Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems with p

This paper deals with p-Laplacian systems ut−div(|∇u|p−2∇u)=∫Ωvα(x, t)dx, x∈Ω, t>0, vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx, x∈Ω, t>0, with null Dirichlet boundary conditions in a smooth bounded domain Ω⊂ℝN, where p,q≥2, α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω=BR={x∈ℝN:|x|0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time

Approximate Analytical Solutions of Fractional Perturbed Diffusion Equation by Reduced Differential Transform Method and the Homotopy Perturbation Method

The approximate analytical solutions of differential equations with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization, or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The two different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance

Uniform Blow-Up Rates and Asymptotic Estimates of Solutions for Diffusion Systems with Nonlocal Sources

This paper investigates the local existence of the nonnegative solution and the finite time blow-up of solutions and boundary layer profiles of diffusion equations with nonlocal reaction sources; we also study the global existence and that the rate of blow-up is uniform in all compact subsets of the domain, the blow-up rate of |u(t)|∞ is precisely determined

Resonances of a fractional-order biomedical model with time delay state feedback

In the present paper, the primary resonance of a fractional-order Willis aneurysm system with time-delay state feedback control is studied. Using the multiple scale method, the amplitude and phase equations are obtained. The first order approximate solution is derived and the influence of time delay on resonance is studied. The concept of equivalent damping related to time-delay feedback is proposed, and the reasonable selection of feedback gain and time delay is discussed from the point of view of vibration control. The frequency response and external excitation response curves of the system are given. In order to test the stability of the system, bifurcation analysis is carried out. The obtained results are very useful in the clinical diagnosis and treatment of cerebral aneurysms

Approximate Analytical Solutions of Fractional Perturbed Diffusion Equation by Reduced Differential Transform Method and the Homotopy Perturbation Method

The approximate analytical solutions of differential equations with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization, or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The two different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance

Existence Results for a Class of the Quasilinear Elliptic Equations with the Logarithmic Nonlinearity

In this paper, the nonlinear quasilinear elliptic problem with the logarithmic nonlinearity −div∇up−2∇u=axφpulogu+hxψpu in Ω⊂Rn was studied. By means of a double perturbation argument and Nehari manifold, the authors obtain the existence results