Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 21, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLUTIONS OF KIRCHHOFF PLATE EQUATIONS WITH INTERNAL DAMPING AND LOGARITHMIC NONLINEARITY DUCIVAL PEREIRA, SEBASTIÃO CORDEIRO, CARLOS RAPOSO, CELSA MARANHÃO Abstract. In this article we study the existence of weak solutions for the nonlinear initial boundary value problem of the Kirchhoff equation utt + ∆2u+M(‖∇u‖2)(−∆u) + ut = u ln |u|2, in Ω× (0, T ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂u ∂η (x, t) = 0, x ∈ ∂Ω, t ≥ 0, where Ω is a bounded domain in R2 with smooth boundary ∂Ω, T > 0 is a fixed but arbitrary real number, M(s) is a continuous function on [0,+∞) and η is the unit outward normal on ∂Ω. Our results are obtained using the Galerkin method, compactness approach, potential well corresponding to the logarithmic nonlinearity, and the energy estimates due to Nakao. 1. Introduction In this article we study the existence and decay properties of global solutions for the nonlinear initial boundary value problem utt + ∆2u+M(‖∇u‖2)(−∆u) + ut = u ln |u|2, in Ω× (0, T ), (1.1) u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, (1.2) u(x, t) = ∂u ∂η (x, t) = 0, x ∈ ∂Ω, t ≥ 0, (1.3) where Ω is a bounded domain in R2 with smooth boundary ∂Ω, T > 0 is a fixed but arbitrary real number, M(s) is a continuous function on [0,+∞) and η is the unit outward normal on ∂Ω. The boundary conditions (1.3) mean that boundary is clamped. We do not imposed a priori conditions on the function space, and it turns out that a weak solution automatically satisfies the boundary conditions. The physical origin of this problem without logarithmic source term leads to the study of dynamic buckling of the hinged extensible beam which is either stretched or compressed by an axial force. The readers can see in Burgreen [10] and Eisley [16] for more physical justifications and the model background. From the mathematical point of view, we cite the pioneer works of Kirchhoff [21], Woinowsky-Krieger [33] and Berger [7]. For logarithmic source term, to the best of our knowledge, the first 2010 Mathematics Subject Classification. 35L15, 35L70, 35B40, 35A01. Key words and phrases. Extensible beam; existence of solutions; asymptotic behavior; logarithmic source term. c©2021 Texas State University. Submitted May 19, 2020. Published March 29, 2021. 1 2 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 contribution in literature was given by Birula and Mycielski [9], where they studied the problem utt − uxx + u = εu ln |u|2, (x, t) ∈ [a, b]× (0, T ), u(x, 0) = u0(x), u′(x, 0) = u1(x), x ∈ [a, b], u(a, t) = u(b, t) = 0, t ∈ (0, T ). (1.4) Problem (1.1)-(1.3) is also called nonlocal because of the presence of the term M ( ‖∇u(t)‖2 ) = M (∫ Ω |∇u(x, t)|2dx ) , which implies that the equation is no longer has a pointwise dependence. The non- local term provokes some mathematical difficulties which make the study of such a problem particularly interesting. See the work of Arosio-Panizzi [3]. Nonlocal initial boundary value problems are important in the framework of their practical application to the modeling and investigation of various phenomena. In particu- lar, the type of problems (1.1)-(1.3) has applications in nuclear physics, optics and geophysics, see for instance [6, 9, 18]. With logarithmic nonlinearity u ln |u|2 it ap- pears naturally in inflation cosmology and supersymmetric field theories, quantum mechanics and nuclear physics, see [5, 17]. Now, we focus on a chronological literature overview. The one-dimensional non- linear equation (1.5) of motion of an elastic string was proposed by Kirchhoff (1883) [21], in connection with some problems in nonlinear elasticity, and rediscovered by Carrier (1945) [11], ∂2u ∂t2 − (τ0 m + k 2mL ∫ L 0 (∂u ∂x )2 dx )∂2u ∂x2 = 0, (1.5) where τ0 is the initial tension, m the mass of the string and k the Young’s modulus of the material of the string. This model describes small vibrations of a stretched string of the length L when only the transverse component of the tension is con- sidered. For mathematical aspects of (1.5) see Bernstein (1940) [8]. The model (1.5) is a generalization of the linearized problem ∂2u ∂t2 − τ0 m ∂2u ∂x2 = 0, obtained by Euler (1707− 1783) and d’Alembert (1714− 1793). A particular case of (1.5) can be written, in general, as ∂2u ∂t2 −M (∫ Ω |∇u(x, t)|2dx ) ∆u = 0, (1.6) or ∂2u ∂t2 +M ( ‖u(t)‖2 ) Au = 0, (1.7) in operator notation, where we consider the Hilbert spaces V ↪→ H ↪→ V ′, where V ′ is the dual of V with the immersions continuous and dense. By ‖ · ‖ we denote the norm in V and A : V → V ′ a bounded linear operator. For M : [0,∞) → R real function, M(λ) ≥ m0 > 0, M ∈ C1(0,∞), the global solution for operator given in (1.7) can be found in Lions (1969) [22]. Later, Po- hozhaev (1974) [30] proved that the mixed problem for (1.6) has global solution in t when the initial data u(x, 0), ut(x, 0) are restricted the class of functions called Pohozhaev’s Class. For the case M(λ) ≥ 0 we cite the works of Hazoya-Yamada EJDE-2021/21 KIRCHHOFF PLATE EQUATION 3 (1991) [20], Arosio-Spagnolo (1996) [4] and Medeiros-Ĺımaco-Menezes (2002) [23] with reference therein. Considering Ω a bounded domain in R2, Cavalcanti et al. (2004) [13], studied the equation utt + ∆2u+M(|∇u|2)(−∆u) + g(ut) + f(u) = 0 (1.8) with g(s) = |s|ρ−1s and f(s) = |s|γ−1s where ρ and γ are positive constants such that 1 < ρ, γ ≤ n/(n − 2) if n ≥ 3; ρ, γ > 1 if n = 1, 2. The global existence and asymptotic stability were obtained using the fixed point theorem and continuity arguments. The problem studied in (1.8) was investigated more generally by Zhijian (2013) [36] as follows utt + ∆2u+M(|∇u|2)(−∆u) + g(ut) + f(u) = h(x) (1.9) where the source terms f, g ∈ C1(R), |f ′(s)| ≤ C(1 + |s|p−1) and K0|s|q−1 < g′(s) ≤ C(1 + |s|q−1), K0, C > 0 with 1 ≤ p < ∞, 1 ≤ q < ∞ if n ≤ 4; 1 ≤ p ≤ p∗ = (n + 4)/(n − 4) and p ≤ q if N ≥ 5. By Galerkin approximation combined with the monotone arguments, the author proved the existence of a global solution. Milla Miranda et al. (2017) [26], investigated the existence and uniqueness of local solutions of the initial value problem for the nonlinear mixed problem 1.10 of Kirchhoff type, u′′ −M ( t, ∫ Ω |∇u|2dx ) ∆u+ |u|ρ = f, in Ω× (0, T0), u(x, 0) = u0(x), u′(x, 0) = u1(x), x ∈ Ω, u = 0 on Γ0 × (0, T0), ∂u ∂ν + δ(x)h(u′) = 0 on Γ1 × (0, T0), (1.10) where Ω is a bounded domain of Rn with its boundary consisting of two disjoint parts Γ0 and Γ1; ρ > 1 is a real number; ν(x) is the exterior unit normal vector at x ∈ Γ1 and δ(x), h(s) are real functions defined in Γ1 and R, respectively. The authors used the Galerkin method with a special basis, a modification of the Tartar approach, compactness method and fixed-point theorem. Mohammad et al. (2018) [1], by using the Galerkin method, established the existence of solutions for a plate equation with nonlinear damping and a logarithmic source term and proved an explicit and general decay rate result, by using the multiplier method and some properties of the convex functions for the problem u′′ + ∆u2 + u+ h(ut) = ku ln |u|, in Ω× (0, T ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂u ∂η (x, t) = 0, x ∈ ∂Ω, t ≥ 0, (1.11) where Ω is a bounded domain of R2 with a smooth boundary ∂Ω. 4 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 For the study of an extensible beam equation with internal damping and source terms, Pereira et al. (2019) [28], considered the nonlinear beam equation utt + ∆2u+M(|∇u|2)(−∆u) + ut = |u|r−1u, in Ω× (0, T ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂u ∂η (x, t) = 0, x ∈ ∂Ω, t ≥ 0, (1.12) where r > 1 is a real number, M(s) is a continuous function on [0,+∞) and Ω is a bounded domain in Rn with smooth boundary ∂Ω. The authors constructed the global solutions by using the Faedo-Galerkin approximations, taking the initial data in an appropriate set for the stability created from the Nehari manifold. The asymptotic behavior was obtained by the Nakao’s method. Meng et al. (2020) [24], used the Nehari manifold, Ekeland variational principle, and the theory of Lagrange multipliers, to prove that there are at least two positive solutions for the nonlocal biharmonic equation of Kirchhoff type involving concave- convex nonlinearities. They considered the system ∆2u− ( a+ b ∫ RN |∇u|2dx ) ∆u+ V (x)u = λf1(x)|u|q−2u+ f2(x)|u|p−2u. Regarding system (1.12) Pereira et al. (2021) [29], taking initial data suitable for the stability created from the Nehari manifold, proved the existence of global solutions and energy decay estimate when the internal damping is |ut|p−1ut. This article is organized as follows. In section 2 we present some hypothesis needed in the proof of our results. In section 3 we construct global weak solutions by means of the Galerkin approximations. In section 4 we present the potential well corresponding to the logarithmic nonlinearity. In section 5 we apply the results due to Nakao [27] to prove the exponential decay of solutions. 2. Preliminaries In this section, we present some material needed in the proof of our results. For simplicity of notation we denote by ‖ · ‖ the norm in the Lebesgue space L2(Ω), and by ‖ · ‖2 the norm in the Sobolev space H2 0 (Ω). We consider the hypothesis (H1) M ∈ C([0,∞)) with M(λ) ≥ −β, for all λ ≥ 0, 0 < β < λ1, where λ1 is the first eigenvalue of the problem ∆2u = λ(−∆u). Remark 2.1 (see Miklin [25]). The first eigenvalue λ1 of ∆2u = λ(−∆u) with the clamped boundary conditions u ∣∣ ∂Ω = 0, ∂u ∂η ∣∣ ∂Ω = 0, satisfies λ1 = inf u∈H2 0 (Ω) ‖∆u‖2 ‖∇u‖2 > 0, and ‖∇u‖2 ≤ 1 λ1 ‖∆u‖2. Now, we enunciate the preliminary results. Lemma 2.2 (Logarithmic Sobolev inequality [14, 19]). Let u be a function in H1 0 (Ω) and a > 0. Then∫ Ω u2 ln |u| dx ≤ ‖u‖2 ln ‖u‖2 + a2 2π ‖∇u‖2 − (1 + ln a)‖u‖2. (2.1) EJDE-2021/21 KIRCHHOFF PLATE EQUATION 5 Corollary 2.3. Let u be a function in H2 0 (Ω) and a > 0. Then∫ Ω u2 ln |u| dx ≤ ‖u‖2 ln ‖u‖2 + a2 2λ1π ‖∆u‖2 − (1 + ln a)‖u‖2. (2.2) Lemma 2.4 (Logarithmic Gronwall inequality [12]). Let γ ∈ L1(0, T ;R+) and c > 0. Also assume that the function w : [0, T ]→ [1,∞) satisfies w(t) ≤ c ( 1 + ∫ t 0 γ(s)w(s) lnw(s) ds ) , 0 ≤ t ≤ T . (2.3) Then w(t) ≤ c exp ( c ∫ t 0 γ(s) ds ) , 0 ≤ t ≤ T. (2.4) Lemma 2.5 (Nakao’s Lemma [27]). Suppose that φ(t) is a bounded nonnegative function on R+ satisfying ess supt≤s≤t+1 φ(s) ≤ C0[φ(t)− φ(t+ 1)], ∀t ≥ 0, where C0 is a positive constant. Then φ(t) ≤ Ce−αt for all t ≥ 0, where C and α are positive constants. 3. Existence of global weak solutions Theorem 3.1. Let u0 ∈ H1 0 (Ω), u1 ∈ L2(Ω), and assume (H1) holds. Then there exists a function u : [0, T ]→ L2(Ω) with u ∈ L∞(0, T ;H2 0 (Ω)), ut ∈ L∞(0, T ;L2(Ω)), (3.1) such that for all w ∈ H2 0 (Ω), d dt (ut(t), w) + 〈∆u(t),∆w〉+M(‖∇u(t)‖2)(−∆u(t), w) + (ut(t), w)− ( u(t) ln |u(t)|2, w ) = 0 in D′(0, T ), (3.2) u(0) = u0, ut(0) = u1. (3.3) Proof. We use Faedo-Galerkin’s method to prove the global existence of solutions. 3.1. Approximated problem. Let (wν)ν∈N be a basis of H2 0 (Ω) consisting of eigenvectors of the operator −∆ and Vm = span{w1, w2, . . . , wm}. For w ∈ Vm, let um(t) = m∑ j=1 kjm(t)wj be a solution of the approximated problem (umtt (t), w) + (∆um(t),∆w) +M(‖∇um(t)‖2)(−∆um(t), w) + (umt (t), w)− ( um(t) ln |um(t)|2, w ) = 0, (3.4) um(0) = u0m → u0 strongly in H2 0 (Ω, (3.5) umt (0) = u1m → u1 strongly in L2(Ω). (3.6) System (3.4)-(3.6) has a local solution in [0, tm), 0 < tm ≤ T , by Carathéodory’s theorem [15]. The extension of the solution to the whole interval [0, T ] is a conse- quence of a priori estimates. 6 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 3.2. A priori estimates. Replacing w = umt (t) in (3.4), we obtain d dt 1 2 ‖umt (t)‖2 + d dt 1 2 ‖∆um(t)‖2 + d dt 1 2 M̂(‖∇um(t)‖2 + d dt 1 2 ‖um(t)‖2 − d dt 1 2 ∫ Ω (um(t))2 ln |um(t)|2 dx = −‖umt (t)‖2, (3.7) where M̂(s) = ∫ s 0 M(ξ)dξ. Integrating (3.7) from 0 to t, 0 ≤ t ≤ tm, we obtain 1 2 ‖umt (t)‖2 + 1 2 ‖∆um(t)‖2 + 1 2 M̂(‖∇um(t)‖2) + 1 2 ‖um(t)‖2 + ∫ t 0 ‖umt (s)‖2ds = 1 2 ‖u1m‖2 + 1 2 ‖∆u0m‖2 + 1 2 M̂(‖∇u0m‖2)− 1 2 ∫ Ω (u0m)2 ln |u0m|2 dx (3.8) + 1 2 ∫ Ω (um(t))2 ln |um(t)|2 dx. Now, by hypothesis (H1), we have M̂(‖∇um(t)‖2) ≥ − β λ1 ‖∆um(t)‖2, (3.9) M̂(‖∇u0m‖2) ≤ m0‖∇u0m‖2 ≤ m0 λ1 ‖∆u0m‖2 (3.10) where m0 = max 0≤s≤‖∇u0m‖2≤C0 M(s), with C0 a positive constant. Replacing (3.9) and (3.10) in (3.8), and using loga- rithmic Sobolev inequality 2.1 we obtain 1 2 ‖umt (t)‖2 + 1 2 ( 1− β λ1 − a2 2λ1π ) ‖∆um(t)‖2 + (3 2 + ln a ) ‖um(t)‖2 ≤ C + ‖um(t)‖2 ln ‖um(t)‖2, (3.11) where C is a positive constant, independent of m and t by (3.5) and (3.6). Choosing e−3/2 < a < √ 2π(λ1 − β) we have 1− β λ1 − a2 2πλ1 > 0, 3 2 + ln a > 0. Taking C1 = min {1 2 , 1 2 ( 1− β λ1 − a2 2λ1π ) , (3 2 + ln a )} we have the estimate ‖umt (t)‖2 + ‖∆um(t)‖2 + ‖um(t)‖2 ≤ C2 ( 1 + ‖um(t)‖2 ln ‖um(t)‖2 ) . (3.12) Now, observe that um(·, t) = um(·, 0) + ∫ t 0 ∂um ∂s (·, s) ds, so, using Cauchy-Schwarz’s inequality, we obtain ‖um(t)‖2 ≤ 2‖u0m‖2 + 2 ∣∣∣∣∣∣ ∫ t 0 ∂um ∂s (s) ds ∣∣∣∣∣∣2 ≤ 2‖u0m‖2 + 2T ∫ t 0 ‖∂u m ∂s (s)‖2 ds. (3.13) EJDE-2021/21 KIRCHHOFF PLATE EQUATION 7 Using estimates (3.12) and (3.13) we have ‖um(t)‖2 ≤ ‖u0m‖2 + 2TC2 ∫ t 0 ( 1 + ‖um(t)‖2 ln ‖um(t)‖2 ) ds and choosing C3 = max{‖u0m‖2, TC2}, we obtain ‖um(t)‖2 ≤ 2C3 ( 1 + ∫ t 0 ‖um(t)‖2 ln ‖um(t)‖2ds ) . Without loss of generality, we take C3 ≥ 1 which gives ‖um(t)‖2 ≤ 2C3 ( 1 + ∫ t 0 ( C3 + ‖um(t)‖2 ) ln ( C3 + ‖um(t)‖2 ) ds ) and then by Lemma 2.4, we obtain ‖um(t)‖2 ≤ 2C3 e 2C3T ≤ C4. Hence, from inequality (3.12) it follows that ‖umt (t)‖2 + ‖∆um(t)‖2 + ‖um(t)‖2 ≤ C5, (3.14) with C4, C5 positive constants independent of m and t. Therefore, we can extend the approximate solutions umt (t) to the whole interval [0, T ]. Then by (3.14) we have that (um) is bounded in L∞(0, T ;H2 0 (Ω)) ∩ L∞(0, T ;L2(Ω)), (3.15) (umt ) is bounded in L∞(0, T ;L2(Ω)). (3.16) 3.3. Passage to the limit. From the estimates (3.15)-(3.16), there exists a sub- sequence of (um), still denoted by (um), such that um ∗ ⇀ u in L∞(0, T ;H2 0 (Ω)), (3.17) um ∗ ⇀ u in L∞(0, T ;L2(Ω)), (3.18) umt ∗ ⇀ ut in L∞(0, T ;L2(Ω)). (3.19) Applying the Lions-Aubin compactness lemma [22], we have from (3.17)-(3.18) that um → u strongly in L2(0, T ;H1 0 (Ω)), (3.20) um → u a.e. in Ω× (0, T ). (3.21) Taking into account that M is continuous and the convergences (3.20), (3.21), we have that M(‖∇um‖2)→M(‖∇u‖2) strongly in L2(0, T ). Therefore, M(‖∇um‖2)(−∆um) ⇀M(‖∇u‖2)(−∆u) weakly in L2(0, T ;L2(Ω)). (3.22) Since the map s→ s ln |s|2 is continuous, we have assured the convergence um ln |um|2 → u ln |u|2 a.e. in (Ω)× (0, T ). By using the immersion of H1 0 (Ω) in L∞(Ω) because (Ω ⊂ R2), it is clear that um ln |um|2 is bounded in L∞(Ω× (0, T )). So, by the Lebesgue dominated conver- gence theorem, um ln |um|2 → u ln |u|2 strongly in L2(0, T ;L2(Ω)). (3.23) 8 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 By the convergence (3.17), (3.18) and (3.23), we can pass to the limit in the approximate problem (3.4) and obtain the equation d dt (ut(t), w) + 〈∆u(t),∆w〉+M(‖∇u‖2)(−∆u,w) + (ut(t), w) − ( um(t) ln |um(t)|2, w ) = 0, (3.24) for all w ∈ Vm, in D′(0, T ). Since the Vm is dense in H2 0 (Ω) it follows that (3.24) is valid for all w ∈ H2 0 (Ω). The verification of the initial data can be obtained in a standard way. � 4. Potential well In this section, we present the potential well corresponding to the logarithmic nonlinearity. It is well known that the energy of a PDE system is, in some sense, split into kinetic and potential energy. Following the idea by Ye [35] and [31], we are able to construct a set of stability as follows. We will prove that there is a valley or a “well” of depth d created in the potential energy. If this height d is strictly positive, we find that for solutions with initial data in the “good part” of the well, the potential energy of the solution can never escape the well. In general, it is possible for the energy from the source term to cause the blow-up in finite time. However, in the good part of the well, it remains bounded. As a result, the total energy of the solution remains finite on any time interval [0, T ), which provides the global existence of the solution. We started by introducing the functionals J, I : H2 0 (Ω)→ R by J(u) := 1 2 ( 1− β λ1 ) ‖∆u‖2 − 1 2 ∫ Ω |u|2 ln |u|2 dx+ 1 2 ‖u‖2, I(u) := ( 1− β λ1 ) ‖∆u‖2 − ∫ Ω |u|2 ln |u|2 dx. From the above definitions, it is clear that J(u) = 1 2 I(u) + 1 2 ‖u‖2. For u ∈ H2 0 (Ω) we define the functional J(λu) = 1 2 I(λu) + λ2 2 ‖u‖2, Associated with the J we have the Nehari Manifold, N := {u ∈ H2 0 (Ω) : I(u) = 0, ‖∆u‖ 6= 0}. Lemma 4.1. For u ∈ H2 0 (Ω) with ‖u‖ 6= 0, let g(λ) = J(λu). Then we have I(λu) = λg′(λ), where λg′(λ)  > 0, 0 < λ < λ∗, = 0, λ = λ∗, < 0, 0 ≤ λ∗ < λ < +∞. with λ∗ = exp ( (1− β λ1 )‖∆u‖2 − ∫ Ω |u|2 ln |u|2 dx 2‖u‖2 ) . EJDE-2021/21 KIRCHHOFF PLATE EQUATION 9 Proof. Note that g(λ) = J(λu) = λ2 2 ( 1− β λ1 ) ‖∆u‖2 − λ2 2 ∫ Ω |u|2 lnλ2|u|2 dx+ λ2 2 ‖u‖2 = λ2 2 ( 1− β λ1 ) ‖∆u‖2 − λ2 2 ∫ Ω |u|2 lnλ2 dx− λ2 2 ∫ Ω |u|2 ln |u|2 dx+ λ2 2 ‖u‖2 = λ2 2 [( 1− β λ1 ) ‖∆u‖2 + (1− 2 lnλ) ‖u‖2 − ∫ Ω |u|2 ln |u|2 dx ] and that g′(λ) = λ [( 1− β λ1 ) ‖∆u‖2 + (1− 2 lnλ) ‖u‖2 − ∫ Ω |u|2 ln |u|2 dx ] + λ2 2 −2‖u‖2 λ = λ [( 1− β λ1 ) ‖∆u‖2 + (1− 2 lnλ) ‖u‖2 − ∫ Ω |u|2 ln |u|2 dx ] − λ‖u‖2 = λ [( 1− β λ1 ) ‖∆u‖2 − 2‖u‖2 lnλ− ∫ Ω |u|2 ln |u|2 dx ] . Then λg′(λ) = λ2 [( 1− β λ1 ) ‖∆u‖2 − 2‖u‖2 lnλ− ∫ Ω |u|2 ln |u|2 dx ] . So that I(λu) = 0 implies λ2 [( 1− β λ1 ) ‖∆u‖2 − 2‖u‖2 lnλ− ∫ Ω |u|2 ln |u|2 dx ] = 0, (4.1) λ = exp ((1− β λ1 ) ‖∆u‖2 − ∫ Ω |u|2 ln |u|2 dx 2‖u‖2 ) = λ∗. (4.2) Now, observe that, for 0 < λ < λ∗ we have − lnλ > − lnλ∗, and then I(λu) = λg′(λ) = λ2 [( 1− β λ1 ) ‖∆u‖2 − 2‖u‖2 lnλ− ∫ Ω |u|2 ln |u|2 dx ] > λ2 [( 1− β λ1 ) ‖∆u‖2 − 2‖u‖2 lnλ∗ − ∫ Ω |u|2 ln |u|2 dx ] = 0. (4.3) In the same way, for λ∗ < λ, we obtain I(λu) = λg′(λ) < 0. (4.4) Finally, from (4.2), (4.3) and (4.4) the proof is complete. � The potential well depth is defined as d := inf{sup λ≥0 J(λu);u ∈ H2 0 (Ω), ‖∆u‖ 6= 0}. (4.5) From the Mountain Pass theorem due to Ambrosetti and Rabinowitz [2], it is well- known that the depth of the well d is a strictly positive constant, see [34, Theorem 4.2], and that d = inf u∈N J(u). (4.6) With this approach, we introduce the potential well W = {u ∈ H2 0 (Ω) : I(u) 6= 0, J(u) < d} ∪ {0} (4.7) 10 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 and from lemma 4.1 we can partition W into two sets W = {u ∈ H2 0 (Ω) : I(u) > 0, J(u) < d} ∪ {0}, U = {u ∈ H2 0 (Ω) : I(u) < 0, J(u) < d}. We will refer to W as the “good” part of the well. Then we define the stability set for problem (1.1)-(1.3) by W = { u ∈ H2 0 (Ω) : ( 1− β λ1 ) ‖∆u‖2 > ∫ Ω u2 ln |u|2 dx, J(u) < d } ∪ {0}. The following lemma establishes a criterion for the solution u to remain in the stability set W . Lemma 4.2. Let u0 ∈ H1 0 (Ω) and u1 ∈ L2(Ω) such that 0 < E(0) < d and I(u0) > 0. (4.8) Then every solution of (1.1)-(1.3) belongs to W. Proof. Let T be maximal existence time of a weak solution u. From (3.7), we defined the energy E(t) = 1 2 ( ‖ut‖2 + ‖∆u‖2 + M̂(‖∇u‖2) + ‖u‖2 − ∫ Ω u2 ln |u|2dx ) , (4.9) where M̂(s) = ∫ s 0 M(ξ) dξ. Differentiating (4.9), and using (1.1)-(1.3), lead to d dt E(t) = − ∫ Ω ‖ut‖2 dx ≤ 0, (4.10) so 1 2 ‖ut(t)‖2 + J(u) ≤ 1 2 ‖u1‖2 + J(u0) < d, (4.11) for all t ∈ [0, T ]. We claim that u(t) ∈ W for all t ∈ [0.T ]. If not, then there exists a t0 ∈ (0, T ) such that u(t0) ∈ ∂W, so either I(u(t0)) = 0 and ‖∆u(t0)‖ 6= 0, or J(u(t0)) = d. By (4.11), J(u(t0)) < 0, thus we have I(u0) = 0 and ‖∆u0(t0)‖ 6= 0. However, (4.6) implies J(u(t0)) > d, which contradicts (4.11). So, we conclude that u(t) ∈ W. � 5. Exponential decay We prove the exponential decay of the problem (1.1)-(1.3), using the Nakao’s lemma. Theorem 5.1. Let u0 ∈ W and u1 ∈ L2(Ω), and 0 < E(0) < d. If (H1) holds, then the energy associated with (1.1)-(1.3) satisfies E(t) ≤ Ce−αt, ∀t ≥ 0, where C and α are positive constants. Proof. Let w = ut(t) in equation (3.24). Then d dt ‖ut(t)‖2 + 1 2 d dt ‖∆u(t)‖2 + 1 2 d dt M̂(‖∇u(t)‖2) + 1 2 d dt ‖u(t)‖2 − 1 2 d dt ∫ Ω u2(t) ln |u(t)|2 dx+ ‖ut(t)‖2 = 0; EJDE-2021/21 KIRCHHOFF PLATE EQUATION 11 that is, d dt E(t) + ‖ut(t)‖2 < 0 where, E(t) is define by (4.9). Integrating from t to t+ 1, we obtain∫ t+1 t ‖ut(s)‖2 ds ≤ E(t)− E(t+ 1) := F 2(t). (5.1) Then there exists t1 ∈ [t, t+ 1 2 ] and t2 ∈ [t+ 3 2 , t+ 1] such that ‖ut(ti)‖ ≤ 2F (ti), i = 1, 2. (5.2) Let w = u(t) in equation (3.24). Then we have ‖∆u(t)‖2 +M(‖∇u(t)‖2)‖∇u(t)‖2 − ∫ Ω u2(t) ln |u(t)|2 dx = − d dt ( ut(t), u(t) ) − ( ut(t), u(t) ) . (5.3) Now by (H1) we obtain M(‖∇u(t)‖2)‖∇u(t)‖2 ≥ − β λ1 ‖∆u(t)‖2. Integrating (5.3) from t1 to t2 we obtain∫ t2 t1 [( 1− β λ1 ) ‖∆u(s)‖2 − ∫ Ω u2(s) ln |u(s)|2 dx ] ds ≤ (ut(t1), u(t1))− (ut(t2), u(t2))− ∫ t2 t1 (ut(s), u(s)) ds ≤ C1 ess supt≤s≤t+1 ‖∆u(s)‖[‖ut(t1)‖+ ‖ut(t2)‖] + C2 1 δ ∫ t2 t1 ‖ut(s)‖2 ds+ δ ∫ t2 t1 ‖∆ut(s)‖2 ds, where 0 < δ < 1− β γ1 and C1 > 0 is a constant such that ‖ut(s)‖ ≤ C1‖∆ut(s)‖. Then, by (5.1) and (5.2), we obtain∫ t2 t1 [( 1− β λ1 − δ ) ‖∆u(s)‖2 − ∫ Ω u2(s) ln |u(s)|2 dx ] ds ≤ 4C1 F (t) ess supt≤s≤t+1 ‖∆u(s)‖+ C2 1 δ F 2(t). Whence ∫ t2 t1 [( 1− β λ1 − δ ) ‖∆u(s)‖2 − ∫ Ω u2(s) ln |u(s)|2 dx ] ds ≤ C2 [ F (t) ess supt≤s≤t+1 ‖∆u(s)‖+ F 2(t) ] =: G2(t). (5.4) Thanks to (5.1), we have∫ t2 t1 [ ‖ut(s)‖2 + ( 1− β λ1 − δ ) ‖∆u(s)‖2 − ∫ Ω u2(s) ln |u(s)|2dx ] ds ≤ F 2(t) +G2(t). Hence, there exists t∗ ∈ [t1, t2] such that ‖ut(t∗)‖2 + ( 1− β λ1 − δ ) ‖∆u(t∗)‖2 − ∫ Ω u2(t∗) ln |u(t∗)|2dx ≤ 2[F 2(t) +G2(t)]; 12 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 that is, ‖ut(t∗)‖2 + ‖∆u(t∗)‖2 − ∫ Ω u2(t∗) ln |u(t∗)|2dx ≤ C3[F 2(t) +G2(t)]. (5.5) By (H1), we obtain ‖ut(t∗)‖2 + M̂‖∆u(t∗)‖2 ≤ C2 1‖∆u(t∗)‖2 +m0‖∇u(t∗)‖2 ≤ ( C2 1 + m0 λ1 ) ‖∆u(t∗)‖2 ≤ C4[F 2(t) + G2(t)]. (5.6) From (4.9), (5.5) and (5.6), it follows that E(t∗) ≤ C5[F 2(t) + G2(t)]. (5.7) Now, by (5.1),(5.4) and (5.7), we have ess supt≤s≤t+1 E(s) ≤ E(t∗)+ ∫ t+1 t ‖ut(s)‖2 ds ≤ C6F 2(t)+ 1 2 ess supt≤s≤t+1E(s). Therefore, ess supt≤s≤t+1 E(s) ≤ C7[E(t)− E(t+ 1)], where Ci = 1, 2, . . . , 7 are positive constant. Finally, from lemma (2.5), we have E(t) ≤ Ce−αt for all t ≥ 0, where C and α are positive constants. � Acknowledgments. The authors thank the anonymous referees for their com- ments and suggestions. References [1] M. M. Al-Gharabli, S. A. Messaoudi; Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term, J. Evol. Equ., 18 (2018), 105–125. [2] A. Ambrosetti, P. H. 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Ye, Global existence and asymptotic behavior of solutions for a class of nonlinear degen- erate wave equations, Differ. Equ. Nonlinear Mech., (2007), Article ID 019685 [36] Y. Zhijian; On an extensible beam equation with nonlinear damping and source terms, J. Differential Equations, 254 (2013), 3903–3927. Ducival Pereira Department of Mathematics, State University of Pará, Belém, PA 66113-200, Brazil Email address: ducival@uepa.br Sebastião Cordeiro Faculty of Exact Sciences and Technology, Federal University of Pará, Abaetetuba, PA 68440-000, Brazil Email address: sebastiao@ufpa.br 14 D. PEREIRA, S. CORDEIRO, C. RAPOSO, C. MARANHÃO EJDE-2021/21 Carlos Raposo Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, MG 36307-352, Brazil Email address: raposo@ufsj.edu.br Celsa Maranhão Department of Mathematics, Federal University of Pará, Belém, PA 66075-110, Brazil Email address: celsa@ufpa.br 1. Introduction 2. Preliminaries 3. Existence of global weak solutions 3.1. Approximated problem 3.2. A priori estimates 3.3. Passage to the limit 4. Potential well 5. Exponential decay Acknowledgments References