Blowup for nonlinearly damped viscoelastic equations with logarithmic source and delay terms

In this work, we investigate blowup phenomena for nonlinearly damped viscoelastic equations with logarithmic source effect and time delay in the velocity. Owing to the nonlinear damping term instead of strong or linear dissipation, we cannot apply the concavity method introduced by Levine. Thus, utilizing the energy method, we show that the solutions with not only non-positive initial energy but also some positive initial energy blow up at a finite point in time.

Many researchers have studied parabolic or hyperbolic equations with logarithmic nonlinearity [2,4,7,13,15]. For the physical application of this nonlinearity, we refer to [1,6]. In [4], the authors discussed a strongly damped equation, u ttuu t = |u| p-2 u ln |u| with Dirichlet boundary condition. They showed that the solutions with subcritical and critical initial energy blow up in a finite point under suitable conditions. Moreover, they estimated bounds of the blowup time. The authors of [7] proved similar results to those of [4] for the equation with memory. Most work dealing with wave equations with logarithmic nonlinearity is associated with a strongly or linearly damped mechanism, and blowup results are investigated by virtue of the potential well method and Levine's concavity technique [12].
On the other hand, time delay effect arises in many natural phenomena depending not only on the present state but also on some past occurrences. Thus, partial differential equations with time delay have become an active area of research in resent years. For the physical application of the time delay, we refer to [3,18]. Recently, Kafini and Messaoudi [8] considered the wave equation with linear damping and delay terms with Dirichlet boundary condition. They established a blowup result of the solution with negative initial energy by adapting the energy method. While there are many studies on the existence and asymptotic stability of the solutions of wave equations with delay, there are relatively few studies on blowup. We refer to [5,10,16,17,19] and [9] for stability and blowup of equations with delay, respectively. Motivated by this pioneering work [8], in this article, we study blowup phenomena for the nonlinearly damped viscoelastic wave equation (1.1)-(1.4) with logarithmic source effect and time delay in the velocity. Due to the presence of nonlinear dissipation instead of strong or linear damping terms, we cannot apply the concavity method. Thus, by applying the energy method, we establish a blowup result of solutions with not only non-positive initial energy but also some positive initial energy. And, it is worth to mention that there are few works dealing with viscoelastic wave equations with nonlinear damping and logarithmic source terms.
Here is the outline of this paper. In Sect. 2, we present notations, hypotheses, and auxiliary functions and lemmas. In Sect. 3, we establish a blowup criterion of solutions with not only non-positive initial energy but also some positive initial energy.

Preliminaries
Throughout this article, (·, ·) denotes the scalar product in Hilbert space L 2 ( ). · r represents the norm in the space L r ( ). Moreover, · Y denotes the norm of a normed space Y . C > 0 represents a generic constant. If there is no ambiguity, we omit the variables t and x.
By the arguments of [7,9], we can state the well-posedness.
Our goal is to find a blowup result to problem (2.1)-(2.5). For this, we will often use the embedding and Young's inequality Also, we need the lemmas below, which are proved by Kafini and Messaoudi [8], to estimate logarithmic nonlinearity.
To establish our desired blowup result, we impose the following assumptions: (2.10) This implies Put D 0 be the embedding constant with and define a continuously differentiable function K as and put (2.13) (2.14) Lemma 2.4 For p > 2 and μ > 0, the function K satisfies Proof The results (i) and (ii) are clear. Since (2.16) We also need the following auxiliary result in the proof of our main theorem.

Blowup results
In this part, we search a blowup result of the solution to (2.1)-(2.5) inspired by the ideas in [8,14]. We define the energy to problem (2.1)-(2.5) by is fulfilled for some γ 1 > 0.

Conclusion
In this paper, we considered a viscoelastic wave equation with nonlinear damping and time delay terms and logarithmic source effect. Under the conditions (H n ), (H p ), (H q ), and (H k ), we showed that the solutions with not only a non-positive initial energy but also some positive initial energy blow up after a finite time by utilizing the energy method.