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This paper investigates a coupled system of nonlinear hyperbolic equations with weak damping. It is shown that the global solution and finite time blow-up solution exist at subcritical or critical initial energy. Moreover, the finite time blow-up result at arbitrarily high initial energy is obtained. Also, we derive the exponential energy decay estimation of the global solution and estimate the upper and lower bounds of the blow-up time. [ABSTRACT FROM AUTHOR]

This work is concerned with a Kirchhoff wave equation with strong nonlinear damping. We establish the local existence of the solution and provide some sufficient conditions for the global existence and finite time blow‐up of the solution. Moreover, we derive the exponential decay estimation of the energy functional and the upper and lower bounds of the blow‐up time. [ABSTRACT FROM AUTHOR]

In this paper, we study the following thermoelastic system with p-Laplacian { u t t − Δ p u + θ = | u | r − 1 u , θ t − Δ θ = u t in a bounded domain Ω ⊂ R n , where u and θ denote the elongation of a plate and the temperature difference to the equilibrium state, respectively; Δ p denotes the classical p-Laplacian operator. By employing the potential well theory, we discuss the properties of global existence and finite time blow-up for the solutions. Moreover, we give the lower and upper bounds of blow-up time to the blow-up solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Kirchhoff-type parabolic equation with logarithmic nonlinearity. By making a more general assumption about the Kirchhoff function, we establish a new finite time blow-up criterion. In particular, the blow-up rate and the upper and lower bounds of the blow-up time are also derived. These results generalize some recent ones in which the blow-up results were obtained when the Kirchhoff function was assumed to be a very special form. [ABSTRACT FROM AUTHOR]

This paper investigates the well‐posedness of solutions for the following quasilinear wave equation with strong damping and logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary: utt−Δu−div(∇u1+|∇u|2)−Δut=|u|r−2ulog|u|$u_{tt}-\Delta u-{\rm div}(\frac{\nabla u}{\sqrt {1+|\nabla u|^2}})-\Delta u_t=|u|^{r-2}u\log |u|$, where r≥2$r\ge 2$. By virtue of the classical Faedo–Galerkin method and some technical efforts, we first establish the local well‐posedness of solutions. Then we discuss the dynamical behaviors of solutions in detail: 1.When r≥2$r\ge 2$ and I(u0)>0$I(u_0)>0$, we show that the solutions exist globally with subcritical and critical initial energy, where I(u0)$I(u_0)$ denotes the Nehari functional with the initial value u0. Especially, under further suitable assumptions about the initial data, we show that the energy functional decays exponentially. 2.When r>2$r>2$ and I(u0)<0$I(u_0)<0$, we show that the solutions blow up in finite time with subcritical and critical initial energy. Moreover, by removing the restriction I(u0)<0$I(u_0)<0$, we prove that the solutions may blow up in finite time with arbitrary high initial energy. In particular, we derive the upper and lower bounds of the blow‐up time. 3.When r=2$r=2$ and I(u0)<0$I(u_0)<0$, we show that the maximal existence time of solutions can be extended to infinity and the solutions blow up at infinity with subcritical, critical, and arbitrary high initial energy. [ABSTRACT FROM AUTHOR]

This paper investigates the infinite time blow‐up of solutions with arbitrary high initial energy to wave equations with weak damping term, strong damping term, and logarithmic nonlinearity. This problem has been studied previously with the assumptions that there is no strong damping term and the initial displacement and initial velocity have the same sign. However, from the physical point of view, it is obvious that the initial displacement and initial velocity may have different signs, and it is very necessary to consider the effects of the strong damping term. For example, the strong damping term indicates that the stress is not only proportional to the strain as with the Hooke law, but also proportional to the strain rate as in a linearized Kelvin–Voigt material. In this paper, by providing a completely different method from previous studies, we show that the solutions may blow up at infinity with arbitrary high initial energy when the model involves the strong damping term and the initial displacement and initial velocity may have different signs. Moreover, in this paper, we prove for the first time how to extend the solution over time (the whole half line) in studying the infinite time blow‐up phenomena for hyperbolic equations with logarithmic nonlinearity. These results fill in the gaps in previous studies on this type of models. [ABSTRACT FROM AUTHOR]

This paper deals with the following mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity u t − Δ u t − d i v (| ∇ u | p − 2 ∇ u) = | u | q − 2 u log ⁡ | u | in a bounded domain with zero Dirichlet boundary condition, which was studied in our previous paper [J Math Anal Appl. 2019;478(2):393-420]. In view the results of [J Math Anal Appl. 2019;478(2):393-420], for the case (1) 1 < p ≤ q ≤ 2 , i f n ≤ p , ≤ 2 , i f 2 n n + 2 < p < n , < n p n − p , i f p ≤ 2 n n + 2 , the global existence and blow-up results were got when J (u 0) ≤ d , where d denotes the mountain-pass level. But for the case (2) 1 < p ≤ q a n d 2 < q < ∞ , i f n ≤ p , n p n − p , i f 2 n n + 2 < p < n , the blow-up results were got when J (u 0) ≤ M , where M ≤ d is a constant. In this paper, we extend and complete the results of [J Math Anal Appl. 2019;478(2):393-420] on the following three aspects: First, the blow-up results are got when J (u 0) ≤ d and (2) are satisfied. Second, the upper and lower bounds of blow-up time are estimated. Third, the global existence and blow-up results are got when J (u 0) > d. [ABSTRACT FROM AUTHOR]

In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity: u t + M ([ u ] s 2) L K u = | u | p - 2 u log | u | , in Ω × (0 , + ∞) , u (x , t) = 0 , in (R N \ Ω) × (0 , + ∞) , u (x , 0) = u 0 (x) , in Ω , where [ u ] s is the Gagliardo seminorm of u, Ω ⊂ R N is a bounded domain with Lipschitz boundary, 0 < s < 1 , L K is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator (- Δ) s , u 0 is the initial function, and M : [ 0 , + ∞) → [ 0 , + ∞) is continuous. Let J (u 0) be the initial energy (see (2.1) for the definition of J), d > 0 be the mountain-pass level given in (2.4), and M ~ ∈ (0 , d ] be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for J (u 0) ≤ d . Then we study the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for J (u 0) ≤ M ~ , the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as t → T is also discussed, where T is the blow-up time. In addition, for J (u 0) ≤ d , we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution. [ABSTRACT FROM AUTHOR]

In this note, the decay estimate of solutions for a class of pseudo-parabolic p -Laplacian equation with logarithmic nonlinearity is studied, which revises and improves a recent result obtained in [He et al. (2018), Theorem 2]. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the following nonlocal Kirchhoff diffusion problem:∂tu+M([u]s2)LKu=|u|p−2u,inΩ×R+,u(x,t)=0,in(RN\Ω)×R+,u(x,0)=u0(x),inΩ,where is a bounded domain with Lipschitz boundary, [u]s is the Gagliardo seminorm of u, 0 < s < min{1, N/2}, is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator (−Δ)s, u0 : Ω → [0, +∞) is the initial function, M : [0, +∞) → [0, +∞) is a continuous function and there exist two constants θ > 1 and m0 > 0 such thatM(σ)⩾m0σθ−1,∀σ∈[0,+∞).This problem has been investigated by Xiang, Rădulescu and Zhang in [30], and Ding and Zhou in [8] by using potential well method. If , in [8], the authors showed the existence of a nontrivial, nonnegative global weak solution, where. However, if , these two papers only studied the model in special cases, the details are as follows: in [30], the blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) < 0; in [8], the global existence and blow-up conditions for nontrivial, nonnegative weak solution were obtained when J(u0) ⩽ d and M(σ) = m0σθ−1, where J(u0) denotes the initial energy and d > 0 denotes the depth of the potential well (see (1.14)). The main purpose of this paper is to extend the above results to the general case M(σ) ⩾ m0σθ−1, , and the conditions on global existence and finite time blow-up are obtained. Furthermore, the decay estimates for global weak solutions, the growth estimates for blow-up solutions, the upper and lower bounds of blow-up time to blow-up solutions, the behavior of the energy functional as t → T (where T denotes the blow-up time) are studied. Moreover, some blow-up conditions independent of d and some equivalent conditions for the weak solutions existing globally or blowing up in finite time are investigated. Finally, the global existence and finite time blow-up results with high initial energy (i.e., J(u0) > d) are obtained. [ABSTRACT FROM AUTHOR]

The finite time blow-up of solutions to a nonlinear Timoshenko-type equation with variable exponents is studied. More concretely, we prove that the solutions blow up in finite time with positive initial energy. Also, the existence of finite time blow-up solutions with arbitrarily high initial energy is established. Meanwhile, the upper and lower bounds of the blow-up time are derived. These results deepen and generalize the ones obtained in [Nonlinear Anal. Real World Appl., 61: Paper No. 103341, 2021]. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of Boussinesq-type equations with logarithmic nonlinearity. By employing the classical Faedo–Galerkin method, we first establish the local well-posedness of solutions. Then we investigate the dynamical behaviors of solutions. More precisely, for the solutions with subcritical or critical initial energy, we prove that they exist globally and are uniformly bounded when I (u 0) > 0 , where I (u 0) denotes the Nehari functional with the initial value u 0. Moreover, under further appropriate assumptions about the initial data, we derive the exponential energy decay estimates of global solutions. In particular, for the solutions with subcritical or critical initial energy, we show that they can be extended over time (the whole half line) and then blow up at infinite time when I (u 0) < 0. Last but not least, by developing some new methods, we prove the existence of infinite time blow-up solutions with arbitrary high initial energy. [ABSTRACT FROM AUTHOR]

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as t → T (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]). [ABSTRACT FROM AUTHOR]