Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 06, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND BLOW UP OF SOLUTIONS FOR A STRONGLY DAMPED PETROVSKY EQUATION WITH VARIABLE-EXPONENT NONLINEARITIES STANISLAV ANTONTSEV, JORGE FERREIRA, ERHAN PIŞKIN Abstract. In this article, we consider a nonlinear plate (or beam) Petrovsky equation with strong damping and source terms with variable exponents. By using the Banach contraction mapping principle we obtain local weak solutions, under suitable assumptions on the variable exponents p(·) and q(·). Then we show that the solution is global if p(·) ≥ q(·). Also, we prove that a solution with negative initial energy and p(·) < q(·) blows up in finite time. 1. Introduction Let be Ω a bounded domain in Rn (n ≥ 1) with a smooth boundary ∂Ω. We consider the initial boundary value problem utt + ∆2u−∆ut + |ut|p(x)−2ut = |u|q(x)−2u, (x, t) ∈ Ω× (0, T ) u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω u(x, t) = ∂vu(x, t) = 0, x ∈ ∂Ω (1.1) where v is the unit outer normal to ∂Ω, and the exponents p(·) and q(·) are mea- surable functions on Ω satisfying 2 ≤ p− ≤ p(x) ≤ p+ ≤ p∗ 2 ≤ q− ≤ q(x) ≤ q+ ≤ q∗, (1.2) where p− = ess infx∈Ω p(x), p+ = ess supx∈Ω p(x) q− = ess infx∈Ω q(x), q+ = ess supx∈Ω q(x) and 2 < p∗, q∗ <∞ if n ≤ 4, 2 < p∗, q∗ < 2n n− 4 if n > 4 . 2010 Mathematics Subject Classification. 35A01, 35B44, 35L55. Key words and phrases. Global solution; blow up; Petrovsky equation; variable-exponent nonlinearities. c©2021 Texas State University. Submitted May 20, 2020. Published January 29, 2021. 1 2 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 When p(x) and q(x) are constant and without strong damping (−∆ut), problem (1.1) becomes to the Petrovsky equation utt + ∆2u+ |ut|m−2ut = |u|p−2u, in QT = Ω× (0, T ) u = ∂u/∂v = 0, on ΓT = ∂Ω× [0, T ) u(x, 0) = u0(x), ut(x, 0) = u1(x) in Ω. (1.3) Messaoudi [24] studied this problem and established an existence result and showed that the solution continues to exist globally if m ≥ p, and that it blows up in finite time if m < p and the initial energy is negative. This result was later improved by Chen and Zhou [9]. For more results related to the plate equations, we refer the reader to Lagnese [19], Horn and Lasiecka [16, 20]. Problem (1.1) with strong damping and p and q constants becomes utt +42u−∆ut + |ut|p−2ut = |u|q−2u. (1.4) Liu et al [21] showed the existence, decay and blow up of the solutions of (1.4) and proved global existence and blow up. In 2013 Pişkin and Polat [34] showed the global existence and the decay of the solutions for (1.4). A considerable effort has been devoted to the study of (1.1) in case of constant and variable-exponent nonlinearities. In recent years, plate equations with lower order perturbation of p-Laplacian type in the form utt + ∆2 xu− div(φ(∇xu)) = F (u, ut) where φ(z) ≈ |s|(p−2)s, p ≥ 2, and F (u, ut) represents additional damping and forcing terms. This attracted attention of several authors. It is a prototype for some important models in real-world applications. In the absence of the viscoelastic term (g = 0) and replacing the ~p(x, t)-Laplacian by ∆pu = div(|∇u|p−2∇u) (p is constant and p ≥ 2), the equation utt + ∆2u− div(|∇u|p−2∇u)−∆ut = h(x, u, ut) (1.5) has been extensively studied and results concerning existence, nonexistence and long-time behaviour have been established; see [39, 40]. In one-dimension, (1.4) without damping or forcing terms is related to the model ρutt + ζuxxxx + a(u2 x)x = 0, a > 0 ζ = const > 0, which describes elastoplastic-microstructure flows as discussed in [2, 3]. In two dimensions, with p = 4 and weak damping, (1.4) corresponds to the so called model for nonlinear plates utt + ∆2u− div[|∇u|2∇u] + kut = σ∆(u2)− f(u). This is indeed a limit of the Mindlin-Timoshenko plates as the shear modulus tends to infinity, as shows in [10]. Remarkable results were obtained in [10, 11], where the existence of finite- dimensional global attractors under a weak damping kut, instead of −∆ut, was proved. Recently, the authors in [31] proved the blow up of solutions for a nonlinear viscoelastic wave equations with variable exponents, utt −∆u+ ∫ t 0 g(t− τ)∆u(τ)dτ + |ut|p(x)−2ut = |u|q(x)−2u (x, t) ∈ Ω× (0, T ). In the presence of the viscoelastic term (g 6= 0), equation (1.1) with memory was first studied in [4]. EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 3 The general decay of weak solutions u = u(x, t) for plate equations with memory term and lower order perturbation of ~p(x, t)-Laplacian type has been studied (see [14]). More precisely, we considered the problem utt = div(|∇u|p∇u) (1.6) with constant exponent of nonlinearity p ∈ (1,∞). Equation (1.6) was intensively studied During the previous decades, and was casted for the role of a touchstone in the nonlinear PDEs. The existence of global a solution without an additional dissipation term is an still open problem. We also mention the very important contribution in [5], where the author proved the existence and blow up for the weak solutions of a wave equation with p(x, t)- Laplacian and damping terms. utt = div ( a(x, t)|∇u|p(x,t)−2∇u+ ε∇ut ) + b(x, t)|u|σ(x,t)−2u+ f(x, t), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u|ΓT = 0 ΓT = ∂Ω× (0, T ), where the coefficients a, b, f and the exponents p, σ are given measurable functions and ε = const > 0. Such equations (with variable exponents of nonlinearities) are usually referred as equations with nonstandard growth conditions. Equations with nonstandard growth conditions occur in the mathematical mod- eling of various physical phenomena, e.g., the flows of electro-rheological fluids or fluids with temperature-dependent viscosity, nonlinear viscoelasticity, processes of filtration through a porous media and the image processing see [6] and references therein. Note that in all papers (referring to the case p 6= 0) the viscous term ε∆ut plays a key role in the proof of the existence of local and global solutions (even if p = const 6= 2). The principal difficulty remains in proving an existence theorem by considering the term −∆~p(x,t)u. The viscous term ε∆ut (with ε > 0) facilitates the proof of existence theorems. The authors in [7] improved the results from [4] by establishing local and global existence, as well as the uniqueness of the weak solution u(x, t) to (1.1). Recently in [26], the author established the decay of solutions of a damped quasilinear wave equation with variable-exponent nonlinearities. Rivera et al. [28] considered the equation utt − γ∆utt + ∆2u− ∫ t 0 g(t− s)∆2u(s)ds = 0 in QT = Ω× (0, T ), with initial and dynamical boundary conditions and proved that the sum of the first and second energies decays exponentially (respectively polynomially) if the kernel g decays exponentially (respectively polynomially). Alabau-Boussouira et al. [1] worked on the problem utt + ∆2u− ∫ t 0 g(t− s)∆2u(s) ds = f(u) in QT = Ω× (0, T ) u = ∂u/∂v = 0 on ΓT = ∂Ω× [0, T ) u(x, 0) = u0(x), ut(x, 0) = u1(x) in Ω 4 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 and established exponential and polynomial decay results for sufficiently small ini- tial data. Lin and Li in [22] studied utt − γ∆utt + ∆2u− ∫ t 0 g(t− s)∆2u(s) ds = div(C(f(∇u)∇u)) in QT = Ω×(0, T ), with initial and dynamical boundary conditions similar to those imposed by Rivera et al. [28], and established similar decay results. Yang in [39], considered the problem utt + ∆2u+ λut = n∑ i=1 ∂ ∂xi σi( ∂u ∂xi ) in Qτ = Ω× (0, T ) u = ∂u/∂v = 0 on ΓT = ∂Ω× [0, T ) u(x, 0) = u0(x), ut(x, 0) = u1(x) in Ω for λ ≥ 0 and σi nonlinear functions. He proved, under some conditions on nonlinear terms and initial data, that the problem admits a global weak solution and the solution decays exponentially to zero as t→∞ . Motivated by [8, 11, 16], we considered the existence of local and global solutions, and their blow up for nonlinear Petrovsky equation with variable exponents and strong damping. To the best of our knowledge, this is the first work dealing with equation (1.1) subject to the variable exponents and strong damping. Our aim in this work is to prove the existence of local and global solutions, and to find sufficient conditions on p, q for which the blow up takes place. This article consists of five sections in addition to the introduction. In Section 2, we recall the definitions of the Lp(·)(Ω), the Sobolev spaces W 1,p(·)(Ω), as some of their properties. In Section 3, we prove the local existence of weak solutions for Problem (1). In Section 4, we establish a global existence. In Section 5,we state and prove our blow up result for solutions with negative initial energy are given. 2. Preliminaries In this section, we state some results about the variable exponent Lebesgue and Sobolev spaces Lp(x)(Ω) and W 1,p(x)(Ω) (see [12, 13, 18, 30]). Let p : Ω → [1,∞] be a measurable function, where Ω is a domain of Rn. We define the variable exponent Lebesgue space by Lp(x)(Ω) = {u : Ω→ R : u is measurable in Ω and ρp(·)(λu) <∞ for some λ > 0}, where ρp(·)(u) = ∫ Ω |u(x)|p(x)dx. The space Lp(·)(Ω) equipped with the Luxemburg-type norm ‖u‖p(·) = inf { λ > 0 : ∫ Ω |u(x) λ |p(x)dx ≤ 1 } becomes a Banach space [12]. The relation between the modular ∫ Ω |f |p(x) dx and the norm follows from min(‖f‖p − p(·), ‖f‖ p+ p(·)) ≤ ∫ Ω |f |p(x) dx ≤ max(‖f‖p − p(·), ‖f‖ p+ p(·)). EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 5 In the case p(·) = const > 1, these inequalities transform into equalities. For all f ∈ Lp(·)(Ω), g ∈ Lp′(·)(Ω) with p(x) ∈ (1,∞), p′(x) = p(x) p(x)− 1 the generalized Hölder inequality holds,∫ Ω |f g| dx ≤ ( 1 p− + 1 (p′)− ) ‖f‖p(·)‖g‖p′(·) ≤ 2‖f‖p(·)‖g‖p′(·) . The variable exponent Sobolev space is defined by W 1,p(·)(Ω) = {u ∈ Lp(·)(Ω) : ∇u exists and |∇u| ∈ Lp(·)(Ω)} with respect to the norm ‖u‖1,p(·) = ‖u‖p(·) + ‖∇u‖p(·). The space W 1,p(·) 0 (Ω) is defined as the closure of C∞0 (Ω) in W 1,p(·)(Ω) with respect to the norm ‖u‖1,p(·). For u ∈W 1,p(·) 0 (Ω), we can define an equivalent norm ‖u‖1,p(·) = ‖∇u‖p(·). Let the variable exponent p(·) satisfy the log-Hölder continuity condition |p(x)− p(y)| ≤ A log 1 |x−y| , for all x, y ∈ Ω with |x− y| < δ, (2.1) where A > 0 and 0 < δ < 1. Lemma 2.1 (Poincare inequality [12]). Let Ω be a bounded domain of Rn and p(·) satisfies log-Hölder condition, then ‖u‖p(x) ≤ c‖∇u‖p(x), for all u ∈W 1,p(x) 0 (Ω), (2.2) where C = C(p−, p+, |Ω|) > 0. Lemma 2.2 ([12]). Let p(·) ∈ C(Ω) and q : Ω → [1,∞) be a measurable function that satisfy ess infx∈Ω(p∗(x)− q(x)) > 0. Then the Sobolev embedding W 1,p(x) 0 (Ω) ↪→ Lq(x)(Ω) is continuous and compact. Where p∗(x) = { np− n−p− , if p− < n any number in [1,∞), if p− ≥ n. If in addition p(·) satisfies log-Hölder condition, then p∗(x) = { np(x) n−p(x) , if p(x) < n any number in [1,∞), if p(x) ≥ n. Remark 2.3. We denote by c various positive constants which may be different at different occurrences. Also, throughout this paper, we use the embedding H2 0 (Ω) ↪→ H1 0 (Ω) ↪→ Lp(Ω) which implies ‖u‖p ≤ C‖∇u‖ ≤ C‖∆u‖, where 2 ≤ p <∞ (n = 1, 2), 2 ≤ p ≤ 2n n−2 (n ≥ 3). Moreover, ‖u‖p ≤ C‖∆u‖, 6 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 p =  ∞ if n < 4, any number in [1,∞), if n = 4, 2n n−4 if n > 4. We will use also the Young inequality ab ≤ 1 p (εa)p + p− 1 p ( b ε a) p p−1 , a, b ≥ 0, ε ∈ (0, 1), 1 < p <∞. (2.3) 3. Existence of weak solutions In this part, we prove a local existence result for (1.1). Firstly, we state the following lemma which can be obtained by exploiting the Feado-Galerkin method and using the similar arguments as in [27, 29]. Lemma 3.1. Suppose that p(·) satisfies (1.2) and (2.1), and that initial data sat- isfies u0 ∈ H2 0 (Ω), u1 ∈ L2(Ω). Then there exists a unique local solution u of utt + ∆2u−∆ut + |ut|p(x)−2ut = f(t, x), (x, t) ∈ Ω× (0, T ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂υu(x, t) = 0, x ∈ ∂Ω, (3.1) satisfying u ∈ L∞((0, T ), H2 0 (Ω)), ut ∈ L∞((0, T ), L2(Ω)) ∩ Lp(·)(Ω× (0, T )), where f ∈ L2(Ω× (0, T )). Theorem 3.2. Suppose that p(·) satisfies (1.2) and 2 ≤ p− ≤ p(x) ≤ p+ ≤ 2 + 4 n− 4 (n > 4). (3.2) Furthermore assume that q(·) satisfies (1.2) and 2 ≤ q− ≤q+ <∞ if n ≤ 4, and 2 ≤ q− ≤q+ ≤ 2 + 4 n− 4 , if n > 4, (3.3) u0 ∈ H2 0 (Ω), u1 ∈ L2(Ω). Then (1.1) has a unique local solution u ∈ L∞((0, T ), H2 0 (Ω)), ut ∈ L∞((0, T ), L2(Ω)) ∩ Lp(·)(Ω× (0, T )). Proof. (Existence) Let v ∈ L∞((0, T ), H1 0 (Ω)) and f(v) = |v|q(x)−2v. We have ‖f(v)‖2 = ∫ Ω∩(|v|≤1) |v|2(q(x)−1)dx+ ∫ Ω∩(1<|v|) |v|2(q(x)−1)dx ≤ |Ω|+ ∫ Ω |v|2(q+−1)dx <∞, since 2(q− − 1) ≤ 2(q+ − 1) ≤ 2n n− 2 . Thus, for each v ∈ L∞((0, T ), H1 0 (Ω)), there exists a unique u ∈ L∞((0, T ), H2 0 (Ω)), ut ∈ L∞((0, T ), L2(Ω)) ∩ Lp(·)(Ω× (0, T )), EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 7 solving the problem utt + ∆2u−∆ut + |ut|p(x)−2ut = f(v), (x, t) ∈ Ω× (0, T ), u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂υu(x, t) = 0, x ∈ ∂Ω. (3.4) We define the space XT := {w ∈ L∞((0, T );H2 0 (Ω)) : w ∈ L∞((0, T );L2(Ω))} which is a Banach space with respect to the norm ‖w‖XT = ‖w‖L∞((0,T );H2 0 (Ω)) + ‖w‖L∞((0,T );L2(Ω)). We define the nonlinear map S as follows. For v ∈ XT , Sv = u is the unique solution (3.4). We shall show that there exist T > 0, such that (i) S : XT → XT (ii) S is a contraction mapping in XT . To show (i), multiplying (3.4) by ut and integrating over Ω× (0, t), we obtain 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ + ∫ t 0 ∫ Ω |uτ |p(x) dx dτ = 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + ∫ t 0 ∫ Ω |v|q(x)−2vuτ dx dτ. (3.5) By the Young’s, Sobolev-Poincare’s inequalities and (3.3), we obtain∫ Ω |v|q(x)−2vuτdx ≤ δ 4 ∫ Ω u2 tdx+ 1 δ ∫ Ω |v|2q(x)−2dx ≤ δ 4 ‖ut‖2 + 1 δ [ ∫ Ω |v|2(q−−1)dx+ ∫ Ω |v|2(q+−1)dx ] ≤ δ 4 ‖ut‖2 + C δ ( ‖∆v‖2(q−−1) + ‖∆v‖2(q+−1) ) . (3.6) Thus, by (3.5) and (3.6), we have 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ + ∫ t 0 ∫ Ω |uτ |p(x) dx dτ ≤ 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + δ 4 ∫ t 0 ‖ut‖2dτ + C δ ∫ t 0 ( ‖∆v‖2(q−−1) + ‖∆v‖2(q+−1) ) dτ, which implies sup t∈(0,T ) [‖ut‖2 + ‖∆u‖2] ≤ ‖u1‖2 + ‖∆u0‖2 + δT 2 sup t∈(0,T ) ‖ut‖2 + CT δ [‖v‖2(q−−1) XT + ‖v‖2(q+−1) XT ]. By taking δT/2 ≤ 1, we have ‖u‖2XT ≤ λ+ CT δ [ ‖v‖2(q−−1) XT + ‖v‖2(q+−1) XT ] , 8 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 where λ = ‖u1‖2 + ‖∆u0‖2. At this point we choose M large enough, such that ‖v‖XT ≤M . Then ‖u‖2XT ≤ λ+ CT δ M2(q+−1) ≤M2 if λ < M2 and T ≤ T0 < δ(M2−λ) CM2(q+−1) . Thus we have S : XT → XT . Next, we show S is a contraction mapping in XT . For this purpose, we let u1 = Sv1 and u2 = Sv2, then u = u1 − u2 satisfies utt + ∆2u−∆ut + [|u1t|p(x)−2u1t − |u2t|p(x)−2u2t] = |v1|q(x)−2v1 − |v2|q(x)−2v2, u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) = ∂υu(x, t) = 0, x ∈ ∂Ω. (3.7) Multiplying by ut = u1t − u2t and integrating over Ω× (0, t), we obtain 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ + ∫ t 0 ∫ Ω [ |u1t|p(x)−2u1t − |u2t|p(x)−2u2t ] (u1t − u2t) dx dτ ≤ 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + ∫ t 0 ∫ Ω (f(v1)− f(v2))ut dx dτ. (3.8) Since [|u1t|p(x)−2u1t − |u2t|p(x)−2u2t](u1t − u2t) ≥ 0, inequality (3.8) yields 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ ≤ 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + ∫ t 0 ∫ Ω (f(v1)− f(v2))ut dx dτ. (3.9) We estimate the right-most term of as follows:∫ Ω |f(v1)− f(v2)| |ut|dx = ∫ Ω |f ′(ξ)| |v| |ut|dx, where v = v1−v2 and ξ = αv1+(1−α)v2, 0 ≤ α ≤ 1. Thanks to Young’s inequality, (2.3), and since f(v) = |v|q(x)−2v, we obtain∫ Ω |f(v1)− f(v2)| |ut|dx ≤ δ 2 ∫ Ω |ut|2dx+ 1 2δ ∫ Ω |f ′(ξ)|2|v|2dx ≤ δ 2 ‖ut‖2 + (q+ − 1)2 2δ ∫ Ω |αv1 + (1− α)v2|2(q(x)−2)|v|2dx ≤ δ 2 ‖ut‖2 + c (∫ Ω |v| 2n n−2 dx )n−2 2 (∫ Ω |αv1 + (1− α)v2|n(q(x)−2)dx )2/n ≤ δ 2 ‖ut‖2 + c (∫ Ω |v| 2n n−2 dx )n−2 2 [ ( ∫ Ω |αv1 + (1− α)v2|n(q+−2)dx)2/n EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 9 + (∫ Ω |αv1 + (1− α)v2|n(q−−2)dx )2/n] . (3.10) Thus by (3.2) and (3.3), we obtain∫ Ω |f(v1)− f(v2)|, |ut|dx ≤ δ 2 ‖ut‖2 + C‖∆v‖2 [ ‖∆v1‖2(q+−2) + ‖∆v1‖2(q−−2) + ‖∆v2‖2(q+−2) + ‖∆v2‖2(q−−2) ] ≤ δ 2 ‖ut‖2 + 4CM2(q+−2)‖∆v‖2 . By the combining this inequality with (3.9), we obtain 1 2 ‖u‖2XT ≤ δ 2 T0‖u‖2XT + 4CM2(q+−2)T0‖v‖2XT . By choosing δ small enough, we have ‖u‖2XT ≤ 8CM2(q+−2)T0‖v‖2XT . Now, we choose T0 sufficiently enough so that 0 < 8CM2(q+−2)T0 < 1. Thus, the map S is contraction. The Banach fixed point theorem implies the existence of a unique u ∈ XT satisfying S(u) = u. Obviously, it is a solution of (1.1). (Uniqueness) Suppose that (1.1) have two solutions u and v. Then w = u − v satisfies wtt + ∆2w −∆wt + |ut|p(x)−2ut − |vt|p(x)−2vt = |u|q(x)−2u− |v|q(x)−2v, (x, t) ∈ Ω× (0, T ), w(x, 0) = 0, wt(x, 0) = 0, x ∈ Ω, w(x, t) = ∂υw(x, t) = 0, x ∈ ∂Ω. Multiplying by wt and integrate over Ω× (0, t), we obtain 1 2 ‖wt‖2 + 1 2 ‖∆w‖2 + ∫ t 0 ‖∇wτ‖2dτ + ∫ t 0 ∫ Ω ( |ut|p(x)−2ut − |vt|p(x)−2vt ) wt dx dτ = ∫ t 0 ∫ Ω ( |u|q(x)−2u− |v|q(x)−2v ) wt dx dτ. By using the inequality (|a|p−2 − |b|p−2b)(a− b) ≥ 0, for all a, b ∈ Rn, 1 < p <∞ and similarly (3.10), we have ‖wt‖2 + ‖∆w‖2 ≤ C ∫ t 0 ∫ Ω (|wt(τ)|2 + |∆w(τ)|2) dx dτ. By Gronwall’s inequality, we obtain ‖wt‖2 + ‖∆w‖2 = 0. Thus w = 0. The proof is complete. � 10 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 4. Existence of global solutions In this section, we obtain a global solution for (1.1) under suitable conditions on p(·) and q(·). In the presence of additional estimates, the proven local solution can be continued to an finite time interval. Theorem 4.1. Let the assumptions of Theorem 3.2 hold. If u0 ∈ H2 0 (Ω), u ∈ L2(Ω) and the exponents p(·) and q(·) satisfy one of the following two conditions 1 < q(x) ≤ 2, 1 < p(x) <∞ or 2 ≤ q(x) ≤ p(x) <∞. Then problem (1.1) has a global solution, with u ∈ L∞((0, T ), H2 0 (Ω))), ut ∈ L∞((0, T )L2(Ω)) ∩ Lp(·)(Ω× (0, T )). Proof. To achieve the global existence of a solution, it suffices to show that sup t∈[0,T ] (‖ut‖2 + ‖∆u‖2) + ∫ T 0 ‖∇uτ‖2dτ + ∫ T 0 ∫ Ω |uτ |p(x) dx dτ ≤ C for any finite T < ∞. Multiplying (1.1) by ut and integrating over Ω × (0, t), we obtain 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ + ∫ t 0 ∫ Ω |uτ |p(x) dx dτ = 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + ∫ t 0 ∫ Ω |u|q(x)−2uuτ dx dτ . (4.1) First let us consider the case q(x) ≤ 2⇔ 2(q(x)− 1) ≤ 2, 1 < p(x) <∞. (4.2) We evaluate the term |I| = ∣∣ ∫ t 0 ∫ Ω |u|q(x)−2uuτ dx dτ ∣∣ ≤ 1 2 ∫ t 0 ( ‖ut‖2 + ∫ Ω |u|2(q(x)−1) dx dτ ) ≤ 1 2 ∫ t 0 ( ‖ut‖2 + ‖u‖2 ) dτ + 1 2 T |Ω| ≤ c 2 ∫ t 0 (‖ut‖2 + ‖∆u‖2)dτ + 1 2 T |Ω| (4.3) 2(q(x)− 1) ≤ 2⇔ q(x) ≤ 2. Introducing the function Y (t) = ‖ut‖2 + ‖∆u‖2 we arrive at integral inequality Y (t) ≤ C ∫ t 0 Y (τ)dτ +B, B = ‖u1‖2 + ‖∆u0‖2 + T |Ω|. Applying the Granwall inequality we derive the estimate sup t∈[0,T ] ( ‖ut‖2 + ‖∆u‖2 ) + ∫ T 0 ‖∇uτ‖2dτ + ∫ T 0 ∫ Ω |uτ |p(x) dx dτ ≤ C (4.4) which holds for any finite T <∞. EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 11 Now we consider the case 2 ≤ q(x) ≤ p(x) <∞ . (4.5) Applying the Young inequality (2.3) and (3.5) we derive 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ + ∫ t 0 ∫ Ω |uτ |p(x) dx dτ ≤ 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + ∫ t 0 ∫ Ω εp − p− |uτ |p(x) dx dτ + I. We evaluate the term I as follows, I = ∫ t 0 ∫ Ω p− 1 p ε p p−1 |u| (q−1)p p−1 dx dτ ≤ C(ε, p±) ∫ t 0 ∫ Ω |u| (q−1)p p−1 dx dτ ≤ ≤ C(ε, p±, T, |Ω|) (∫ t 0 ∫ Ω |u|p(x) dx dτ + 1 ) , (q − 1)p p− 1 ≤ p ⇔ q ≤ p. Choosing εp/p ≤ εp−/p− ≤ 1/2 we obtain 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 + ∫ t 0 ‖∇uτ‖2dτ + 1 2 ∫ t 0 ∫ Ω |uτ |p(x) dx dτ ≤ 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + ∫ t 0 ∫ Ω p− 1 p ε p p−1 |u| (q−1)p p−1 dx dτ ≤ 1 2 ‖u1‖2 + 1 2 ‖∆u0‖2 + C(ε, p±) ( ∫ t 0 ∫ Ω |u|p(x) dx dτ + 1 ) , (q − 1)p p− 1 ≤ p⇔ q ≤ p. Next we use the inequality∫ Ω |u|p(x)dx = ∫ Ω ∣∣∣ ∫ t 0 utds+ u0 ∣∣∣p(x) dx ≤ C(p±) ∫ Ω ( tp−1 ∫ t 0 |ut|p(x)ds+ |u0|p(x)p(x) ) ≤ C(p±) (∫ t 0 ∫ Ω tp(x)−1|ut|p(x) dx ds+ ∫ Ω |u0|p(x)dx ) ≤ C(p±, T ) (∫ t 0 ∫ Ω |ut|p(x) dx ds+ ∫ Ω |u0|p(x)dx ) ≤ C(p±, T ) (∫ t 0 ∫ Ω |ut|p(x) dx ds+ ∫ Ω |u0|p(x)dx ) . (4.6) We introduce the function Y (t) = ∫ t 0 ∫ Ω |ut(x, s)|p(x) dx ds. 12 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 Then (4.6), (4.6) lead us to the integral inequality Y (t) ≤ C (∫ t 0 Y (s)ds+ ∫ Ω |u0|p(x)dx+ ‖u1‖2 + ‖∆u0‖2 + 1 ) . (4.7) Applying the Granwall inequality we arrive at estimate (4.4). This estimate permits us to continue local solution for any finite interval of time. This completes the proof. � 5. Blow up of solutions In this part, we consider the blow up of the solution for problem (1.1). Firstly, we give following lemma. Lemma 5.1 ([27]). If q : Ω→ [1,∞) is a measurable function and 2 ≤ q− ≤ q(x) ≤ q+ <∞ for n ≤ 4, 2 ≤ q− ≤ q(x) ≤ q+ < 2n n− 2 for n > 4 (5.1) holds. Then, we have following inequalities: ρ s q− q(·)(u) ≤ C ( ‖∆u‖2 + ρq(·)(u) ) , (5.2) ‖u‖sq− ≤ C ( ‖∆u‖2 + ‖u‖q − q− ) , (5.3) ρ s q− q(·)(u) ≤ C ( |H(t)|+ ‖ut‖2 + ρq(·)(u) ) , (5.4) ‖u‖sq− ≤ C(|H(t)|+ ‖ut‖2 + ‖u‖q − q−), (5.5) C‖u‖q − q− ≤ ρq(·)(u) = ∫ Ω |u|q(·)dx (5.6) for any u ∈ H2 0 (Ω) and 2 ≤ s ≤ q−. Where C > 1 a positive constant and H(t) = −E(t). The functions H(t), E(t) will be defined later. Now, we state and prove our blow up result. Theorem 5.2. Let the assumptions of Theorem 3.2, and Lemma 5.1 hold. Also let initial energy satisfy E(0) < 0, and the exponents p(·) and q(·) satisfy 2 ≤ p− ≤ p(x) ≤ p+ < q− ≤ q(x) ≤ q+ ≤ 2 + 4 n− 4 , if n > 4 . Then the solution of (1.1) blows up in a finite time T ∗, in the following sense Ψ(t)→∞ as t→ T ∗ ≤ 1− σ ξσΨ σ 1−σ (0) , (5.7) where ξ ∈ (0, 1), and Ψ(t) and σ are given in (5.11) and (5.12) respectively. Proof. Multiplying both sides of the equation in (1.1) by ut, and integrating by parts, we have d dt [1 2 ‖ut‖2 + 1 2 ‖∆u‖2 − ∫ Ω 1 q(x) |u|q(x)dx ] = − ∫ Ω |ut|p(x)dx− ‖∇ut‖2, E′(t) = − ∫ Ω |ut|p(x)dx− ‖∇ut‖2 (5.8) EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 13 where E(t) = 1 2 ‖ut‖2 + 1 2 ‖∆u‖2 − ∫ Ω 1 q(x) |u|q(x)dx. (5.9) Set H(t) = −E(t) then E(0) < 0 and (5.8) gives H(t) ≥ H(0) > 0. Also, by the definition H(t), we have H(t) = −1 2 ‖ut‖2 − 1 2 ‖∆u‖2 + ∫ Ω 1 q(x) |u|q(x)dx ≤ ∫ Ω 1 q(x) |u|q(x)dx ≤ 1 q− ρq(·)(u). (5.10) We define Ψ(t) = H1−σ(t) + ε ∫ Ω uut dx+ ε 2 ‖∇u‖2, (5.11) where ε small to be chosen later and 0 < σ ≤ min { q− − p+ (p+ − 1)q− , q− − 2 2q− } . (5.12) Differentiating Ψ(t) with respect to t, and using (1.1), we have Ψ′(t) = (1− σ)H−σ(t)H ′(t) + ε ∫ Ω (u2 t + uutt)dx+ ε ∫ Ω ∇u∇ut dx = (1− σ)H−σ(t)H ′(t) + ε‖ut‖2 − ε‖∆u‖2 + ε ∫ Ω |u|q(·)dx− ε ∫ Ω uut|ut|p(·)−2dx. (5.13) By using the definition of the H(t), it follows that −εq−(1− ξ)H(t) = εq−(1− ξ) 2 ‖ut‖2 + εq−(1− ξ) 2 ‖∆u‖2 − εq−(1− ξ) ∫ Ω 1 q(x) |u|q(·)dx, (5.14) where 0 < ξ < 1. Adding and subtracting (5.14) into (5.13), we obtain Ψ′(t) ≥ (1− σ)H−σ(t)H ′(t) + εq−(1− ξ)H(t) + ε (q−(1− ξ) 2 + 1 ) ‖ut‖2 + ε (q−(1− ξ) 2 − 1 ) ‖∆u‖2 + εξ ∫ Ω |u|q(·)dx− ε ∫ Ω uut|ut|p(·)−2dx. (5.15) Then, for ξ small enough, we obtain Ψ′(t) ≥ εβ[H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u)] + (1− σ)H−σ(t)H ′(t)− ε ∫ Ω uut|ut|p(·)−2dx (5.16) where β = min { q−(1− ξ), ξ, q −(1− ξ) 2 − 1, q−(1− ξ) 2 + 1 } > 0 14 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 and ρq(·)(u) = ∫ Ω |u|q(·)dx. To estimate the last term in (5.16), we use the Young inequality (2.3). Consequently, applying the previous we have ∫ Ω u|ut|p(·)−1dx ≤ ∫ Ω 1 p(x) δp(x)|u|p(x)dx+ ∫ Ω p(x)− 1 p(x) δ− p(x) p(x)−1 |ut|p(x)dx ≤ 1 p− ∫ Ω δp(x)|u|p(x)dx+ p+ − 1 p+ ∫ Ω δ− p(x) p(x)−1 |ut|p(x)dx, (5.17) where δ is constant depending on the time t and specified later. Inserting estimate (5.17) into (5.16), we obtain Ψ′(t) ≥ εβ[H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u)] + (1− σ)H−σ(t)H ′(t)− ε 1 p− ∫ Ω δp(x)|u|p(x)dx − εp + − 1 p+ ∫ Ω δ− p(x) p(x)−1 |ut|p(x)dx. (5.18) Let us choose δ so that δ− p(x) p(x)−1 = k1H −σ(t), where k1, k2 > 0 are specified later, we obtain Ψ′(t) ≥ εβ[H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u)] + (1− σ)H−σ(t)H ′(t)− εk2H −σ(t)H ′(t) − ε 1 p− ∫ Ω k 1−p(x) 1 Hσ(p(x)−1)(t)|u|p(x)dx− εp + − 1 p+ ∫ Ω k1H −σ(t)|ut|p(x)dx ≥ εβ[H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u)] + (1− σ − εk2)H−σ(t)H ′(t) − εk 1−p− 1 p− Hσ(p+−1)(t) ∫ Ω |u|p(x)dx− ε (p+ − 1 p+ ) k1H −σ(t) ∫ Ω |ut|p(x)dx ≥ εβ[H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u)] + [ (1− σ − εk2)− ε (p+ − 1 p+ ) k1 ] H−σ(t)H ′(t) − εk 1−p− 1 p− Hσ(p+−1)(t) ∫ Ω |u|p(x)dx. (5.19) EJDE-2021/06 PETROVSKY EQUATION WITH VARIABLE-EXPONENT 15 By using (5.6) and (5.10), we obtain Hσ(p+−1)(t) ∫ Ω |u|p(x)dx ≤ Hσ(p+−1)(t) [ ∫ Ω− |u|p − dx+ ∫ Ω+ |u|p + dx ] ≤ Hσ(p+−1)(t)C [( ∫ Ω− |u|q − dx ) p− q− + (∫ Ω+ |u|q − dx ) p+ q− ] = Hσ(p+−1)(t)C [ ‖u‖p − q− + ‖u‖p + q− ] ≤ C ( 1 q− ρq(·)(u) )σ(p+−1)[ (ρq(·)(u)) p− q− + (ρq(·)(u)) p+ q− ] = C1 [ (ρq(·)(u)) p− q− +σ(p+−1) + (ρq(·)(u)) p+ q− +σ(p+−1)] (5.20) where Ω− = {x ∈ Ω : |u| < 1} and Ω+ = {x ∈ Ω : |u| ≥ 1}. We then use Lemma 5.1 and (5.12), for s = p− + σq−(p+ − 1) ≤ q− and for s = p+ + σq−(p+ − 1) ≤ q−, to deduce, from (5.20), that Hσ(p+−1)(t) ∫ Ω |u|p(x)dx ≤ C1 [ ‖∆u‖2 + ρq(·)(u) ] . (5.21) Thus, inserting estimate (5.21) into (5.19), we have Ψ′(t) ≥ ε ( β − k1−p− 1 p− C1 ) [H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u)] + [ (1− σ − εk2)− ε (p+ − 1 p+ ) k1 ] H−σ(t)H ′(t). (5.22) Let us choose k1 large enough so that γ = β − k1−p− 1 p− C1 > 0, and picking ε small enough such that (1− σ − εk2)− ε (p+ − 1 p+ ) k1 > 0 and Ψ(t) ≥ Ψ(0) = H1−σ(0) + ε ∫ Ω u0u1dx+ ε 2 ‖∇u0‖2 > 0, ∀t ≥ 0. (5.23) Consequently, (5.22) yields Ψ′(t) ≥ εγ [ H(t) + ‖ut‖2 + ‖∆u‖2 + ρq(·)(u) ] ≥ εγ [ H(t) + ‖ut‖2 + ‖∆u‖2 + ‖u‖q − q− ] , (5.24) because of (5.6). Therefore Ψ(t) ≥ Ψ(0) > 0, for all t ≥ 0. 16 S. ANTONTSEV, J. FERREIRA, E. PIŞKIN EJDE-2021/06 On the other hand, applying Hölder inequality, we obtain∣∣∣ ∫ Ω uut dx ∣∣∣ 1 1−σ ≤ ‖u‖ 1 1−σ ‖ut‖ 1 1−σ ≤ C ( ‖u‖ 1 1−σ q− ‖ut‖ 1 1−σ ) . Young inequality gives∣∣∣ ∫ Ω uut dx ∣∣∣ 1 1−σ ≤ C ( ‖u‖ µ 1−σ q− + ‖ut‖ θ 1−σ ) , (5.25) for 1 µ + 1 θ = 1. We take θ = 2(1 − σ), to obtain µ 1−σ = 2 1−2σ ≤ q− by (5.12). Therefore, (5.25) becomes∣∣∣ ∫ Ω uut dx ∣∣∣ 1 1−σ ≤ C ( ‖ut‖2 + ‖u‖sq− ) , where 2 1−2σ ≤ q −. By using (5.5), we obtain∣∣∣ ∫ Ω uut dx ∣∣∣ 1 1−σ ≤ C(‖ut‖2 + ‖u‖q − q− +H(t)). Thus, using the inequality (a1 + a2 + ...+ am)λ ≤ 2(m−1)/(λ−1)(aλ1 + aλ2 + ...+ aλm), (for a1, a2, . . . , am ≥ 0, λ ≥ 1), we have Ψ 1 1−σ (t) = [ H1−σ(t) + ε ∫ Ω uut dx+ ε 2 ‖∇u‖2 ] 1 1−σ ≤ 2 σ 1−σ ( H(t) + ε 1 1−σ | ∫ Ω uut dx| 1 1−σ ) ≤ C ( ‖ut‖2 + ‖u‖q − q− +H(t) ) ≤ C ( H(t) + ‖ut‖2 + ‖∆u‖2 + ‖u‖q − q− ) . (5.26) By combining of (5.24) and (5.26), we arrive at Ψ′(t) ≥ ξΨ 1 1−σ (t), (5.27) where ξ is a positive constant. A simple integration of (5.27) over (0, t) yields Ψ σ 1−σ (t) ≥ 1 Ψ− σ 1−σ (0)− ξσt 1−σ , which implies that the solution blows up in a finite time T ∗, with T ∗ ≤ 1− σ ξσΨ σ 1−σ (0) . This completes the proof. � Remark 5.3. Estimate (5.7) shows that the larger ψ(0) is, the quicker the blow up takes place. Conclusion. In this work, we obtained the local and global solutions and blow up in finite time for a nonlinear plate(or beam) Petrovsky equations with strong damping and source terms with variable exponents in a bounded domain. 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Stanislav Antontsev Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russia. CMAF-CIO, University of Lisbon, Portugal Email address: antontsevsn@mail.ru Jorge Ferreira Federal Fluminense University - UFF - VCE, Department of Exact Sciences, Av. dos Trabalhadores, 420 Volta Redonda RJ, Brazil Email address: ferreirajorge2012@gmail.com Erhan Pişkin Dicle University, Department of Mathematics, 21280 Diyarbakir, Turkey Email address: episkin@dicle.edu.tr 1. Introduction 2. Preliminaries 3. Existence of weak solutions 4. Existence of global solutions 5. Blow up of solutions Conclusion Acknowledgements References