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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 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 paper, we obtain the existence of a global attractor for the higher-order evolution type equation. Moreover, we discuss the asymptotic behavior of global solution.

In this paper, we investigate a nonlinear system of viscoelastic equations with degenerate damping and source terms in a bounded domain. Under appropriate assumptions on the parameters, degenerate damping terms and the relaxation functions 𝜛𝑖, (𝑖 = 1, 2), we prove local existence and uniqueness of the solution by using the Faedo–Galerkin method with a new scenario. Then, we prove the blow-up of weak solutions to problem (1.1). This improves earlier results in the literature [6,23,25].

In this paper, we consider the following Timoshenko equation utt + ( 2u ( M ((( ( u (( 2( ( u + ut = (u( q-1 u associated with initial and Dirichlet boundary conditions. We prove the non-existence of solutions with positive and negative initial energy.

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]