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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]

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]

In this paper, we study an initial-boundary value problem for the following mixed pseudo-parabolic p -Laplacian type equation with logarithmic nonlinearity in a bounded domain Ω ⊂ R n : u t − Δ u t − div (| ∇ u | p − 2 ∇ u) = | u | q − 2 u log ⁡ | u | , 1 < p ≤ q { < ∞ , if n ≤ p ; < n p n − p , if n > p , which is referred to as Showalter equation [1]. We focus on the initial conditions, which ensure the solutions of the model exist global, blow up in finite time and blow up at infinite time. Explicitly, we classify the ranges of p , q into two cases: (0.1) 1 < p ≤ q { ≤ 2 , if n ≤ p , ≤ 2 , if 2 n n + 2 < p < n , < n p n − p , if p ≤ 2 n n + 2 , and (0.2) 1 < p ≤ q and 2 < q < { ∞ , if n ≤ p , n p n − p , if 2 n n + 2 < p < n. Let J (u 0) be the initial energy (see (2.1) for the definition of J), I (u 0) be the vale of Nehari functional at u 0 (see (2.2) for the definition of I), d > 0 be the mountain-pass level given in (2.4) , and M ∈ [ 0 , d ] be the constant given in (2.6) , we get (i) If J (u 0) ≤ d and I (u 0) ≥ 0 , then the solution exists globally; (ii) If J (u 0) ≤ d , I (u 0) < 0 and p , q satisfies (0.1) , then the solution blows up at infinite time. Moreover, if in addition J (u 0) { ≤ M , if M < d , < M , if M = d , the growth rate of the solution is got; (iii) If J (u 0) ≤ M , I (u 0) < 0 and p , q satisfies (0.2) , then the solution blows up in finite time. [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]