The main results of this paper are the global existence and long time behavior of solutions of a fractional wave equation with a nonlocal nonlinearity. The techniques in this work rely on norm estimates of the solutions of ε u tt + u t + ( - Δ ) β u = 0 , u ( 0 , x ) = φ ( x ) , u t ( 0 , x ) = ψ ( x ) , which we derive particularly to observe the roles of ε and β in the long time behavior of solutions. Moreover, we apply these estimates to obtain local in time weak solutions, and global solutions, under the influence of a nonlocal non-power nonlinear term.

This paper examines the critical exponents for the existence of global solutions to the equation u t − ∆ H u = ∫ 0 t ( t − s ) − γ | u ( s ) | p − 1 u ( s ) ds , 0 ≤ γ < 1 , η ∈ H n , t > 0 , on the Heisenberg groups H n . There exists a critical exponent p c = max { 1 γ , n + 2 n + γ } ∈ ( 0 , + ∞ ] , such that for all 1 < p ≤ p c , no global solution exists regardless of the non-negative initial data, while for p > p c , a global positive solution exists if the initial data is sufficiently small. The results obtained are a natural extension of the results of Cazenave et al. [Nonlinear Analysis 68 (2008), 862-874], where similar studies were carried out in R n . Furthermore, several theorems are presented that provide lifespan estimates for local solutions under various initial data conditions. The proofs of the main results are based on test function methods and Banach fixed point principle.

In this paper, we investigate the existence and stability properties of both classical and fractional-order generalized Brusselator reaction-diffusion systems. By analyzing the system in both ODEs and PDEs forms, we establish the global existence of solutions in the classical case by using an appropriate Lyapunov functional, we also show the local existence of solutions in the fractional-order case. Additionally, we derive sufficient conditions for local asymptotic stability. To illustrate our theoretical findings, we consider the Brusselator reaction-diffusion model as a particular case of our generalized system and employ numerical methods to explore its solutions under the proposed conditions.

The purpose of this paper is to investigate the non-existence of global weak solutions of the following degenerate inequality on the Heisenberg group { u t – ∆ H u ≥ ( K ∗ H | u | p ) | u | q , η ∈ H n , t > 0 , u ( η , 0 )= u 0 ( η ), η ∈ H n , where n≥1, p, q>0, u 0 ∈ L loc 1 ( H n ) , ∆ H is the Heisenberg Laplacian, and K:(0,∞)→(0,∞) is a continuous function satisfying K ( | · | H ) ∈ L loc 1 ( H n ) which decreases in a vicinity of infinity. In addition, ∗ H denotes the convolution operation in H n . Our approach is based on the non-linear capacity method.

We consider a non-linear wave equation with an internal fractional damping, a polynomial source and an infinite memory. Using the semi-group theory, we get the existence of a local weak solution. Moreover, we show under some conditions, local solutions may blow up a in finite time; this is achieved by constructing a suitable Lyapunov functional.

In this work, we consider a mathematical model of viscoelastic incompressible fluid governed by the Navier-Stokes-Voigt equations in a three dimensional thin domain Ω ε , with damping term and Tresca friction law. First, we give the problem statement and the weak variational formulation of the considered problem. Then we study the asymptotic analysis of the problem when a dimension of the domain tends to zero. The limit problem and the specific equation of Reynolds are obtained.

The Moore-Gibson-Thompson equation with a viscoelastic memory and a forcing is considered. The existence and uniqueness of a local solution is obtained via the Faedo-Galerkin's method. Furthermore, blowing-up solutions with or without a positive initial energy exist due to the nonlinear forcing.