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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 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 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 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 investigate a nonlinear Petrovsky equation with variable exponent and damping terms. First, we establish the local existence using the Faedo–Galerkin approximation method under the conditions of positive initial energy and appropriate constraints on the variable exponents p ⋅ and q ⋅ . Finally, we prove a finite-time blow-up result for negative initial energy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Kirchhoff-type viscoelastic equation with degenerate damping term have initial and Dirichlet boundary conditions. We obtain the blow up and exponential growth of solutions with negative initial energy. [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 article, we deal with a logarithmic wave equation with distributed delay. Firstly, we establish the well-posedness 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 distributed delay and that of strong damping, we get the exponential decay results. [ABSTRACT FROM AUTHOR]

This paper deal with the initial boundary value problem for a higher-order heat equation with logarithmic source term ut + (−∆)mu − ∆ut = uk−2u ln |u| . We obtain blow-up of weak solutions in the finite time, by employing potential well technique and concave technique. In addition, the upper bound of blow-up time is considered. This improves and extends some previous studies. [ABSTRACT FROM AUTHOR]

In this work, we investigate generalized coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms and initial-boundary value conditions, in a bounded domain. We obtain decay of solutions by use of Nakao inequality. The blow up of solutions with negative initial energy is also established. [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 deals with a parabolic-type Kirchhoff equation with variable exponents. Firstly, we obtain the global existence of solutions by Faedo-Galerkin method. Later, we prove the decay of solutions by Komornik's inequality. [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 consider the existence, both locally and globally in time, the decay and the blow up of the solution for the extensible beam equation with nonlinear damping and source terms. We prove the existence of the solution by Banach contraction mapping principle. The decay estimates of the solution are proved by using Nakao’s inequality. Moreover, under suitable conditions on the initial datum, we prove that the solution blow up in finite time.

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]