Homogeneous Nonlinearity Research Articles

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

Исследование периодических решений системы обыкновенных дифференциальных уравнений с квазиоднородной нелинейностью

The article considers a system of ordinary differential equations in which the main nonlinear part, which is a quasi-homogeneous mapping, is distinguished. The question of the existence of periodic solutions is investigated. Consideration of a quasi-homogeneous mapping allows us to generalize previously known results on the existence of periodic solutions for a system of ordinary differential equations with the main positively homogeneous non-linearity. An a priori estimate for periodic solutions is proved under the condition that the corresponding unperturbed system of equations with a quasi-homogeneous right-hand side does not have nonzero bounded solutions. Under the conditions of an a priori estimate, the following results were obtained: 1) the invariance of the existence of periodic solutions under continuous change (homotopy) of the main quasi-homogeneous non-linear part was proved; 2) the problem of homotopy classification of two-dimensional quasi-homogeneous mappings satisfying the a priori estimation condition has been solved; 3) a criterion for the existence of periodic solutions for a two-dimensional system of ordinary differential equations with the main quasi-homogeneous non-linearity is proved.

Ob ogranichennykh traektoriyakh avtonomnoy sistemy s vydelennoy polozhitel'no odnorodnoy nelineynost'yu

Bounded trajectories of an autonomous system with an isolated positively homogeneous nonlinearity that is the gradient of a smooth function are studied. We prove the existence of nonstationary bounded trajectories lying in connected components of the set of points where the positively homogeneous function is negative and nonzero stationary points in those connected components whose closure has nonzero Euler characteristic. The existence of nonstationary bounded trajectories is substantiated using the Waűewski method; and the existence of stationary points, using methods for calculating the winding number of finite-dimensional vector fields.

On Bounded Trajectories of an Autonomous System with an Isolated Positively Homogeneous Nonlinearity

On Bounded Trajectories of an Autonomous System with an Isolated Positively Homogeneous Nonlinearity

Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials

This paper is concerned with the final state problem for the homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26,29].

Positive solution for a class of p-Laplacian systems with critical homogeneous nonlinearity

Positive solution for a class of p-Laplacian systems with critical homogeneous nonlinearity

On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity

We study the solvability of a periodic problem for a system of ordinary differential equations in which we separate the main nonlinear part that is positive homogeneous mapping (of order greater than unity), with the rest called a perturbation. It is proved that if the unperturbed system of equations has no nonzero bounded solutions, then the periodic problem is solvable under any perturbation if and only if the degree of the positive homogeneous mapping on the unit sphere is nonzero. The result obtained is of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations.

On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity

On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity

Stability of Peaked Solitary Waves for a Class of Cubic Quasilinear Shallow-Water Equations

AbstractThis paper is concerned with two classes of cubic quasilinear equations, which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. One class of the models has homogeneous cubic nonlinearity and includes the integrable modified Camassa–Holm (mCH) equation and Novikov equation, and the other class encompasses both quadratic and cubic nonlinearities. It is demonstrated here that both these models possess localized peaked solutions. By constructing a Lyapunov function, these peaked waves are shown to be dynamically stable under small perturbations in the natural energy space $H^1$, without restriction on the sign of the momentum density. In particular, for the homogeneous cubic nonlinear model, we are able to further incorporate a higher-order conservation law to conclude orbital stability in $H^1\cap W^{1,4}$. Our analysis is based on a strong use of the conservation laws, the introduction of certain auxiliary functions, and a refined continuity argument.

Complex planar Hamiltonian systems: Linearization and dynamics

<p style='text-indent:20px;'>Global dynamics of complex planar Hamiltonian polynomial systems is difficult to be characterized. In this paper, for general complex quadratic Hamiltonian systems of one degree of freedom, we obtain some sufficient conditions on the existence of family of invariant tori. We also complete characterization on locally analytic linearizability of complex planar Hamiltonian systems with homogeneous nonlinearity of degrees either 2 or 3 at a nondegenerate singularity, and present their global dynamics. For these classes of systems we also prove existence of families of invariant tori, together with isochronous periodic orbits.

Strongly localized semiclassical states for nonlinear Dirac equations

<p style='text-indent:20px;'>We study semiclassical states of the nonlinear Dirac equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -i\hbar{\partial}_t\psi = ic\hbar\sum\limits_{k = 1}^3{\alpha}_k{\partial}_k\psi - mc^2{\beta} \psi - M(x)\psi + f(|\psi|)\psi, \quad t\in \mathbb{R}, \ x\in \mathbb{R}^3, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a bounded continuous potential function and the nonlinear term <inline-formula><tex-math id="M2">\begin{document}$ f(|\psi|)\psi $\end{document}</tex-math></inline-formula> is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity <inline-formula><tex-math id="M3">\begin{document}$ f(s) = s^p $\end{document}</tex-math></inline-formula>. We develop a variational method for the strongly indefinite functional associated to the problem.

Normalized solutions to the fractional Schrödinger equations with combined nonlinearities

We study the normalized solutions of the fractional nonlinear Schrodinger equations with combined nonlinearities $$\begin{aligned} (-\Delta )^s u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \quad \text{ in }~{\mathbb {R}}^N, \end{aligned}$$ and we look for solutions which satisfy prescribed mass $$\begin{aligned} \int _{\mathbb {R}^N}|u|^2=a^2, \end{aligned}$$ where $$N\ge 2,s\in (0,1),\mu \in \mathbb {R}$$ and $$2<q<p<2_s^*=2N/(N-2s)$$ . Under different assumptions on $$q 0$$ and $$\mu \in \mathbb {R}$$ , we prove some existence and nonexistence results about the normalized solutions. More specifically, in the purely $$L^2$$ -subcritical case, we overcome the lack of compactness by virtue of the monotonicity of the least energy value and obtain the existence of ground state solution for $$\mu >0$$ . While for the defocusing situation $$\mu <0$$ , we prove the nonexistence result by constructing an auxiliary function. We emphasis that the nonexistence result is new even for Laplacian operator. In the purely $$L^2$$ -supercritical case, we introduce a fiber energy functional to obtain the boundedness of the Palais–Smale sequence and get a mountain-pass type solution. In the combined-type cases, we construct different linking structures to obtain the saddle type solutions. Finally, we remark that we prove a uniqueness result for the homogeneous nonlinearity ( $$\mu =0$$ ), which is based on the Morse index of ground state solutions.

Liouville theorems for superlinear parabolic problems with gradient structure

We improve one of the methods for obtaining Liouville theorems for superlinear parabolic problems. In particular, if we consider a positively homogeneous gradient nonlinearity $$F:{{\mathbb {R}}}^m\rightarrow {{\mathbb {R}}}^m$$ of degree $$p>1$$, where $$n>2$$, $$p<n/(n-2)$$ and F satisfies some additional hypotheses, then the method introduced in [Math. Ann. 364 (2016), 269–292] guarantees the nonexistence of positive solutions of the problem $$\begin{aligned} U_t-\varDelta U=F(U), \qquad x\in {{\mathbb {R}}}^n,\ t\in {{\mathbb {R}}}. \end{aligned}$$We show that the condition $$p<n/(n-2)$$ can be weakened to $$p\le n/(n-2)$$: This covers the important case of cubic or quadratic nonlinearities if $$n=3$$ or $$n=4$$, respectively. Our improvement also applies to problems with nonlinear boundary conditions and other problems.

Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions

In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [SIAM J. Math. Anal. 50 (2018), pp. 3251–3270], the first and second authors consider one- and two-dimensional cases and give a sufficient condition on the nonlinearity so that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converge to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in the aforementioned article. Moreover, we present a candidate for the second asymptotic profile of the solution.

Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearity

We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u|^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.

Long Range Scattering for Nonlinear Schrödinger Equations with Critical Homogeneous Nonlinearity

In this paper, we consider the final state problem for the nonlinear Schrodinger equation with a homogeneous nonlinearity which is of the long range critical order and is not necessarily a polynomi...

Multiple symmetric results for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in [formula omitted

This paper is dedicated to studying the existence and multiplicity of symmetric solutions for a class of biharmonic elliptic systems with critical homogeneous nonlinearity in RN. By virtue of variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on the parameters and the weighted functions.

Multiple symmetric results for singular quasilinear elliptic systems with critical homogeneous nonlinearity

This paper deals with the existence and multiplicity of symmetric solutions for a class of singular quasilinear elliptic systems with critical homogeneous nonlinearity in a bounded symmetric domain. Applying variational methods and the symmetric criticality principle of Palais, we establish several existence and multiplicity results of G‐symmetric solutions under some appropriate assumptions on the weighted functions and the parameters. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Extinction of solutions of higher order parabolic equations with double nonlinearity and degenerate absorption potential

We study the extinction property of solutions to the Cauchy–Dirichlet problem for nonlinear parabolic equations of the order 2m with absorption potential in a semibounded cylinder (0,+∞) × Ω, where Ω is a bounded domain in ℝN, N ≥ 1. The sufficient conditions ensuring the extinction of a solution in a finite time, which depend on N,m, and q (where q is a parameter of the homogeneous nonlinearity in the main part of the equation), are obtained.

Exact solutions for bright and dark solitons in spatially inhomogeneous nonlinearity

Abstract We present exact analytical results for bright and dark solitons in a type of one-dimensional spatially inhomogeneous nonlinearity. We show that the competition between a homogeneous self-defocusing (SDF) nonlinearity and a localized self-focusing (SF) nonlinearity supports stable fundamental bright solitons. For a specific choice of the nonlinear parameters, exact analytical solutions for fundamental bright solitons have been obtained. By applying both variational approximation and Vakhitov-Kolokolov stability criterion, it is found that exact fundamental bright solitons are stable. Our analytical results are also confirmed numerically. Additionally, we show that a homogeneous SF nonlinearity modulated by a localized SF nonlinearity allows the existence of exact dark solitons, for certain special cases of nonlinear parameters. By making use of linear stability analysis and direct numerical simulation, it is found that these exact dark solitons are linearly unstable.