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In this paper, we deal with a p(x)-biharmonic heat equation with variable exponent under Dirichlet boundary and initial condition. We prove the blow up of solutions under suitable conditions.

Our aim here is to collect and to compare two definitions of the fractional powers of non-negative operators that can be found in the literature; we will present the proof of an equivalence and compare properties of that notions in different approaches. Then we will apply next this equivalence in the study of global and local existence of solution for the semilinear Cauchy problem in R N with fractional Laplacian ut = -(-∆)au + f(x, u), u(0, x) = u0(x), x ∈ R N .

In this work, we deal with the higher-order hyperbolic-type equation. Firstly, we prove the global existence and blow up of solutions with the subcritical initial energy. Later, we prove the global existence and blow up of solutions with the critical initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a plate viscoelastic p(x)–Kirchhoff type equation with variable-exponent nonlinearities of the form associated with initial and boundary feedback. Under appropriate conditions on p(·), m(·) and q(·), general decay result along the solution energy is proved. By introducing a suitable auxiliary function, it is also shown that regarding negative initial energy and a suitable range of variable exponents, solutions blow up in a finite time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weighted m -biharmonic equation with nonlinear damping and source terms. We proved the global existence of solutions. Later, the decay of the energy is established by using Nakao's inequality. Finally, we proved the blow-up of solutions in finite time. [ABSTRACT FROM AUTHOR]

In this article, we deal with a logarithmic wave equation with strong damping and delay. Firstly, we prove the local existence by utilizing the semigroup theory. Later, we obtain the global existence of solutions by using the well‐depth method. Moreover, under appropriate assumptions on the weight of the delay and that of strong damping, we get the exponential decay. [ABSTRACT FROM AUTHOR]

In this research, we take into account the Kirchhoff type equation with variable exponent. The Kirchhoff type equation is known as a kind of evolution equations, namely, PDEs, where t is an independent variable. This type problem can be extensively used in many mathematical models of various applied sciences such as flows of electrorheological fluids, thin liquid films, and so on. This research, we investigate the upper bound for blow up time under suitable conditions. [ABSTRACT FROM AUTHOR]

This paper contains results about the existence, uniqueness and stability of solutions for the damped nonlinear extensible beam equation$ u_{tt}+\Delta ^2u-M(\|\nabla u(t)\|^2)\Delta u+\|\Delta u(t)\|^{2\alpha}\,|u_t|^{\gamma}u_t = 0\ \mbox{ in } \ \Omega \times \mathbb{R}^+, $where $ \alpha>0 $, $ \gamma\ge 0 $, $ \Omega\subset \mathbb{R}^n $ is a bounded domain with smooth boundary $ \Gamma = \partial \Omega $, and $ M $ is a nonlocal function that represents beam's extensibility term. The novelty of the work is to consider the damping as a product of a degenerate and nonlocal term with a nonlinear function. This work complements the recent article by Cavalcanti et al. [8] who treated this model with degenerate nonlocal weak (and strong) damping. The main result of the work is to show that for regular initial data the energy associated with the problem proposed goes to zero when $ t $ goes to infinity. [ABSTRACT FROM AUTHOR]

We investigate a Kirchhoff-type plate equation with a degenerate damping term. We prove blow up and exponential growth of the solution under some sufficient conditons on the initial data. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear beam equation with a strong damping and the p(x)-biharmonic operator. The exponent p(∙) of nonlinearity is a given function satisfying some condition to be specified. Applying Faedo-Galerkin's method, the existence of weak solutions is proved. Using Nakao's lemma, the asymptotic behavior of weak solutions is established with mild assumptions on the variable exponent p(∙). We show the asymptotic behavior of the weak solution is exponentially and algebraically depending on the variable exponent. This work improves and extends many other results in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with the initial boundary value problem for a higher-order parabolic equation with logarithmic source term ... . By employing the potential wells technique we show the global existence of the weak solution. Also, we obtain the exponential decay for the weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear beam equation with a strong damping and the p(x)-biharmonic operator. The exponent p(·) of nonlinearity is a given function satisfying some condition to be specified. Using Faedo-Galerkin method, the local and global existence of weak solutions is established with mild assumptions on the variable exponent p(·). This work improves and extends many other results in the literature. [ABSTRACT FROM AUTHOR]

In this article, we consider a logarithmic viscoelastic plate equation with distributed delay. Firstly, we study the local and global existence of solutions by using the energy method combined with Faedo-Galerkin method. Then, by introducing a suitable Lyapunov functional, we prove the asymptotic behavior of the solution. Our results are more general than the earlier results. [ABSTRACT FROM AUTHOR]

In this paper, we aim to understand the characteristics of dynamical behaviour for a higher order Kirchhoff type systems with logarithmic nonlinearities. Based on the potential well method, the main ingredient of this study is to construct several conditions for initial data leading to the solution global existence in case of E (0) < d. On the other hand, we proved that if the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of energy. [ABSTRACT FROM AUTHOR]