Decay Estimates Of Solutions Research Articles

Global solution for semilinear σ$$ \sigma $$‐evolution models with memory term

In this paper, the initial value problem of the semilinear ‐evolution equations with a memory term is concerned. Firstly, using the energy method in the Fourier space, the decay estimates for the solutions to the corresponding linear problem are established. Additionally, assuming small initial data in suitable time‐weighted Sobolev spaces, the global‐in‐time existence of the solutions to the semilinear issue is proved by contraction mapping. Finally, the decay estimates of solutions are obtained under the additional regularity assumption on the initial data.

Global Existence and Decay Estimates of Energy of Solutions for a New Class of p -Laplacian Heat Equations with Logarithmic Nonlinearity

The present research paper is related to the analytical studies of p -Laplacian heat equations with respect to logarithmic nonlinearity in the source terms, where by using an efficient technique and according to some sufficient conditions, we get the global existence and decay estimates of solutions.

L^{infty } decay estimates of solutions of nonlinear parabolic equation

In this paper, we are interested in L^{infty } decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain L^{infty } decay estimates of weak solutiona.

Energy and Large Time Estimates for Nonlinear Porous Medium Flow with Nonlocal Pressure in $$\mathbb {R}^N$$

We study the general family of nonlinear evolution equations of fractional diffusive type $$\partial _t u-\text{ div }\left( |u|^{m_1}\nabla (-\Delta )^{-s} [ |u|^{m_2-1}u]\right) =f$$ . Such nonlocal equations are related to the porous medium equations with a fractional Laplacian pressure. Our study concerns the case in which the flow takes place in the whole space. We consider $$m_1, m_2>0$$ , and $$s\in (0,1)$$ , and prove the existence of weak solutions. Moreover, when $$f\equiv 0$$ we obtain the $$L^p$$ - $$L^\infty $$ decay estimates of solutions, for $$p\geqq 1$$ . In addition, we also investigate the finite time extinction of solution.

On the Cauchy Problem for IMBq System Arising from DNA

In this article, we focus on the Cauchy problem for the generalized IMBq system in n-dimensional space, which arises from DNA. We show the global existence and decay estimates of solution for a class of initial velocity, provided that the initial value is suitably small.

The Effect of Gain and Strong Dissipative Structures on Nonlinear Schrödinger Equations in Optical Fiber

We consider the initial value problem for the nonlinear Schrödinger equation satisfying the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ in one space dimension. Our purpose in this paper is to study how the gain coefficient μ(t) and strong dissipative nonlinearity λvp-1v affect solutions to the nonlinear Schrödinger equation for large initial data. We prove global existence of solutions and present some time decay estimates of solutions for large initial data.

Global existence and time decay estimate of solutions to the Keller–Segel system

Global existence and time decay estimate of solutions to the Keller–Segel system

Global well-posedness and polynomial decay for a nonlinear Timoshenko–Cattaneo system under minimal Sobolev regularity

We consider the nonlinear stability of the Timoshenko–Cattaneo system in the one-dimensional whole space. The Timoshenko system consists of two coupled wave equations with non-symmetric relaxation, and describes vibrations of the beam with shear deformation and rotational inertia effect. Generally, if the relaxation is not symmetric, the dissipation is produced through the complicated interaction of the components of the system, and their decay estimates and the energy estimates are of regularity-loss type. In this paper, we introduce the mathematical method to control such a weak dissipativity by investigating the Timoshenko system with Cattaneo’s law, which is the first order approximation of Fourier’s law with its time-delay effect. Racke & Said-Houari (2012), showed the global existence and the decay estimate of solutions by assuming high regularity H8∩L1 on the small initial data to control their weak dissipativity. In contrast, we prove the global existence in H2 by energy methods without any negative weights. Our regularity assumption is the same as that needed to show the local existence. That is, we do not need to assume the extra higher regularity on the initial data. Besides, the optimal decay estimate in H2∩L1 is shown by using the time decay inequality of Lp-Lq-Lr type.

Optimal decay estimates for solutions to damped second order ODE's

Optimal decay estimates for solutions to damped second order ODE's

Decay estimate of global solutions to the generalized double dispersion model in Morrey spaces

In this paper, we investigate the initial value problem for the generalized double dispersion model in Morrey spaces. Based on the decay properties of the solution operator in Morrey spaces, global existence and decay estimates of solutions are proved by Banach fixed point theorem.

Semilinear nonlocal elliptic equations with critical and supercritical exponents

We study the problem $\left\{ \begin{align} &{{(-\Delta lta )}^{s}}u={{u}^{p}}-{{u}^{q}}\ \text{in}\ \text{ }{{\mathbb{R}}^{N}}, \\ &u\in {{{\dot{H}}}^{s}}({{\mathbb{R}}^{N}})\cap {{L}^{q+1}}({{\mathbb{R}}^{N}}), \\ &u>0\ \ \text{in}\ \ {{\mathbb{R}}^{N}}, \\ \end{align} \right.$ where $s∈(0,1)$ is a fixed parameter, $(-Δ)^s$ is the fractional Laplacian in $\mathbb{R}^N$, $q>p≥q \frac{N+2s}{N-2s}$ and $N>2s$. For every $s∈(0,1)$, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all $s∈(0,1)$. Using those decay estimates, we prove Pohozaev type identity in ${{\mathbb{R}}^{N}}$ and we show that the above problem does not have any solution when $p=\frac{N+2s}{N-2s}$. We also discuss radial symmetry and decreasing property of the solution and prove that when $p>\frac{N+2s}{N-2s}$, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every $p≥q \frac{N+2s}{N-2s}$ and every solution is a classical solution.

The optimal decay estimates on the framework of Besov spaces for the Euler–Poisson two-fluid system

In this paper, we are concerned with the optimal decay estimates for the Euler–Poisson two-fluid system. It is first revealed that the irrotationality of the coupled electronic field plays a key role such that the two-fluid system has the same dissipative structure as generally hyperbolic systems satisfying the Shizuta–Kawashima condition. This fact inspires us to obtain decay properties for linearized systems in the framework of Besov spaces. Furthermore, various decay estimates of solution and its derivatives of fractional order are deduced by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As the direct consequence, the optimal decay rates of Lp(ℝ3)-L2 (ℝ3) (1 ≤ p < 2) type for the Euler–Poisson two-fluid system are also shown. Compared with previous works in Sobolev spaces, a new observation is that the difference of variables exactly consists of a one-fluid Euler–Poisson equations, which leads to the sharp decay estimates for velocities.

Decay estimate of solutions for a semi-linear wave equation

In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Under a smallness condition on the initial value, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates for the spatial derivatives of the solution are provided. The proof is carried out by means of the decay property of the solution operator and a fixed point-contraction mapping argument.

Extinction and decay estimates of solutions for a p-Laplacian evolution equation with nonlinear gradient source and absorption

We investigate the extinction properties of non-negative nontrivial weak solutions of the initial-boundary value problem for a p-Laplacian evolution equation with nonlinear gradient source and absorption terms.

Decay estimates of solutions for mildly degenerate Kirchhoff type dissipative wave equations in unbounded domains

Decay estimates of solutions for mildly degenerate Kirchhoff type dissipative wave equations in unbounded domains

Global existence and decay estimate of solution to one dimensional convection-diffusion equation

We study the global existence of solution to one dimensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation. Based on the global bounds of the solution, we extend the local one to a global one that decays in Hl space.

On the asymptotic behaviour of solution for the generalized double dispersion equation

In this article, we investigate the Cauchy problem for the generalized double dispersion equation in n-dimensional space. We establish the decay estimates of solution to the corresponding linear equation. Under smallness condition on the initial data, we prove the global existence and asymptotic behaviour of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.

Extinction and decay estimates of solutions for a porous medium equation with nonlocal source and strong absorption

Abstract In this paper, we investigate extinction properties of the solutions for the initial Dirichlet boundary value problem of a porous medium equation with nonlocal source and strong absorption terms. We obtain some sufficient conditions for the extinction of nonnegative nontrivial weak solutions and the corresponding decay estimates which depend on the initial data, coefficients, and domains.

Pointwise Estimates for Solutions to a System of Radiating Gas

In this paper we focus on the initial value problem of a hyperbolic-elliptic coupled system in multi-dimensional space of a radiating gas. By using the method of Green function combined with Fourier analysis, we obtain the pointwise decay estimates of solutions to the problem.

On the asymptotic behavior of solution for the generalized IBq equation with Stokes damped term

In this paper, we investigate the Cauchy problem for the generalized improved Boussinesq equation with Stokes damped term in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. Based on the decay estimates of solutions to the corresponding linear equation and smallness condition on the initial data, we prove the global existence and asymptotic of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.