This paper is concerned with the final value problem for a system of nonlinear wave equations. The main issue is to solve the problem for the case where the nonlinearity is of a long range type. By assuming that the solution is spherically symmetric, we shall show global solvability of the final value problem around a suitable final state, and hence the generalized wave operator and long range scattering operator can be constructed.
We will study solvability of nonlinear second-order elliptic system of partial differential equations with nonlinear boundary conditions. We study the generalized Steklov Robin eigensystem (with possibly matrices weights) in which the spectral parameter is both in the system and on the boundary. We prove the existence of solutions for nonlinear system when both nonlinearities in the differential system and on the boundary interact, in some sense, with the generalized spectrum of Steklov Robin. The method of proof makes use of Leray-Schauder degree.
In this article, we establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz condition.
A class of generalized Schrödinger elliptic problems involving concave-convex and other types of nonlinearities is studied. A reasonable overview about the set of solutions is provided when the parameters involved in the equation assume different real values.
Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential equations and second extend the previous results for linear differential equations to a much broader class of nonlinear differential equations.
We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.
We study the orbital instability of solitary waves for a generalized derivative nonlinear Schrödinger equation. We give sufficient conditions for instability of a two-parameter family of solitary waves in a degenerate case.
A nonlinear profile decomposition is established for solutions of supercritical generalized Korteweg-de Vries equations. As a consequence, we obtain a concentration result for finite time blow-up solutions that are of Type II.
In this paper, a generalized nonlinear Camassa-Holm equation with time- and space-dependent coefficients is considered. We show that the control of the higher order dispersive term is possible by using an adequate weight function to define the energy. The existence and uniqueness of solutions are obtained via a Picard iterative method.
In this paper we consider the Cauchy problem for linear dissipative generalized Klein-Gordon equations with nonlinear memory in the right hand side. Our goal is to study the effect of this nonlinearity on both the decay estimates of global solutions as well as the admissible range of the exponent p.
In this paper we consider a generalized Kirchhoff? equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools.
In this paper, we deal with equations of variational form which Nahari manifolds can contain more than two different types of critical points. We introduce a method of separating critical points on the Nahari manifold, based on the use of nonlinear generalized Rayleigh quotients. The method is illustrated by establishing the existence of positive solutions, ground states and multiplicity results for a two-parameter nonlinear elliptic boundary problem with polynomial nonlinearities.
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schrödinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite $L^{2}$ norm.
We are concerned with the initial value problem governed by generalized Rayleigh-Stokes equations, where the nonlinearity depends on history states and takes values in Hilbert scales of negative order. The solvability and Hölder regularity of solutions are proved by using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.
We study the generalized eigenvalue problem in $\mathbb{R}^N$ for a general convex nonlinear elliptic operator which is locally elliptic and positively $1$-homogeneous. Generalizing article of Berestycki and Rossi in [Comm. Pure Appl. Math. 68 (2015), no. 6, 1014-1065] we consider three different notions of generalized eigenvalues and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.
In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.
We prove global stability for a system of nonlinear wave equations satisfying a generalized null condition. The generalized null condition allows for null forms whose coefficients have bounded $C^k$ norms. We prove both pointwise decay and improved decay of good derivatives using bilinear energy estimates and duality arguments. Combining this strategy with the $r^p$ estimates of Dafermos--Rodnianski then allows us to prove global stability. The proof requires analyzing the geometry of intersecting null hypersurfaces adapted to solutions of wave equations.
We revisit the properties of Bessel-Riesz operators and refine the proof of the boundedness of these operators on generalized Morrey spaces using Young's inequality. We also obtain an estimate for the norm of these operators on generalized Morrey spaces. In addition, we reprove the boundedness of fractional integral operators on generalized Morrey spaces and estimate their norm.
We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic functions involving an operator having generalized Orlicz-growth embracing besides Orlicz case also natural ranges of variable exponent and double-phase cases. In particular, Harnack's Principle and Minimum Principle are provided for $\mathcal{A}$-superharmonic functions and boundary Harnack inequality is proven for $\mathcal{A}$-harmonic functions.
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability of a large number of nonlinear scalar boundary value problems. We begin by giving our general strategy which involves the reformulation of our boundary value problem as an operator equation. We then proceed to establish our results and compare them to closely related previous work.
In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $\mathscr{T}_{\!\!\ell} u = |\partial_t u|^p$, where $ \mathscr{T}_{\!\!\ell} = \partial_t^2-t^{2\ell}Δ$. Smooth solutions blow up in finite time for positive Cauchy data when the exponent $p$ of the nonlinear term is below $\frac{\mathscr{Q}}{\mathscr{Q}-2}$, where $\mathscr{Q}=(\ell+1)n+1$ is the quasi-homogeneous dimension of the generalized Tricomi operator $\mathscr{T}_{\!\!\ell}$. Furthermore, we get also an upper bound estimate for the life…
In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.
We consider an inverse optimization spectral problem for the Sturm-Liouville operator $$\mathcal{L}[q] u:=-u''+q(x)u$$ subject to the separated boundary conditions. In the main result, we prove that this problem is related to the existence of solutions of boundary value problems for the nonlinear equations of the form $$-u"+q_0(x) u=λu+σu^3$$ with $σ=1$ or $σ=-1$. The key outcome of this relationship is a generalized Sturm's nodal theorem for the nonlinear boundary value problems.
In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems in unbounded domains, which involves a general variable exponent elliptic operator. Under some suitable conditions on the nonlinearities, we establish existence of at least three weak solutions for the problem. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano.
The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equation…
The aim of this paper is to get the boundedness of certain multi-sublinear operators generated by multilinear fractional integral operators on the product generalized local Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutators of multilinear operators generated by local campanato functions and multilinear fractional integral operators are also bounded on the product generalized local Morrey spaces.
This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized $ψ$-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its properties, from which we obtain a new Lyapunov-type inequality for our problem. A lower bound for the possible eigenvalues of our problem is derived. Furthermore, we apply some properties of the Green function to obtain some existence results for our problem. It is worth mentioning that our results still work with some source functions including singularities.
This paper is concerned with the local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity. The equation arises as a non-integrable and lower nonlinearity version of the higher-order KdV equation. As for the lower nonlinearity model of the KdV equation, Linares, the author and Ponce [11] prove the local well-posedness under a non-degenerate condition introduced by Cazenave and Naumkin [1]. In this paper, it turns out that the well-posedness result can be extended into the higher-order equation. We also give a lower bound for the lifespan of the sol…
This article explains the relationship between analytic and algebraic order in case of abstract pseudo-differential operators for a regular spectral triple.
We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, their velocity. We illustrate these results with numerical simulations.
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of ${\mathcal G}^\infty$-regularity does not hold.
A generalized nonlinear Schrödinger equation admits WTC formal solutions as was shown by Clarkson. We show that they are convergent and determine real-analytic solutions near a noncharacteristic, movable singularity manifold.
A class of generalized nonlinear Kolmogorov equations is investigated. We present the group classification of Lie symmetries of the class with respect to the group of equivalence transformations. We find a number of exact solutions of nonlinear Kolmogorov equations which has the maximal symmetry properties.
The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-Laplacian. The eigenvalue and the corresponding eigenfunction are represented in terms of common parameters, and a complete description of the spectra and a closed form representation of the corresponding eigenfunctions are obtained. As a by-product of the representation, it turns out that a kind of solution is also a solution of another eigenvalue problem with $p/2$-Laplacian.
We consider the semi-classical limit of nonlinear Schrodinger equations in the presence of both a polynomial nonlinearity and thederivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolicstructure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time dependent analyticregularity.
We present new solutions in terms of elementary functions of the multi-component nonlinear Schrödinger equations and known solutions of the Davey-Stewartson equations such as multi-soliton, breather, dromion and lump solutions. These solutions are given in a simple determinantal form and are obtained as limiting cases in suitable degenerations of previously derived algebro-geometric solutions. In particular we present for the first time breather and rational breather solutions of the multi-component nonlinear Schrödinger equations.
Reduction operators of generalized Burgers equations are studied. A connection between these equations and potential fast diffusion equations with power nonlinearity -1 via reduction operators is established. Exact solutions of generalized Burgers equations are constructed using this connection and known solutions of the constant-coefficient potential fast diffusion equation.
We survey the correct definition of a generalized Dirac operator on a Space--Time and the classical result about propagation of singularities. This says that light travels along light--like geodesics. Finally we show this is also true for generalized Dirac operators.
Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in {\em Mathematica}. The codes {\sc InvariantsSymmetries.m} and {\sc DDERecursionOperator.m} can aid researchers interested in properties of nonlinear DDEs.
