We study the existence and asymptotic behavior of the global solutions of the nonlinear equation
$$u_tt-\Delta_p u+(-\Delta)^\alpha u_t+g(u)=f$$
where $0<\alpha\leq 1$ and $g$ does not satisfy the sign condition $g(u)u \geq 0$

Gao, Hongjun

Ma, T.F.

Electronic journal of qualitative theory of differential equations

Hongjun Gao

T. F. Ma

Crossref

Global solutions for a nonlinear wave equation with the p-laplacian operator

To Fu Ma

Directory of Open Access Journals

Gao Hongjun

Ma To Fu

University of Szeged

Global Solutions for a Nonlinear Wave Equationwith the p-Laplacian Operator ∗Hongjun GaoInstitute of Applied Physics and Computational Mathematics100088 Beijing, ChinaTo Fu MaDepartamento de Matemática - Universidade Estadual de Maringá87020-900 Maringá-PR, BrazilAbstractWe study the existence and asymptotic behaviour of the globalsolutions of the nonlinear equationutt − ∆pu + (−∆)αut + g(u) = fwhere 0 < α ≤ 1 and g does not satisfy the sign condition g(u)u ≥ 0.Key words: Quasilinear hyperbolic equation, asymptotic behaviour, small data.AMS Subject Classification: 35B40, 35L70.1. IntroductionThe study of global existence and asymptotic behaviour for initial-boundaryvalue problems involving nonlinear operators of the typeutt −n∑i=1{σi(uxi)}xi − ∆ut = f(t, x) in (0, T ) × Ωgoes back to Greenberg, MacCamy & Mizel [3], where they considered theone-dimensional case with smooth data. Later, several papers have appearedin that direction, and some of its important results can be found in, for∗Work done while the authors were visiting LNCC/CNPq, Brazil.EJQTDE, 1999 No. 11, p. 1example, Ang & Pham Ngoc Dinh [1], Biazutti [2], Nakao [8], Webb [10]and Yamada [11]. In all of the above cited papers, the damping term −∆utplayed an essential role in order to obtain global solutions. Our objective isto study this kind of equations under a weaker damping given by (−∆)αutwith 0 < α ≤ 1. This approach was early considered by Medeiros andMilla Miranda [6] to Kirchhoff equations. We also consider the presence ofa forcing term g(x, u) that does not satisfy the sign condition g(x, u)u ≥ 0.Our study is based on the pseudo Laplacian operator−∆pu = −n∑i=1∂∂xi(∣∣∣∣∂u∂xi∣∣∣∣p−2 ∂u∂xi)which is used as a model for several monotone hemicontinuous operators.Let Ω be a bounded domain in Rn with smooth boundary ∂Ω. Weconsider the nonlinear initial-boundary value problem(1.1)utt − ∆pu + (−∆)αut + g(x, u) = f(t, x) in (0, T ) × Ω,u = 0 on (0, T ) × ∂Ω,u(0, x) = u0 and ut(0, x) = u1 in Ω,where 0 < α ≤ 1 and p ≥ 2. We prove that, depending on the growth ofg, problem (1.1) has a global weak solution without assuming small initialdata. In addition, we show the exponential decay of solutions when p = 2and algebraic decay when p > 2. The global solutions are constructedby means of the Galerkin approximations and the asymptotic behaviour isobtained by using a difference inequality due to M. Nakao [7]. Here we onlyuse standard notations. We often write u(t) instead u(t, x) and u′(t) insteadut(t, x). The norm in Lp(Ω) is denoted by ‖ · ‖p and in W 1,p0 (Ω) we use thenorm‖u‖p1,p =n∑j=1‖uxj‖pp.For the reader’s convenience, we recall some of the basic properties of theoperators used here. The degenerate operator −∆p is bounded, monotoneand hemicontinuous from W 1,p0 (Ω) to W−1,q(Ω), where p−1 + q−1 = 1. Thepowers for the Laplacian operator is defined by(−∆)αu =∞∑j=1λαj (u, ϕj)ϕj ,where 0 < λ1 < λ2 ≤ λ3 ≤ · · · and ϕ1, ϕ2, ϕ3, · · · are, respectively, thesequence of the eigenvalues and eigenfunctions of −∆ in H 10 (Ω). Then‖u‖D((−∆)α) = ‖(−∆)αu‖2 ∀u ∈ D((−∆)α)EJQTDE, 1999 No. 11, p. 2and D((−∆)α) ⊂ D((−∆)β) compactly if α > β ≥ 0. In particular, forp ≥ 2 and 0 < α ≤ 1, W 1,p0 (Ω) ↪→ D((−∆)α/2) ↪→ L2(Ω).2. Existence of Global SolutionsLet g : Ω×R → R be a continuous function satisfying the growth condition|g(x, u)| ≤ a|u|σ−1 + b ∀(x, u) ∈ Ω × R, (2.1)where a, b are positive constants, 1 < σ < pn/(n−p) if n > p and 1 < σ < ∞if n ≤ p.Theorem 2.1 Let us assume condition (2.1) with σ < p. Then given u0 ∈W 1,p0 (Ω), u1 ∈ L2(Ω) and f ∈ L2(0, T ;L2(Ω)), there exists a function u :(0, T ) × Ω → R such thatu ∈ L∞(0, T ;W 1,p0 (Ω)), (2.2)u′ ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;D((−∆)α2 )), (2.3)u(0) = u0 and u′(0) = u1 a.e. in Ω, (2.4)utt − ∆pu + (−∆)αut + g(x, u) = f in L2(0, T ;W−1,q(Ω)), (2.5)where p−1 + q−1 = 1.Next we consider an existence result when σ ≥ p. In this case, the globalsolution is obtained with small initial data. For each m ∈ N we putγm =12‖u1m‖22 +32p‖u0m‖p1,p +aCσσ‖u0m‖σ1,p +2( p√2bC1)qqwhere Ck > 0 is the Sobolev constant for the inequality ‖u‖k ≤ Ck‖u‖1,p,when W 1,p0 (Ω) ↪→ Lk(Ω). We also define the polynomial Q byQ(z) =12pzp − aCσσzσ,which is increasing in [0, z0], wherez0 = (2aCσ)−1σ−pis its unique local maximum. We will assume that‖u0‖1,p < z0 (2.6)EJQTDE, 1999 No. 11, p. 3andγ +14λα1∫ T0‖f(t)‖22dt < Q(z0), (2.7)where γ = limm→∞ γm.Theorem 2.2 Suppose that condition (2.1) holds with σ > p. Suppose inaddition that inital data satisfy (2.6) and (2.7). Then there exists a functionu : (0, T ) × Ω → R satisfying (2.2)-(2.5).Proof of Theorem 2.1: Let r be an integer for which H r0(Ω) ↪→ W1,p0 (Ω)is continuous. Then the eigenfunctions of −∆rwj = αjwj in Hr0(Ω) yieldsa “Galerkin” basis for both Hr0(Ω) and L2(Ω). For each m ∈ N, let us putVm = Span{w1, w2, · · · , wm}. We search for a functionum(t) =m∑j=1kjm(t)wjsuch that for any v ∈ Vm, um(t) satisfies the approximate equation∫Ω{u′′m(t) − ∆pum(t) + (−∆)αu′m(t) + g(x, um(t)) − f(t, x)}v dx = 0 (2.8)with the initial conditionsum(0) = u0m and u′m(0) = u1m,where u0m and u1m are chosen in Vm so thatu0m → u0 in W 1,p0 (Ω) and u1m → u1 in L2(Ω). (2.9)Putting v = wj , j = 1, · · · ,m, we observe that (2.8) is a system of ODEsin the variable t and has a local solution um(t) in a interval [0, tm). In thenext step we obtain the a priori estimates for the solution um(t) so that itcan be extended to the whole interval [0, T ].A Priori Estimates: We replace v by u′m(t) in the approximate equation(2.8) and after integration we have12‖u′m(t)‖22 +1p‖um(t)‖p1,p +∫ t0‖(−∆)α2 u′m(s)‖22 ds +∫ΩG(x, um(t))dx≤∫ t0‖f(s)‖2‖u′m(s)‖2 ds +12‖u1m‖22 +1p‖u0m‖p1,p +∫ΩG(x, u0)dx,EJQTDE, 1999 No. 11, p. 4where G(x, u) =∫ u0 g(x, s) ds. Now from growth condition (2.1) and theSobolev embedding, we have that∫Ω|G(x, um(t))|dx ≤aσCσ‖um(t)‖σ1,p + bC1‖um(t)‖1,p. (2.10)But since p > σ, there exists a constant C > 0 such that∫Ω|G(x, um)|dx ≤12p‖um(t)‖p1,p + C,and then we have12‖u′m(t)‖22 +12p‖um(t)‖p1,p +∫ t0‖(−∆)α2 u′m(s)‖22 ds≤∫ t0‖f(s)‖2‖u′m(s)‖2 ds +12‖u1m‖22 +32p‖u0m‖p1,p + 2C.Using the convergence (2.9) and the Gronwall’s lemma, there exists a con-stant C > 0 independent of t,m such that‖u′m(t)‖22 + ‖um(t)‖p1,p +∫ t0‖(−∆)α2 u′m(s)‖22 ds ≤ C. (2.11)With this estimate we can extend the approximate solutions um(t) to theinterval [0, T ] and we have that(um) is bounded in L∞(0, T ;W 1,p0 (Ω)), (2.12)(u′m) is bounded in L∞(0, T ;L2(Ω)), (2.13)(u′m) is bounded in L2(0, T ;D((−∆)α2 )), (2.14)and(−∆pum) is bounded in L∞(0, T ;W−1,q(Ω)). (2.15)Now we are going to obtain an estimate for (u′′m). Since our Galerkin basiswas taken in the Hilbert space Hr(Ω) ⊂ W 1,p0 (Ω), we can use the standardprojection arguments as described in Lions [4]. Then from the approximateequation and the estimates (2.12)-(2.15) we get(u′′m) is bounded in L2(0, T ;H−r(Ω)). (2.16)EJQTDE, 1999 No. 11, p. 5Passage to the Limit: From (2.12)-(2.14), going to a subsequence if neces-sary, there exists u such thatum ⇀ u weakly star in L∞(0, T ;W 1,p0 (Ω)), (2.17)u′m ⇀ u′ weakly star in L∞(0, T ;L2(Ω)), (2.18)u′m ⇀ u′ weakly in L2(0, T ;D((−∆)α2 ), (2.19)and in view of (2.15) there exists χ such that−∆pum ⇀ χ weakly star in L∞(0, T ;W−1,q(Ω)). (2.20)By applying the Lions-Aubin compactness lemma we get from (2.12)-(2.13)um → u strongly in L2(0, T ;L2(Ω)), (2.21)and since D((−∆)α2 ) ↪→ L2(Ω) compactly, we get from (2.14) and (2.16)u′m → u′ strongly in L2(0, T ;L2(Ω)). (2.22)Using the growth condition (2.1) and (2.21) we see that∫ T0∫Ω|g(x, um(t, x))|σσ−1 dxdtis bounded andg(x, um) → g(x, u) a.e. in (0, T ) × Ω.Therefore from Lions [4] (Lemma 1.3) we infer thatg(x, um) ⇀ g(x, u) weakly in Lσσ−1 (0, T ;Lσσ−1 (Ω)). (2.23)With these convergence we can pass to the limit in the approximate equationand thenddt(u′(t), v) + 〈χ(t), v〉 + ((−∆)αu′(t), v) + (g(x, u(t)), v) = (f(t), v) (2.24)for all v ∈ W 1,p(Ω), in the sense of distributions. This easily implies that(2.2)-(2.4) hold. Finally, since we have the strong convergence (2.22), wecan use a standard monotonicity argument as done in Biazutti [2] or Ma &Soriano [5] to show that χ = −∆pu. This ends the proof. @AEJQTDE, 1999 No. 11, p. 6Proof of Theorem 2.2: We only show how to obtain the estimate (2.11).The remainder of the proof follows as before. We apply an argument madeby L. Tartar [9]. From (2.10) we have∫Ω|G(x, um)|dx ≤aCσσ‖um‖σ1,p +12p‖um‖p1,p +( p√2bC1)qqand therefore12‖u′m(t)‖22 + Q(‖um(t)‖1,p) +∫ t0‖(−∆)α2 u′m(s)‖22 ds≤ γm +∫ t0‖f(s)‖2‖u′m(s)‖2 ds.Since‖(−∆)α2 u′m(s)‖2 ≥ λα21 ‖u′m(t)‖2, (2.25)where λ1 is the first eigenvalue of −∆ in H10 (Ω), this implies that12‖u′m(t)‖22 + Q(‖um(t)‖1,p) ≤ γm +14λα1∫ T0‖f(t)‖22 dt. (2.26)We claim that there exists an integer N such that‖um(t)‖1,p < z0 ∀ t ∈ [0, tm) m > N (2.27)Suppose the claim is proved. Then Q(‖um(t)‖1,p) ≥ 0 and from (2.7) and(2.26), ‖u′m(t)‖2 is bounded and consequently (2.11) follows.Proof of the Claim: Suppose (2.27) false. Then for each m > N , there existst ∈ [0, tm) such that ‖um(t)‖1,p ≥ z0. We note that from (2.6) and (2.9)there exists N0 such that‖um(0)‖1,p < z0 ∀m > N0.Then by continuity there exists a first t∗m ∈ (0, tm) such that‖um(t∗m)‖1,p = z0, (2.28)from whereQ(‖um(t)‖1,p) ≥ 0 ∀ t ∈ [0, t∗m].Now from (2.7) and (2.26), there exist N > N0 and β ∈ (0, z0) such that0 ≤ 12‖u′m(t)‖22 + Q(‖um(t)‖1,p) ≤ Q(β) ∀ t ∈ [0, t∗m] ∀m > N.EJQTDE, 1999 No. 11, p. 7Then the monotonicity of Q in [0, z0] implies that0 ≤ ‖um(t)‖1,p ≤ β < z0 ∀t ∈ [0, t∗m],and in particular, ‖um(t∗m)‖1,p < z0, which is a contradiction to (2.28). @ARemarks: From the above proof we have the following immediate conclu-sion: The smallness of initial data can be dropped if either condition (2.1)holds with σ = p and coefficient a is sufficiently small, or σ > p and the signcondition g(x, u)u ≥ 0 is satisfied. @A3. Asymptotic BehaviourTheorem 3.1 Let u be a solution of Problem (1.1) given by(a) either Theorem 2.1 with the additional assumption: there exists ρ > 0such thatg(x, u)u ≥ ρG(x, u) ≥ 0, (3.1)(b) or Theorem 2.2.Then there exists positive constants C and θ such that‖u′(t)‖22 + ‖u(t)‖p1,p ≤ C exp(−θt) if p = 2,or‖u′(t)‖22 + ‖u(t)‖p1,p ≤ C(1 + t)−pp−2 if p > 2.The proof of Theorem 3.1 is based on the following difference inequality ofM. Nakao [7].Lemma 3.1 (Nakao) Let φ : R+ → R be a bounded nonnegative functionfor which there exist constants β > 0 and γ ≥ 0 such thatsupt≤s≤t+1(φ(s))1+γ ≤ β(φ(t) − φ(t + 1)) ∀ t ≥ 0.Then(i) If γ = 0, there exist positive constants C and θ such thatφ(t) ≤ C exp(−θt) ∀ t ≥ 0.(ii) If γ > 0, there exists a positive constant C such thatφ(t) ≤ C(1 + t)−1γ ∀ t ≥ 0.EJQTDE, 1999 No. 11, p. 8Let us define the approximate energy of the system (1.1) byEm(t) =12‖u′m(t)‖22 +1p‖um(t)‖p1,p +∫ΩG(x, um(t))dx. (3.2)Then the following two lemmas hold.Lemma 3.2 There exists a constant k > 0 such thatkEm(t) ≤ ‖um(t)‖p1,p +∫Ωg(x, um(t))um(t)dx. (3.3)Proof. In the case (a), condition (3.3) is a direct consequence of (3.1). Inthe case (b) the result follows from (2.27). In fact, from assumption (2.1)with b = 0 we have∫Ω|G(x, um)|dx ≤aCσσ‖um‖σ1,p and∫Ω|g(x, um)um|dx ≤ aCσ‖um‖σ1,p.Then given δ > 0,‖um‖p1,p +∫Ωg(x, um)umdx =δp‖um‖p1,p + δ∫ΩG(x, um)dx−δ∫ΩG(x, um)dx + (1 −δp)‖um‖p1,p +∫Ωg(x, um)um dximplies that‖um(t)‖p1,p +∫Ωg(x, um(t))um(t)dx ≥δp‖um(t)‖p1,p + δ∫ΩG(x, um(t))dx+(1 − δp)‖um(t)‖p1,p − (1 +δσ)aCσ‖um(t)‖σ1,p.Now, since ‖um(t)‖1,p ≤ (2aCσ)−1σ−p = z0 uniformly in t,m, we have that‖um(t)‖σ−p1,p ≤ (2aCσ)−1. Then taking δ ≤ (σp)(p+2σ)−1, we conclude that(1 − δp)‖um(t)‖p1,p − (1 +δσ)aCσ‖um(t)‖σ1,p=[(1 − δp) − (1 + δσ)aCσ‖um(t)‖σ−p1,p]‖um(t)‖p1,p ≥ 0.EJQTDE, 1999 No. 11, p. 9This implies that‖um(t)‖p1,p +∫Ωg(x, um(t))um(t)dx ≥δp‖um(t)‖p1,p + δ∫ΩG(x, um(t))dxand therefore (3.3) holds. @ALemma 3.3 For any t > 0,Em(t) ≥12‖u′m(t)‖22 +12p‖um(t)‖p1,p.Proof. In the case (a), the Lemma is a consequence of (3.1). In the case (b),we use again the smallness of the approximate solutions.Em(t) ≥12‖u′m(t)‖22 +1p‖um(t)‖p1,p −aCσσ‖um(t)‖σ1,p=12‖u′m(t)‖22 +12p‖um(t)‖p1,p − Q(‖um(t)‖1,p).Since Q(‖um(t)‖1,p) ≥ 0, the result follows. @AProof of Theorem 3.1: We first obtain uniform estimates for the approx-imate energy (3.2). Fix an arbitrary t > 0, we get from the approximateproblem (2.8) with f = 0 and v = u′m(t)Em(t) +∫ t0‖(−∆)α2 u′m(s)‖22ds = Em(0),from whereE′m(t) + ‖(−∆)α2 u′m(t)‖22 = 0. (3.4)Integrating (3.4) from t to t + 1 and puttingD2m(t) = Em(t) − Em(t + 1)we have in view of (2.25)D2m(t) =∫ t+1t‖(−∆)α2 u′m(s)‖22 ds ≥ λα1∫ t+1t‖u′m(s)‖22 ds. (3.5)By applying the Mean Value Theorem in (3.5), there exist t1 ∈ [t, t+ 14 ] andt2 ∈ [t + 34 , t + 1] such that‖u′m(ti)‖2 ≤√2λ−α21 Dm(t) i = 1, 2. (3.6)EJQTDE, 1999 No. 11, p. 10Now, integrating the approximate equation with v = um(t) over [t1, t2], weinfer from Lemma 3.2 thatk∫ t2t1Em(s) ds ≤ (u′m(t1), um(t1)) − (u′m(t2), um(t2))+∫ t2t1‖u′m(s)‖22 ds −∫ t2t1((−∆)α2 u′m(s), (−∆)α2 um(s))ds.Then from Lemma 3.3, Hölder’s inequality and Sobolev embeddings, wehave in view of (3.5) and (3.6), there exist positive constants C1 and C2such that∫ t2t1Em(s)ds ≤ C1D2m(t) + C2Dm(t)E1pm(t).By the Mean Value Theorem, there exists t∗ ∈ [t1, t2] such thatEm(t∗) ≤ C1D2m(t) + C2Dm(t)E1pm(t).The monotonicity of Em implies thatEm(t + 1) ≤ C1D2m(t) + C2Dm(t)E1pm(t).Since Em(t + 1) = Em(t) − D2m(t), we conclude thatEm(t) ≤ (C1 + 1)D2m(t) + C2Dm(t)E1pm(t).Using Young’s inequality, there exist positive constants C3 and C4 such thatEm(t) ≤ C3D2m(t) + C4Dpp−1m (t). (3.7)If p = 2 thenEm(t) ≤ (C3 + C4)D2m(t)and since Em is decreasing, then from Lemma 3.1 there exist positive con-stants C and θ (independent of m) such thatEm(t) ≤ C exp(−θt) ∀ t > 0. (3.8)If p > 2, then relation (3.7) and the boundedness of Dm(t) show the existenceof C5 > 0 such thatEm(t) ≤ C5Dpp−1m (t),EJQTDE, 1999 No. 11, p. 11and thenE2(p−1)pm (t) ≤ C2(p−1)p5 D2m(t).Applying Lemma 3.1, with γ = (p − 2)/p, there exists a constant C > 0(independent of m) such thatEm(t) ≤ C(1 + t)−pp−2 ∀ t ≥ 0. (3.9)Finally we pass to the limit (3.8) and (3.9) and the proof is complete in viewof Lemma 3.3. @AAcknowledgement: The authors thank Professor J. E. Muñoz Rivera forhis hospitality during their stay in LNCC/CNPq.References[1] D. D. Ang & A. Pham Ngoc Dinh; Strong solutions of a quasilin-ear wave equation with nonlinear damping, SIAM J. Math. Anal. 19,(1988), 337-347 .[2] A. C. Biazutti; On a nonlinear evolution equation and its applications,Nonlinear Anal. T. M. A. 24, (1995), 1221-1234 .[3] J. Greenberg, R. C. MacCamy & V. J. Mizel; On the existence, unique-ness and stability of solutions of the equation σ ′(ux)uxx + λuxtx =ρ0utt, J. Math. Mech. 17, (1968), 707-728 .[4] J. L. Lions; Quelques méthodes de résolution des problèmes aux limitesnon linéaires, Dunod-Gauthier Villars, Paris, 1969.[5] T. F. Ma & J. A. Soriano; On weak solutions for an evolution equationwith exponential nonlinearities, Nonlinear Anal. T. M. A. 37, (1999),1029-1038.[6] L. A. Medeiros & M. Milla Miranda; On a nonlinear wave equationwith damping, Rev. Mat. Univ. Complutense de Madrid 3, (1990),213-231.[7] M. Nakao; A difference inequality and its applications to nonlinearevolution equations, J. Math. Soc. Japan 30, (1978), 747-762 .[8] M. Nakao; Energy decay for the quasilinear wave equation with vis-cosity, Math Z. 219, (1992), 289-299 .EJQTDE, 1999 No. 11, p. 12[9] L. Tartar; Topics in Nonlinear Analysis, Publications Mathématiquesd’Orsay, Université Paris-Sud, Orsay, 1978.[10] G. F. Webb; Existence and asymptotic for a strongly damped nonlin-ear wave equation, Canadian J. Math. 32, (1980), 631-643 .[11] Y. Yamada; Some remarks on the equation ytt − σ(yx)yxx − yxtx = f ,Osaka J. Math. 17, (1980), 303-323 .EJQTDE, 1999 No. 11, p. 13

Global solutions for a nonlinear wave equation with the p-Laplacian operator

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Global Solutions for a Nonlinear Wave Equationwith the p-Laplacian Operator ∗Hongjun GaoInstitute of Applied Physics and Computational Mathematics100088 Beijing, ChinaTo Fu MaDepartamento de Matema´tica - Universidade Estadual de Maringa´87020-900 Maringa´-PR, BrazilAbstractWe study the existence and asymptotic behaviour of the globalsolutions of the nonlinear equationutt −∆pu + (−∆)αut + g(u) = fwhere 0 < α ≤ 1 and g does not satisfy the sign condition g(u)u ≥ 0.Key words: Quasilinear hyperbolic equation, asymptotic behaviour, small data.AMS Subject Classification: 35B40, 35L70.1. IntroductionThe study of global existence and asymptotic behaviour for initial-boundaryvalue problems involving nonlinear operators of the typeutt −n∑i=1{σi(uxi)}xi −∆ut = f(t, x) in (0, T ) ×Ωgoes back to Greenberg, MacCamy & Mizel [3], where they considered theone-dimensional case with smooth data. Later, several papers have appearedin that direction, and some of its important results can be found in, for∗Work done while the authors were visiting LNCC/CNPq, Brazil.EJQTDE, 1999 No. 11, p. 1example, Ang & Pham Ngoc Dinh [1], Biazutti [2], Nakao [8], Webb [10]and Yamada [11]. In all of the above cited papers, the damping term −∆utplayed an essential role in order to obtain global solutions. Our objective isto study this kind of equations under a weaker damping given by (−∆)αutwith 0 < α ≤ 1. This approach was early considered by Medeiros andMilla Miranda [6] to Kirchhoff equations. We also consider the presence ofa forcing term g(x, u) that does not satisfy the sign condition g(x, u)u ≥ 0.Our study is based on the pseudo Laplacian operator−∆pu = −n∑i=1∂∂xi(∣∣∣∣ ∂u∂xi∣∣∣∣p−2 ∂u∂xi)which is used as a model for several monotone hemicontinuous operators.Let Ω be a bounded domain in Rn with smooth boundary ∂Ω. Weconsider the nonlinear initial-boundary value problem(1.1)utt −∆pu + (−∆)αut + g(x, u) = f(t, x) in (0, T )× Ω,u = 0 on (0, T ) × ∂Ω,u(0, x) = u0 and ut(0, x) = u1 in Ω,where 0 < α ≤ 1 and p ≥ 2. We prove that, depending on the growth ofg, problem (1.1) has a global weak solution without assuming small initialdata. In addition, we show the exponential decay of solutions when p = 2and algebraic decay when p > 2. The global solutions are constructedby means of the Galerkin approximations and the asymptotic behaviour isobtained by using a difference inequality due to M. Nakao [7]. Here we onlyuse standard notations. We often write u(t) instead u(t, x) and u′(t) insteadut(t, x). The norm in Lp(Ω) is denoted by ‖ · ‖p and in W 1,p0 (Ω) we use thenorm‖u‖p1,p =n∑j=1‖uxj‖pp.For the reader’s convenience, we recall some of the basic properties of theoperators used here. The degenerate operator −∆p is bounded, monotoneand hemicontinuous from W 1,p0 (Ω) to W−1,q(Ω), where p−1 + q−1 = 1. Thepowers for the Laplacian operator is defined by(−∆)αu =∞∑j=1λαj (u, ϕj)ϕj ,where 0 < λ1 < λ2 ≤ λ3 ≤ · · · and ϕ1, ϕ2, ϕ3, · · · are, respectively, thesequence of the eigenvalues and eigenfunctions of −∆ in H 10 (Ω). Then‖u‖D((−∆)α) = ‖(−∆)αu‖2 ∀u ∈ D((−∆)α)EJQTDE, 1999 No. 11, p. 2and D((−∆)α) ⊂ D((−∆)β) compactly if α > β ≥ 0. In particular, forp ≥ 2 and 0 < α ≤ 1, W 1,p0 (Ω) ↪→ D((−∆)α/2) ↪→ L2(Ω).2. Existence of Global SolutionsLet g : Ω×R → R be a continuous function satisfying the growth condition|g(x, u)| ≤ a|u|σ−1 + b ∀(x, u) ∈ Ω× R, (2.1)where a, b are positive constants, 1 < σ < pn/(n−p) if n > p and 1 < σ < ∞if n ≤ p.Theorem 2.1 Let us assume condition (2.1) with σ < p. Then given u0 ∈W 1,p0 (Ω), u1 ∈ L2(Ω) and f ∈ L2(0, T ;L2(Ω)), there exists a function u :(0, T ) × Ω → R such thatu ∈ L∞(0, T ;W 1,p0 (Ω)), (2.2)u′ ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;D((−∆)α2 )), (2.3)u(0) = u0 and u′(0) = u1 a.e. in Ω, (2.4)utt −∆pu + (−∆)αut + g(x, u) = f in L2(0, T ;W−1,q(Ω)), (2.5)where p−1 + q−1 = 1.Next we consider an existence result when σ ≥ p. In this case, the globalsolution is obtained with small initial data. For each m ∈ N we putγm =12‖u1m‖22 +32p‖u0m‖p1,p +aCσσ‖u0m‖σ1,p +2( p√2bC1)qqwhere Ck > 0 is the Sobolev constant for the inequality ‖u‖k ≤ Ck‖u‖1,p,when W 1,p0 (Ω) ↪→ Lk(Ω). We also define the polynomial Q byQ(z) =12pzp − aCσσzσ,which is increasing in [0, z0], wherez0 = (2aCσ)−1σ−pis its unique local maximum. We will assume that‖u0‖1,p < z0 (2.6)EJQTDE, 1999 No. 11, p. 3andγ +14λα1∫ T0‖f(t)‖22dt < Q(z0), (2.7)where γ = limm→∞ γm.Theorem 2.2 Suppose that condition (2.1) holds with σ > p. Suppose inaddition that inital data satisfy (2.6) and (2.7). Then there exists a functionu : (0, T ) × Ω → R satisfying (2.2)-(2.5).Proof of Theorem 2.1: Let r be an integer for which H r0(Ω) ↪→ W 1,p0 (Ω)is continuous. Then the eigenfunctions of −∆rwj = αjwj in Hr0(Ω) yieldsa “Galerkin” basis for both Hr0(Ω) and L2(Ω). For each m ∈ N, let us putVm = Span{w1, w2, · · · , wm}. We search for a functionum(t) =m∑j=1kjm(t)wjsuch that for any v ∈ Vm, um(t) satisfies the approximate equation∫Ω{u′′m(t)−∆pum(t) + (−∆)αu′m(t) + g(x, um(t))− f(t, x)}v dx = 0 (2.8)with the initial conditionsum(0) = u0m and u′m(0) = u1m,where u0m and u1m are chosen in Vm so thatu0m → u0 in W 1,p0 (Ω) and u1m → u1 in L2(Ω). (2.9)Putting v = wj , j = 1, · · · ,m, we observe that (2.8) is a system of ODEsin the variable t and has a local solution um(t) in a interval [0, tm). In thenext step we obtain the a priori estimates for the solution um(t) so that itcan be extended to the whole interval [0, T ].A Priori Estimates: We replace v by u′m(t) in the approximate equation(2.8) and after integration we have12‖u′m(t)‖22 +1p‖um(t)‖p1,p +∫ t0‖(−∆)α2 u′m(s)‖22 ds +∫ΩG(x, um(t))dx≤∫ t0‖f(s)‖2‖u′m(s)‖2 ds +12‖u1m‖22 +1p‖u0m‖p1,p +∫ΩG(x, u0)dx,EJQTDE, 1999 No. 11, p. 4where G(x, u) =∫ u0 g(x, s) ds. Now from growth condition (2.1) and theSobolev embedding, we have that∫Ω|G(x, um(t))|dx ≤ aσCσ‖um(t)‖σ1,p + bC1‖um(t)‖1,p. (2.10)But since p > σ, there exists a constant C > 0 such that∫Ω|G(x, um)|dx ≤ 12p‖um(t)‖p1,p + C,and then we have12‖u′m(t)‖22 +12p‖um(t)‖p1,p +∫ t0‖(−∆)α2 u′m(s)‖22 ds≤∫ t0‖f(s)‖2‖u′m(s)‖2 ds +12‖u1m‖22 +32p‖u0m‖p1,p + 2C.Using the convergence (2.9) and the Gronwall’s lemma, there exists a con-stant C > 0 independent of t,m such that‖u′m(t)‖22 + ‖um(t)‖p1,p +∫ t0‖(−∆)α2 u′m(s)‖22 ds ≤ C. (2.11)With this estimate we can extend the approximate solutions um(t) to theinterval [0, T ] and we have that(um) is bounded in L∞(0, T ;W 1,p0 (Ω)), (2.12)(u′m) is bounded in L∞(0, T ;L2(Ω)), (2.13)(u′m) is bounded in L2(0, T ;D((−∆)α2 )), (2.14)and(−∆pum) is bounded in L∞(0, T ;W−1,q(Ω)). (2.15)Now we are going to obtain an estimate for (u′′m). Since our Galerkin basiswas taken in the Hilbert space Hr(Ω) ⊂ W 1,p0 (Ω), we can use the standardprojection arguments as described in Lions [4]. Then from the approximateequation and the estimates (2.12)-(2.15) we get(u′′m) is bounded in L2(0, T ;H−r(Ω)). (2.16)EJQTDE, 1999 No. 11, p. 5Passage to the Limit: From (2.12)-(2.14), going to a subsequence if neces-sary, there exists u such thatum ⇀ u weakly star in L∞(0, T ;W 1,p0 (Ω)), (2.17)u′m ⇀ u′ weakly star in L∞(0, T ;L2(Ω)), (2.18)u′m ⇀ u′ weakly in L2(0, T ;D((−∆)α2 ), (2.19)and in view of (2.15) there exists χ such that−∆pum ⇀ χ weakly star in L∞(0, T ;W−1,q(Ω)). (2.20)By applying the Lions-Aubin compactness lemma we get from (2.12)-(2.13)um → u strongly in L2(0, T ;L2(Ω)), (2.21)and since D((−∆)α2 ) ↪→ L2(Ω) compactly, we get from (2.14) and (2.16)u′m → u′ strongly in L2(0, T ;L2(Ω)). (2.22)Using the growth condition (2.1) and (2.21) we see that∫ T0∫Ω|g(x, um(t, x))|σσ−1 dxdtis bounded andg(x, um) → g(x, u) a.e. in (0, T ) × Ω.Therefore from Lions [4] (Lemma 1.3) we infer thatg(x, um) ⇀ g(x, u) weakly in Lσσ−1 (0, T ;Lσσ−1 (Ω)). (2.23)With these convergence we can pass to the limit in the approximate equationand thenddt(u′(t), v) + 〈χ(t), v〉 + ((−∆)αu′(t), v) + (g(x, u(t)), v) = (f(t), v) (2.24)for all v ∈ W 1,p(Ω), in the sense of distributions. This easily implies that(2.2)-(2.4) hold. Finally, since we have the strong convergence (2.22), wecan use a standard monotonicity argument as done in Biazutti [2] or Ma &Soriano [5] to show that χ = −∆pu. This ends the proof. @AEJQTDE, 1999 No. 11, p. 6Proof of Theorem 2.2: We only show how to obtain the estimate (2.11).The remainder of the proof follows as before. We apply an argument madeby L. Tartar [9]. From (2.10) we have∫Ω|G(x, um)|dx ≤ aCσσ‖um‖σ1,p +12p‖um‖p1,p +( p√2bC1)qqand therefore12‖u′m(t)‖22 + Q(‖um(t)‖1,p) +∫ t0‖(−∆)α2 u′m(s)‖22 ds≤ γm +∫ t0‖f(s)‖2‖u′m(s)‖2 ds.Since‖(−∆)α2 u′m(s)‖2 ≥ λα21 ‖u′m(t)‖2, (2.25)where λ1 is the first eigenvalue of −∆ in H10 (Ω), this implies that12‖u′m(t)‖22 + Q(‖um(t)‖1,p) ≤ γm +14λα1∫ T0‖f(t)‖22 dt. (2.26)We claim that there exists an integer N such that‖um(t)‖1,p < z0 ∀ t ∈ [0, tm) m > N (2.27)Suppose the claim is proved. Then Q(‖um(t)‖1,p) ≥ 0 and from (2.7) and(2.26), ‖u′m(t)‖2 is bounded and consequently (2.11) follows.Proof of the Claim: Suppose (2.27) false. Then for each m > N , there existst ∈ [0, tm) such that ‖um(t)‖1,p ≥ z0. We note that from (2.6) and (2.9)there exists N0 such that‖um(0)‖1,p < z0 ∀m > N0.Then by continuity there exists a first t∗m ∈ (0, tm) such that‖um(t∗m)‖1,p = z0, (2.28)from whereQ(‖um(t)‖1,p) ≥ 0 ∀ t ∈ [0, t∗m].Now from (2.7) and (2.26), there exist N > N0 and β ∈ (0, z0) such that0 ≤ 12‖u′m(t)‖22 + Q(‖um(t)‖1,p) ≤ Q(β) ∀ t ∈ [0, t∗m] ∀m > N.EJQTDE, 1999 No. 11, p. 7Then the monotonicity of Q in [0, z0] implies that0 ≤ ‖um(t)‖1,p ≤ β < z0 ∀t ∈ [0, t∗m],and in particular, ‖um(t∗m)‖1,p < z0, which is a contradiction to (2.28). @ARemarks: From the above proof we have the following immediate conclu-sion: The smallness of initial data can be dropped if either condition (2.1)holds with σ = p and coefficient a is sufficiently small, or σ > p and the signcondition g(x, u)u ≥ 0 is satisfied. @A3. Asymptotic BehaviourTheorem 3.1 Let u be a solution of Problem (1.1) given by(a) either Theorem 2.1 with the additional assumption: there exists ρ > 0such thatg(x, u)u ≥ ρG(x, u) ≥ 0, (3.1)(b) or Theorem 2.2.Then there exists positive constants C and θ such that‖u′(t)‖22 + ‖u(t)‖p1,p ≤ C exp(−θt) if p = 2,or‖u′(t)‖22 + ‖u(t)‖p1,p ≤ C(1 + t)−pp−2 if p > 2.The proof of Theorem 3.1 is based on the following difference inequality ofM. Nakao [7].Lemma 3.1 (Nakao) Let φ : R+ → R be a bounded nonnegative functionfor which there exist constants β > 0 and γ ≥ 0 such thatsupt≤s≤t+1(φ(s))1+γ ≤ β(φ(t)− φ(t + 1)) ∀ t ≥ 0.Then(i) If γ = 0, there exist positive constants C and θ such thatφ(t) ≤ C exp(−θt) ∀ t ≥ 0.(ii) If γ > 0, there exists a positive constant C such thatφ(t) ≤ C(1 + t)−1γ ∀ t ≥ 0.EJQTDE, 1999 No. 11, p. 8Let us define the approximate energy of the system (1.1) byEm(t) =12‖u′m(t)‖22 +1p‖um(t)‖p1,p +∫ΩG(x, um(t))dx. (3.2)Then the following two lemmas hold.Lemma 3.2 There exists a constant k > 0 such thatkEm(t) ≤ ‖um(t)‖p1,p +∫Ωg(x, um(t))um(t)dx. (3.3)Proof. In the case (a), condition (3.3) is a direct consequence of (3.1). Inthe case (b) the result follows from (2.27). In fact, from assumption (2.1)with b = 0 we have∫Ω|G(x, um)|dx ≤ aCσσ‖um‖σ1,p and∫Ω|g(x, um)um|dx ≤ aCσ‖um‖σ1,p.Then given δ > 0,‖um‖p1,p +∫Ωg(x, um)umdx =δp‖um‖p1,p + δ∫ΩG(x, um)dx−δ∫ΩG(x, um)dx + (1− δp)‖um‖p1,p +∫Ωg(x, um)um dximplies that‖um(t)‖p1,p +∫Ωg(x, um(t))um(t)dx ≥ δp‖um(t)‖p1,p + δ∫ΩG(x, um(t))dx+(1− δp)‖um(t)‖p1,p − (1 +δσ)aCσ‖um(t)‖σ1,p.Now, since ‖um(t)‖1,p ≤ (2aCσ)−1σ−p = z0 uniformly in t,m, we have that‖um(t)‖σ−p1,p ≤ (2aCσ)−1. Then taking δ ≤ (σp)(p+2σ)−1, we conclude that(1− δp)‖um(t)‖p1,p − (1 +δσ)aCσ‖um(t)‖σ1,p=[(1− δp)− (1 + δσ)aCσ‖um(t)‖σ−p1,p]‖um(t)‖p1,p ≥ 0.EJQTDE, 1999 No. 11, p. 9This implies that‖um(t)‖p1,p +∫Ωg(x, um(t))um(t)dx ≥ δp‖um(t)‖p1,p + δ∫ΩG(x, um(t))dxand therefore (3.3) holds. @ALemma 3.3 For any t > 0,Em(t) ≥ 12‖u′m(t)‖22 +12p‖um(t)‖p1,p.Proof. In the case (a), the Lemma is a consequence of (3.1). In the case (b),we use again the smallness of the approximate solutions.Em(t) ≥ 12‖u′m(t)‖22 +1p‖um(t)‖p1,p −aCσσ‖um(t)‖σ1,p=12‖u′m(t)‖22 +12p‖um(t)‖p1,p −Q(‖um(t)‖1,p).Since Q(‖um(t)‖1,p) ≥ 0, the result follows. @AProof of Theorem 3.1: We first obtain uniform estimates for the approx-imate energy (3.2). Fix an arbitrary t > 0, we get from the approximateproblem (2.8) with f = 0 and v = u′m(t)Em(t) +∫ t0‖(−∆)α2 u′m(s)‖22ds = Em(0),from whereE′m(t) + ‖(−∆)α2 u′m(t)‖22 = 0. (3.4)Integrating (3.4) from t to t + 1 and puttingD2m(t) = Em(t)−Em(t + 1)we have in view of (2.25)D2m(t) =∫ t+1t‖(−∆)α2 u′m(s)‖22 ds ≥ λα1∫ t+1t‖u′m(s)‖22 ds. (3.5)By applying the Mean Value Theorem in (3.5), there exist t1 ∈ [t, t+ 14 ] andt2 ∈ [t + 34 , t + 1] such that‖u′m(ti)‖2 ≤√2λ−α21 Dm(t) i = 1, 2. (3.6)EJQTDE, 1999 No. 11, p. 10Now, integrating the approximate equation with v = um(t) over [t1, t2], weinfer from Lemma 3.2 thatk∫ t2t1Em(s) ds ≤ (u′m(t1), um(t1))− (u′m(t2), um(t2))+∫ t2t1‖u′m(s)‖22 ds−∫ t2t1((−∆)α2 u′m(s), (−∆)α2 um(s))ds.Then from Lemma 3.3, Ho¨lder’s inequality and Sobolev embeddings, wehave in view of (3.5) and (3.6), there exist positive constants C1 and C2such that ∫ t2t1Em(s)ds ≤ C1D2m(t) + C2Dm(t)E1pm(t).By the Mean Value Theorem, there exists t∗ ∈ [t1, t2] such thatEm(t∗) ≤ C1D2m(t) + C2Dm(t)E1pm(t).The monotonicity of Em implies thatEm(t + 1) ≤ C1D2m(t) + C2Dm(t)E1pm(t).Since Em(t + 1) = Em(t)−D2m(t), we conclude thatEm(t) ≤ (C1 + 1)D2m(t) + C2Dm(t)E1pm(t).Using Young’s inequality, there exist positive constants C3 and C4 such thatEm(t) ≤ C3D2m(t) + C4Dpp−1m (t). (3.7)If p = 2 thenEm(t) ≤ (C3 + C4)D2m(t)and since Em is decreasing, then from Lemma 3.1 there exist positive con-stants C and θ (independent of m) such thatEm(t) ≤ C exp(−θt) ∀ t > 0. (3.8)If p > 2, then relation (3.7) and the boundedness of Dm(t) show the existenceof C5 > 0 such thatEm(t) ≤ C5Dpp−1m (t),EJQTDE, 1999 No. 11, p. 11and thenE2(p−1)pm (t) ≤ C2(p−1)p5 D2m(t).Applying Lemma 3.1, with γ = (p − 2)/p, there exists a constant C > 0(independent of m) such thatEm(t) ≤ C(1 + t)−pp−2 ∀ t ≥ 0. (3.9)Finally we pass to the limit (3.8) and (3.9) and the proof is complete in viewof Lemma 3.3. @AAcknowledgement: The authors thank Professor J. E. Mun˜oz Rivera forhis hospitality during their stay in LNCC/CNPq.References[1] D. D. Ang & A. Pham Ngoc Dinh; Strong solutions of a quasilin-ear wave equation with nonlinear damping, SIAM J. Math. Anal. 19,(1988), 337-347 .[2] A. C. Biazutti; On a nonlinear evolution equation and its applications,Nonlinear Anal. T. M. A. 24, (1995), 1221-1234 .[3] J. Greenberg, R. C. MacCamy & V. J. Mizel; On the existence, unique-ness and stability of solutions of the equation σ ′(ux)uxx + λuxtx =ρ0utt, J. Math. Mech. 17, (1968), 707-728 .[4] J. L. Lions; Quelques me´thodes de re´solution des proble`mes aux limitesnon line´aires, Dunod-Gauthier Villars, Paris, 1969.[5] T. F. Ma & J. A. Soriano; On weak solutions for an evolution equationwith exponential nonlinearities, Nonlinear Anal. T. M. A. 37, (1999),1029-1038.[6] L. A. Medeiros & M. Milla Miranda; On a nonlinear wave equationwith damping, Rev. Mat. Univ. Complutense de Madrid 3, (1990),213-231.[7] M. Nakao; A difference inequality and its applications to nonlinearevolution equations, J. Math. Soc. Japan 30, (1978), 747-762 .[8] M. Nakao; Energy decay for the quasilinear wave equation with vis-cosity, Math Z. 219, (1992), 289-299 .EJQTDE, 1999 No. 11, p. 12[9] L. Tartar; Topics in Nonlinear Analysis, Publications Mathe´matiquesd’Orsay, Universite´ Paris-Sud, Orsay, 1978.[10] G. F. Webb; Existence and asymptotic for a strongly damped nonlin-ear wave equation, Canadian J. Math. 32, (1980), 631-643 .[11] Y. Yamada; Some remarks on the equation ytt − σ(yx)yxx − yxtx = f ,Osaka J. Math. 17, (1980), 303-323 .EJQTDE, 1999 No. 11, p. 13

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Global solutions for a nonlinear wave equation with the p-laplacian operator

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