤ w(0) 2ℓt L2 (Ω) e for all t ∈ [0, T ] This completes the proof REFERENCES [1] R Dautray, J.L Lions, 1985 Mathematical Analysis and Numerical Methods for Science and Technology Vol I: Physical origins and classical methods, Springer-Verlag, Berlin [2] H Li, S Ma and C.K Zhong, 2014 Long-time behavior for a class of degenerate parabolic equations Discrete Contin Dyn Syst 34, 2873-2892 [3] P Caldiroli and R Musina, 2000 On a variational degenerate elliptic problem Nonlinear Diff Equ Appl 7, 187-199 10 Existence and uniqueness of solutions to a class of quasilinear degenerate parabolic equations [4] C.T Anh and T.D Ke, 2009 Long-time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators Nonlinear Anal., 71, 4415-4422 [5] C.T Anh, N.D Binh and L.T Thuy, 2010 Attractors for quasilinear parabolic equations involving weighted p-Laplacian operators Viet J Math 38, 261-280 [6] P.G Geredeli and A Khanmamedov, 2013 Long-time dynamics of the parabolic p-Laplacian equation Commun Pure Appl Anal 12, 735-754 [7] N.I Karachalios and N.B Zographopoulos, 2006 On the dynamics of a degenerate parabolic equation: Global bifurcation of stationary states and convergence Calc Var Partial Differential Equations, 25, 361-393 [8] X Li, C Sun and N Zhang, 2016 Dynamics for a non-autonomous degenerate parabolic equation in D01 (Ω, σ) Discrete Contin Dyn Syst 36, 7063-7079 [9] X Li, C Sun and F Zhou, 2016 Pullback attractors for a non-autonomous semilinear degenerate parabolic equation Topol Methods Nonlinear Anal 47, 511-528 [10] W Tan, 2018 Dynamics for a class of non-autonomous degenerate p-Laplacian equations J Math Anal Appl 458, 1546-1567 [11] M.H Yang, C.Y Sun and C.K Zhong, 2007 Global attractors for p-Laplacian equations J Math Anal Appl 327, 1130-1142 [12] J.-L Lions, 1969 Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires Dunod, Paris 11 ... 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