Nonlinear Memory Term Research Articles

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

Higher order evolution inequalities involving Leray–Hardy potential singular on the boundary

We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.

On the Global Nonexistence of a Solution for Wave Equations with Nonlinear Memory Term

The paper is devoted to the problem of the local existence for a solution to a nonlinear wave equation, with the dissipation given by a nonlinear form with the presence of a nonlinear memory term. Moreover, the global nonexistence of a solution is established using the test function method. We combine the Fourier transform and fractional derivative calculus to achieve our goal.

Well posedness for a heat equation with a nonlinear memory term

Well posedness for a heat equation with a nonlinear memory term

Threshold Results for the Existence of Global and Blow-Up Solutions to a Time Fractional Diffusion System with a Nonlinear Memory Term in a Bounded Domain

In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property.

A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term

<p style='text-indent:20px;'>In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, which can be represented by the Riemann-Liouville fractional integral of order <inline-formula><tex-math id="M2">\begin{document}$ 1-\gamma $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \gamma\in(0, 1) $\end{document}</tex-math></inline-formula>. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, on the blow-up range between the effective case and the non-effective case.</p>

Higher order evolution inequalities with convection terms in an exterior domain of [formula omitted

We establish new blow-up results for a higher order (in time) evolution inequality involving a convection term in an exterior domain of RN. We study two types of inhomogeneous boundary conditions: Dirichlet and Neumann. Using a unified approach, we obtain optimal criteria of Fujita type for each case. Our study yields naturally optimal nonexistence results for the corresponding stationary problems. We also investigate the effect of a nonlinear memory term on the critical behaviors of the considered problems. Notice that no restriction on the sign of solutions is imposed.

Dynamics of a colloidal particle coupled to a Gaussian field: from a confinement-dependent to a non-linear memory

The effective dynamics of a colloidal particle immersed in a complex medium is often described in terms of an overdamped linear Langevin equation for its velocity with a memory kernel which determines the effective (time-dependent) friction and the correlations of fluctuations. Recently, it has been shown in experiments and numerical simulations that this memory may depend on the possible optical confinement the particle is subject to, suggesting that this description does not capture faithfully the actual dynamics of the colloid, even at equilibrium. Here, we propose a different approach in which we model the medium as a Gaussian field linearly coupled to the colloid. The resulting effective evolution equation of the colloidal particle features a non-linear memory term which extends previous models and which explains qualitatively the experimental and numerical evidence in the presence of confinement. This non-linear term is related to the correlations of the effective noise via a novel fluctuation-dissipation relation which we derive.

A Non-autonomous Damped Wave Equation with a Nonlinear Memory Term

A Non-autonomous Damped Wave Equation with a Nonlinear Memory Term

The interplay between fractional damping and nonlinear memory for the plate equation

In this paper, we study the interplay between a fractional damping (− Δ)θut with θ ∈ [0, 1/2), and a nonlinear memory term applied to a plate equation: in space dimension . In different scenarios, we prove the global existence of small data solutions in for supercritical powers, where the critical exponent is determined by the fractional power of the damping term and by the order of the fractional integration in the memory term. In one case, we also feel the influence of the regularity‐loss type decay, which is due to the presence of the rotational inertia term −Δutt in the plate equation.

Blow-up rates for a higher-order semilinear parabolic equation with nonlinear memory term

ABSTRACT In this paper, we establish blow-up rates for a higher-order semilinear parabolic equation with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.

Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term

<abstract><p>In this paper, we investigate blow-up conditions for the semilinear generalized Tricomi equation with a general nonlinear memory term in $ \mathbb{R}^n $ by using suitable functionals and employing iteration procedures. Particularly, a new combined effect from the relaxation function and the time-dependent coefficient is found.</p></abstract>

A structurally damped σ‐evolution equation with nonlinear memory

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: with σ>0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m>1.

Global existence and blow-up of solutions for p-Laplacian evolution equation with nonlinear memory term and nonlocal boundary condition

In this paper, we deal with an initial boundary value problem for a p-Laplacian evolution equation with nonlinear memory term and inner absorption term subject to a weighted linear nonlocal boundary condition. We find the effects of a weighted function as regards determining blow-up of nonnegative solutions or not and establish the precise blow-up estimate for the linear diffusion case under some suitable conditions.

Improved equalizer of nonlinear satellite channel

Concerning the defect that Volterra structure equalizer has large amount of computation, an improved adaptive equalizer of nonlinear satellite channel was proposed, and a new cascade structure of a linear equalizer and a nonlinear equalizer were obtained by analyzing the truncated Volterra series. The third-order nonlinear memory term multiplying in the Volterra equalizer expression was converted to second-order nonlinear memory term multiplying in the new structure equalizer.When signals passed through the equalizer, the number of complex multiplications was reduced. The simulation results show that the number of complex multiplications of improved nonlinear satellite channel adaptive equalizer is 1 /9 times as that of Volterra equalizer when channel has strong memory depth. Meanwhile, 16 Amplitude Phase Shift Keying( 16APSK) signal equalized by the improved adaptive equalizer has more compact constellation points.

Global existence and blow-up properties of solutions for porous medium equation with nonlinear memory and weighted nonlocal boundary condition

In this paper, we investigate a initial boundary value problem of porous medium equation with nonlocal boundary condition. We find the effects of weight function in the boundary condition and competitive relationship between nonlinear memory term and inner absorption term on whether determining blow-up of solutions or not, and give the blow-up rate estimate under some suitable condition.

Decay of mass for fractional evolution equation with memory term

We investigate the decay properties of the mass M ( t ) = ∫ R N u ( ⋅ , t ) d x M(t)= \int _{\mathbb {R}^N} u(\cdot ,t)dx of the solutions of a fractional diffusion equation with nonlinear memory term. We show, using a suitable class of initial data and a restriction on the diffusion and nonlinear term, that the memory term determines the large time asymptotics; that is, M ( t ) M(t) tends to zero as t → ∞ . t\to \infty .

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.

Convergence of the Homogenization Process for a Double-Porosity Model of Immiscible Two-Phase Flow

In this paper, we justify by periodic homogenization the double-porosity model for immiscible incompressible, two-phase flow. The volume fraction of the fissured part and the nonfissured part are kept positive constants and of the same order. The scaling is such that, in the final homogenized equations, the less permeable part of the matrix contributes as a nonlinear memory term. To prove the convergence of the total velocity and of the “reduced” pressure, we use the two-scale convergence since it seems to be appropriate for the problem, even though it would be possible to work with periodic modulation. However, in the final step, the degenerate ellipticity prevents the use of the two-scale convergence method and leads us to use periodic modulation.