Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form utt(x,t)−Δu(x,t)+αut(x,t)+βut(x,t−τ)=u(x,t)ln|u(x,t)|γ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$\end{document} There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.


Introduction
We consider the following wave equation with frictional damping, time delay in the velocity, and logarithmic source: u ttu + αu t (t) + βu t (x, tτ ) = u ln |u| γ for (x, t) ∈ × (0, ∞), (1.1) u(x, t) = 0 for (x, t) ∈ ∂ × (0, ∞), (1.2) u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x) for x ∈ , (1.3) u t (x, t) = j 0 (x, t) for (x, t) ∈ × (-τ , 0), (1.4) where ⊂ R N , N ≥ 1, is a bounded domain with smooth boundary ∂ . τ > 0 is time delay, α, β, and γ are real numbers that will be specified later. Equation (1.1) is related to a relativistic version of logarithmic quantum mechanics and many branches of physics such as nuclear physics, optics and geophysics [3,10,15]. One of the important theories addressing the existence and nonexistence of solutions for problems with source terms is the potential well method, which was devised by Sattinger [29]. Based on the method, the interaction between the damping and the source terms was firstly considered by Levine [16]. Since then, the damped wave equation with polynomial nonlinear source of the form u ttu + h(u t ) = |u| p-2 u (1.5) has been studied extensively on existence, nonexistence, stability, and blow-up of solutions (see [4,12,13,30] and the references therein). Recently, much attention has been paid to the study of nonlinear models of hyperbolic and parabolic equations with logarithmic source nonlinearity [1, 2, 5-8, 17, 18, 22]. For the strongly damped wave equation u ttua u t + bu t = u ln |u| γ , Ma and Fang [22] showed the global existence and infinite time blow-up of solutions when γ = 2, a = 1, and b = 0. They used a family of potential wells that is related to the logarithmic nonlinearity, which was introduced by Chen et al. [7]. Lian and Xu [18] proved the global existence, energy decay and infinite time blow-up of solutions when γ = 1, a ≥ 0, and b > -aλ, where λ is the first eigenvalue of the operator -under homogeneous Dirichlet boundary conditions. In [1], the authors considered the plate equation They proved the global existence of solutions and showed that the solutions decay exponentially for a suitable initial data. Later, they extended the results to the case of nonlinear damping in the work [2]. There is not much literature for wave equations with time delay and logarithmic nonlinear source. Thus, in this paper, we intend to study such problem; see (1.1)-(1.4). When γ = 0 in (1.1), Nicaise and Pignotti [24] proved that the energy decays exponentially under the condition 0 < β < α, and then improved the result to the case of time varying delay in [25]. For related work on problems with time delay, we also refer to [9,14,27,31,32] and the references therein. Inspired by these results, we discuss the solutions for problem (1.1)-(1.4). To the best of our knowledge, there is little work that takes into account wave equations with time delay and logarithmic source. Thus, we prove the local existence of solutions for problem (1.1)-(1.4) via Faedo-Galerkin's method and the logarithmic Sobolev inequality, and then show the global existence and energy estimates of solutions using the perturbed energy method. Moreover, we establish an infinite time blow-up result by applying the ideas presented in [20,23,26] with some necessary modification. The outline of this paper is as follows. In Sect. 2, we give some notations and material needed for our work. In Sect. 3, we prove the local existence for problem (1.1)-(1.4). In Sect. 4, we provide the global existence and energy decay rates of solutions. Finally, in Sect. 5, we show that the solution occurs with an infinite time blow-up.

Preliminaries
We denote the norm of X by · X for a Banach space X. We denote the scalar product in L 2 ( ) by (·, ·). For brevity, we denote · 2 by · . Let B 1 be the optimal constant of the embedding inequality With regard to problem (1.1)-(1.4), we impose the following assumptions: (H 1 ) The weights of dissipation and delay satisfy Let us list some lemmas for our work.
Lemma 2.1 (Logarithmic Sobolev inequality [7,11]) For any u ∈ H 1 0 ( ) and any positive real number k, Remark 2.1 Even though the inequality (2.4) holds for all k > 0, for the computations throughout this work, we take the constant k satisfying where μ is any real number with 0 < μ < 1.
Remark 2.2 For a given v ∈ H 1 0 ( ), J(λv) has the absolute maximum value at λ * , that is, Taking the limit k → π γ -, we have Considering this and (2.12), we have and hence Thus, we obtain from (2.13) and (2.9) By the definition of d given in (2.10), we get the desired result.

Local existence of solutions
In this section we prove the local existence of solutions by applying the ideas in [1,24]. Using the function Definition 3.1 Let T > 0. We say that (u, y) is a local solution of problem (3.2)-(3.6) if it satisfies the following: and Since problem (3.7)-(3.9) is a normal system of ordinary differential equations, there exists a solution (u n , y n ) on the interval [0, t n ), t n ∈ (0, T]. The extension of this solution to the whole interval [0, T) is a consequence of the estimate below.
Replacing v by u n t (t) in (3.7) and using the relation Replacing ϕ by ωy n (η, t) in (3.8), one sees Collecting (3.10) and (3.11), we get By Young's inequality and the fact y n (x, 0, t) = u n t (x, t), we get and where  Since the function s → s ln |s| γ is continuous on R, Now, we let 1 = x ∈ | u n (x, t) < 1 and 2 = x ∈ | u n (x, t) ≥ 1 .
Then we have here we used the fact |s ln s| ≤ 1 e for 0 < s < 1 and s -κ ln s ≤ 1 eκ for s ≥ 1 and κ > 0.
From (3.25) and (3.17), we arrive at where B 2 is the best Sobolev imbedding constant of Thus, we have from (3.26) u n ln u n γ is uniformly bounded in L ∞ 0, T; L 2 ( ) . (3.27) By the Lebesgue bounded convergence theorem, (3.24), and (3.27), we infer u n ln u n γ → u ln |u| γ strongly in L 2 0, T; L 2 ( ) . Now, we are ready to pass to the limit m → ∞ in (3.7) and (3.8). The proof of the remainder is standard and can be done as in [1,19].

Global existence and energy decay estimate
In this section, we prove the global existence and energy decay rates of solutions to problem (3.2)- (3.6). For this, we define the energy of problem (3.2)-(3.6) as where ω is the positive constant given in (3.12). It is noted that By the same arguments as of (3.13), we can deduce where C 1 and C 2 are positive constants given in (3.15). Let t 0 be the maximum of t 1 satisfying (4.5). Suppose t 0 < T, then I(u(t 0 )) = 0, that is, Thus, we have from (2.11) But this is contradiction to the following relation: It is noted that E(t) is a nonincreasing positive function from (4.3) and Lemma 4.1.

Theorem 4.1 Under the conditions of Lemma 4.1, the solution u is global.
Proof It suffices to show that u t (t) 2 + ∇u(t) 2 is bounded independent of t. From Lemma 4.1, (4.2), and (4.3), we have Similarly, we see From Lemma 2.1 and (2.8), we infer Taking the limit k → ρ + in this inequality and using (4.7), we get From Lemma 2.4 and (2.5), we get Thus, we observe from (4.8) and (4.7) that 1 - This gives We complete the proof from (4.6) and (4.9).
Proof Young's inequality and Lemma 4.1 imply Taking ε > 0 suitably small, we complete the proof. Proof Using (3.2) and Young's inequality, we have

Infinite time blow-up
In this section, inspired by the ideas in [20,23,26], we establish a blow-up result for problem (1.1)- (1.4). For this, we first give the following lemma. and where T is the maximal existence time of solutions.
Proof Since I(u 0 ) < 0 and u is continuous on [0, T), we know that Let t 0 be the maximal time satisfying (5.3) and suppose t 0 < T, then I(u(t 0 )) = 0, that is, Thus, we have This is in contradiction to Lemma 2.4. So, (5.1) is proved. From Lemma 2.4, (2.13), and (5.1), we find Thus, we complete the proof.
Thus, G(t) blows up at infinity.