Asymptotic Behavior Of Global Solutions Research Articles

Asymptotic Behavior of Global Solution of an Anomalous Gray Scott Model

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. However, the asymptotic behavior is investiged by technique semi group estimates and the Sobolev embedding theorem

Effect of unfavorable regions on the spreading solution in a diffusion equation

We consider a diffusion equation on the real line with growth rates (reaction terms) being negative in a bounded unfavorable region and bistable on two sides. The equation can be used to model a species living in a habitat with a polluted or hunting zone but still tries to survive or even spread to the whole space. Using the zero number argument, we first show the general convergence to stationary solution for any nonnegative global solutions, and then we prove a spreading–transition–vanishing trichotomy result for the asymptotic behavior of global solutions. The key point is to find a quite special stationary solution to distinguish the spreading and transition solutions. Finally, we construct precise upper and lower solutions to estimate the asymptotic speed for the spreading solution.

A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

A class of fractional viscoelastic Kirchhoff equations involving two nonlinear source terms of different signs are studied. Under suitable assumptions on the exponents of nonlinear source terms and the memory kernel, the existence of global solutions in an appropriate functional space is established by a combination of the theory of potential wells and the Galerkin approximations. Furthermore, the asymptotic behavior of global solutions is obtained by a combination of the theory of potential wells and the perturbed energy method.

Asymptotic behavior for solutions to the vector nonlinear Schrödinger equations

In this paper, we study the Cauchy problem for the vector nonlinear Schrödinger equations with cubic nonlinearities. The vector nonlinear Schrödinger equations are also called as Manakov systems. In particular, our main purpose of this study is to consider the asymptotic behavior of global solutions and our main results of this study shows the existence of asymptotically free solutions for space dimensions n=2,3 in the intersection of Sobolev space and weighted Lebesgue space Hs∩FHγ with s,γ∈(0,1]∩(n/2−1,n/2) and existence of modified asymptotically free solutions for space dimension n=1 in the weighted Lebesgue space FHγ with 1/2<γ≤1.

Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

AbstractIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for $$J_{\mu }(u_{0})&lt;0$$ J μ ( u 0 ) &lt; 0 and decay rate of $$H^{1}_{0}$$ H 0 1 and $$L^{2}$$ L 2 -norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

Fisher–KPP equation on the Heisenberg group

AbstractIn this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow‐up solutions. Moreover, we extend the obtained results to the time‐fractional Fisher–KPP equation on the Heisenberg group.

Global existence and finite time blowup for a mixed pseudo-parabolic [formula omitted]-Laplacian type equation

This paper concerns the initial–boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then prove the existence, uniqueness and decay estimate of global solutions and the blowup phenomena of solutions with subcritical initial energy. Next, we extend parallelly these results to the critical initial energy. Lastly, the existence, uniqueness and asymptotic behavior of global solutions with supercritical initial energy are proved by further analyzing the properties of ω-limits of solutions.

Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities \(\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,\) involving time-fractional Caputo derivative \(\mathcal {D}_{0|t}^{\alpha }\) and space-fractional p-Laplacian operator \((-\varDelta )^s_p\). We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of \(\gamma ,\mu ,m\) and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.

Global existence for systems of nonlinear wave and Klein–Gordon equations in two space dimensions under a kind of the weak null condition

We consider coupled systems of nonlinear wave and Klein–Gordon equations in two space dimensions with cubic nonlinearities. For these kinds of systems, the small data global existence is already known if the cubic nonlinearity satisfies a certain condition related to the null condition. In this article, our aim is to investigate the small data global existence under a condition related to the weak null condition. We also make a remark on the asymptotic behavior of global solutions.

Asymptotic behavior of global solutions to some multidimensional quasilinear hyperbolic systems

Asymptotic behavior of global solutions to some multidimensional quasilinear hyperbolic systems

Systems of semilinear wave equations with multiple speeds in two space dimensions and a weaker null condition

&lt;p style='text-indent:20px;'&gt;We consider the Cauchy problem for systems of semilinear wave equations with multiple propagation speeds in two space dimensions. The nonlinear terms are assumed to be cubic and to depend only on first derivatives of the unknowns. The null condition is one of the sufficient condition for the small data global existence for the single-speed case, and it is extended to the multiple-speed case by many authors. Recently, various weaker sufficient conditions for the small data global existence were obtained for the single-speed case. In this paper, we introduce the multiple-speed version of one of such weaker null conditions, and we show the small data global existence. The asymptotic behavior of global solutions is also investigated through associated systems of ODEs called the reduced systems.&lt;/p&gt;

A fractional Gronwall inequality and the asymptotic behaviour of global solutions of Caputo fractional problems

We study the asymptotic behaviour of global solutions of some nonlinear integral equations related to some Caputo fractional initial value problems. We consider problems of fractional order between 0 and 1 and of order between 1 and 2, each in two cases: when the nonlinearity depends only on the function, and when the nonlinearity also depends on fractional derivatives of lower order. Our main tool is a new Gronwall inequality for integrals with singular kernels, which we prove here, and a related boundedness property of a fractional integral of an \(L^1[0,\infty)\) function. For more information see https://ejde.math.txstate.edu/Volumes/2021/80/abstr.html

Asymptotic convergence of solutions to the forest kinematic model

We study the asymptotic behavior of global solutions to forest kinetic model equations composed of young trees, old trees, and air-borne seeds. Under some parameter assumptions, we prove the asymptotic convergence of global solutions to a stationary solution. To this end, we show a non-smooth version of the Łojasiewicz–Simon gradient inequality on a suitable functional space and a certain norm estimate of the time derivative of global solutions.

Energy scattering for radial focusing inhomogeneous bi-harmonic Schrödinger equations

This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$\begin{aligned} i\dot{u}+\Delta ^2 u+F(x,u)=0 . \end{aligned}$$Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson and Murphy (Proc Am Math Soc 145(11):4859–4867, 2017).

Dynamical boundary problem for Dirichlet-to-Neumann operator with critical Sobolev exponent and Hardy potential

We study the Laplacian equation with dynamical boundary condition involving Dirichlet-to-Neumann operator, critical growth, and Hardy potential. We first prove the existence and decay estimates of global solutions and finite time blowup of local solutions under certain assumptions. Then we focus on the asymptotic behavior of global solutions approaching a stationary solution in the long time series. Furthermore, we give a precise bubbling description by the concentration-compactness principle.

Asymptotic behavior of global solutions to a class of mixed pseudo-parabolic Kirchhoff equations

In this paper, we consider the asymptotic behavior of global solutions to the initial boundary value problem of a mixed pseudo-parabolic Kirchhoff equation. We find the convergent relationship between the global solutions of the evolution problem and the ground state solutions of the corresponding stationary problem.

Scattering for Radial Defocusing Inhomogeneous Bi-Harmonic SchrÖDinger Equations

This note studies the asymptotic behavior of global solutions to the fourth-order Schrodinger equation $$ i\dot u+{\Delta}^{2} u-F(x,u)=0 . $$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the defocusing mass super-critical and energy sub-critical regimes, with radial setting.

Asymptotic behavior of global solutions to one-dimension quasilinear wave equations

Asymptotic behavior of global solutions to one-dimension quasilinear wave equations

Stable and unstable sets for damped nonlinear wave equations with variable exponent sources

In this paper, we study a class of nonlinear wave equations with variable exponent sources. By introducing a family of potential wells, we first prove the global existence of solutions with initial data in the potential wells and the finite time blow-up for solutions starting in the unstable sets. The boundedness and asymptotic behavior of global solutions are also concerned. Finally, we obtain the finite time blow-up with high energy initial data.

Global solution and global orbit to reaction–diffusion equation for fractional Dirichlet‐to‐Neumann operator with subcritical exponent

We consider the reaction–diffusion equation for fractional Dirichlet‐to‐Neumann operator with subcritical exponent motivated by electrical impedance tomography (EIT) and a need to overcome the non‐locality of a fractional differential equation for modeling anomalous diffusion. We mainly deal with the asymptotic behavior of global solution and the boundedness of global orbit, which allows us to show that any global solution is classical solution using Moser iteration technique.