A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L¹L^p. Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L¹ of the cut-off equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result. Mathematics Subject Classifications (2000) 60K35, 35S10.     

Weak solutions to a nonlinear variational wave equation and some related problems

In this paper we present some results on the global existence of weak solutions to a nonlinear variational wave equation and some related problems. We first introduce the main tools, the L ^p Young measure theory and related compactness results, in the first section. Then we use the L ^p Young measure theory to prove the global existence of dissipative weak solutions to the asymptotic equation of the nonlinear wave equation, and comment on its relation to Camassa-Holm equations in the second section. In the third section, we prove the global existence of weak solutions to the original nonlinear wave equation under some restrictions on the wave speed. In the last section, we present global existence of renormalized solutions to two-dimensional model equations of the asymptotic equation, which is also the so-called vortex density equation arising from sup-conductivity.     

Nonlinear semigroup approach to transport equations with delayed neutrons

This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on L~p-spaces(1 ≤ p +∞). Local and non local evolution problems are discussed.     

The attractor on viscosity Degasperis–Procesi equation

In this paper the dynamical behaviors of a dispersive shallow water equation with viscosity, viscosity Degasperis–Procesi equation, are investigated. The existence of global solution to viscosity Degasperis–Procesi equation in L² under the periodical boundary condition is studied and the existence of the global attractor of semi-group to solution on viscosity Degasperis–Procesi equation in H² is obtained.     

The mean field kinetic equation for interacting particle systems with non-Lipschitz force

In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the N-particle Newtonian system. For L¹∩L^∞ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the nonlocal nonlinear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.     

On the Cauchy problem for a doubly nonlinear degenerate parabolic equation with strongly nonlinear sources

In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||_f||| _h and 〈f〉_h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered. This work was supported by the National Natural Science Foundation of China (Grant No. 10531020)     

Second Order Nonlinear Evolution Inclusions Ⅰ： Existence and Relaxation Results

This is the first part of a work on second order nonlinear, nonmonotone evolution inclusions defined in the framework of an evolution triple of spaces and with a multivalued nonlinearity depending on both x（t） and x（t）. In this first part we prove existence and relaxation theorems. We consider the case of an usc, convex valued nonlinearity and we show that for this problem the solution set is nonempty and compact in C^1 （T, H）. Also we examine the Isc, nonconvex case and again we prove the existence of solutions. In addition we establish the existence of extremal solutions and by strengthening our hypotheses, we show that the extremal solutions are dense in C^1 （T, H） to the solutions of the original convex problem （strong relaxation）. An example of a nonlinear hyperbolic optimal control problem is also discussed.     

On the periodic Wigner-Poisson-Fokker-Planck system

This paper is concerned with the existence and uniqueness analysis of global classical solutions of a diffusive quantum evolution equation with nonlinear coupling to the Poisson equation. The main technical difficulty in the existence proof is to show that the quantum Fokker-Planck term is a semigroup-generator in a weighted L²-space. The potential term is then a Lipschitz perturbation of it.     

On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.     

Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems with BV data

We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C¹ norm of the initial data is large, we prove that, if the C¹ norm of the boundary data is bounded but possibly large, and the BV norm of the boundary data is sufficiently small, then the solution remains C¹ globally in time. Applications include the equation of time‐like extremal surfaces in Minkowski space R^{1 + (1 + n)} and the one‐dimensional Chaplygin gas equations. Copyright © 2013 John Wiley & Sons, Ltd.     

A nonlinear singular diffusion equation with source

In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which depends on initial data in L¹ continuously.     

Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L²-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H²(R^N), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.     

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (? ^d , 𝔤), d ≥ 3. The metric 𝔤 is non-trapping and approaches the Euclidean metric like ?x?^?ρ. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ ≥ 1. Long time existence means that, for all n > 0, the life time of the solution is a least δ^?n , where δ is the size of the initial data in some appropriate Sobolev space. Moreover, for d ≥ 4 and ρ > 1, we obtain global existence for small data.     

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H^2,2(?) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H^2,2(?) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.     

On the Cauchy Problem of the Camassa-Holm Equation

The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in L ^p frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.     

Nonmonotone, nonlinear evolution inclusions

In this paper, we consider nonlinear evolution problems, defined on an evolution triple of spaces, driven by a nonmonotone operator, and with a perturbation term which is multivalued. We prove existence theorems for the cases of a convex and of a nonconvex valued perturbation term which is defined on all of T × H or only on T × X with values in H or even in X^* (here X - H - X^* is the evolution triple). Also, we prove the existence of extremal solutions, and for the “monotone” problem we have a strong relaxation theorem. Some examples of nonlinear parabolic problems are presented.     

Dissipativity and global attractors for a class of quasilinear parabolic systems

We show that a-priori weak L^p dissipativity implies strong L ^∞ dissipativity for a class of weakly coupled quasilinear parabolic systems satisfies general structure conditions. The existence of global attractors of general nonlinear reaction diffusion systems will be proven.     

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (u_xx ? u)_t + u _xxx + uu_x = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.     

Long-Time Behavior for Second Order Lattice Dynamical Systems

Many researchers examined the existence of global attractors for various types of first and second order lattice dynamical systems. Here we prove the existence of a global attractor for a new type of second order lattice dynamical systems in the Hilbert space l ²×l ². For specific choices of the linear operators this system can be regraded as a spatial discretization of a continuous damped nonlinear Boussinesq equation on ℝ ^m ,m≥1.