pinecone/core-2020-05-10-deduplication huggingface dataset:
Solutions of Nonlinear Planar Elliptic Problems with Triangle Symmetry

AbstractIn this paper we continue the study of the nodal domain structure of doubly periodic solutions of certain nonlinear elliptic problems initiated in Fifeet al.(Physica D100(1997), 257–278). More precisely, we consider small amplitude solutions ofΔu+λf(u)=0 in R2whose nodal domains consist of equilateral triangles tiling the plane. If this equation is suitably perturbed, then for genericfwe prove the existence of unique nearby solutions with triangle symmetry and show how their nodal domain geometry breaks up. Furthermore, we treat the non-generic rectangular cases which had to be excluded in Fifeet al.(Physica D100(1997), 257–278) as well as other nodal domain structures
Existence of Many Nonequivalent Nonradial Positive Solutions of Semilinear Elliptic Equations on Three-Dimensional Annuli

AbstractWe consider a semilinear elliptic equation,Δu+up=0 onΩR≡{x∈Rn∣R−1<|x|<R+1} with zero Dirichlet boundary condition, where 1<p<∞ forn=2, 1<p<(n+2)/(n−2) forn>2. We prove that, when the space dimension n is three, the number of nonequivalent nonradial positive solutions of the equation goes to ∞ asR→∞. The same result has been known forn=2 andn⩾4; in those cases, the result was obtained by showing that the minimal energy solutions in various symmetry classes have different energy levels. As we will show in this paper, this is not true ifn=3. This makes the casen=3 highly exceptional, and explains why past attempts failed in this case. In this paper we will prove the above result by considering local—rather than global—minimizers in some symmetry classes
Averaging for Fundamental Solutions of Parabolic Equations

AbstractHerein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order,ε-dependent parabolic partial differential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an “averaged” parabolic equation, we bring forth a novel approach to comparingx-derivatives of∂tuε(x, t)=∑|k|⩽2pAk(x, t/ε)∂kxuε(x, t)+fε(x, t),uε(x, 0)=ϕε(x)on Rd×[0, T] to like derivatives of∂tu(x, t)=∑|k|⩽2pA0k(x)∂kxuε(x, t)+f(x, t),u(x)=ϕ(x)(asε→0) under general regularity conditions and our basic hypothesis that∫t0Ak(x, s/ε)−A0k(x)ds→ε→00for eachx, t(i.e., pointwise). The flexibility afforded by studing fundamental vis-à-vis specific solutions of these equations not only permitsε-dependent Cauchy data and provides a unified method of treating allx-derivatives ofuεup to order 2p−1 but also proves an invaluable tool when considering related problems of stochastic averaging. Our development was motivated by and retains a strong resemblance to the classical theory of parabolic partial differential equations. However, it will turn out that the classical conditions under which fundamental solutions are known to exist are somewhat unsuitable for our purposes and a modified set of conditions must be used
Dispersive Smoothing Effects for KdV Type Equations

AbstractIn this paper we study the smoothness properties of solutions of some nonlinear equations of Korteweg–de Vries (KdV) type, which are of the form∂tu=a(x, t)u3+f(u2, u1, u, x, t),(1)wherex∈R,uj=∂jxu, andkandjare nonnegative integers. Our principal condition is thata(x, t) be positive and bounded, so that the dispersion is dominant. It is shown under certain additional conditions onaandfthatC∞solutionsu(x, t) are obtained fort>0 if the initial datau(x, 0) decays faster than it does polynomially onR−and has certain initial Sobolev regularity. A quantitative relationship between the rate of decay and the amount of gain of smoothness is given. Lets0be the Sobolev index. If∫Ru2(x, 0)(1+|x−|m)dx<∞(2)for an integerm⩾0 and the solution obeys ‖u‖Hs0<∞ for an existence time 0<t<T, thenu(x, t)∈Hmloc(R) for all 0<t⩽T, andu(x, t)∈L1([0, T];H(m+1)loc(R)). Our method can also be extended to address the fully nonlinear dispersive equations related to (1)
The problem of center for resonant singular points of polynomial vector fields

We generalize the notion of “center” to the case of a p : −q resonant singular point of a polynomial vector field in ℂ2 and to some other situations (resonant node, saddle-node, non-elementary singular point of vector field and resonant fixed point of one-dimensional complex diffeomorphism). We show some similarities and differences with the classical case. In particular, the analogue of Bautin's theorem does not hold. Four small amplitude limit cycles can bifurcate from the center after quadratic perturbation of a quadratic vector field with resonant center
Semi-Hyperbolic Mappings, Condensing Operators, and Neutral Delay Equations

AbstractSemi-hyperbolic mappings in Banach spaces are Lipschitz continuous and not necessarily invertible. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. It is shown that semi-hyperbolic mappings are locallyψ-contracting, whereψis the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it isψ-contracting and has no spectral values on the unit circle. A bishadowing result, which combines both direct and indirect forms of shadowing, is extended to semi-hyperbolic mappings in Banach spaces with locally condensing continuous comparison mappings. The result is applied to linear neutral delay equations with nonsmooth perturbations
Almost Periodic Solutions of First- and Second-Order Cauchy Problems

AbstractAlmost periodicity of solutions of first- and second-order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by Ruess–Vũ and Basit. Our proof uses a new idea. It is based on a factorisation method which also gives a short proof (of the vector-valued version) of Loomis' classical theorem, saying that a bounded uniformly continuous function from R into a Banach spaceXwith countable spectrum is almost periodic ifc0⊄X. Our method can also be used for solutions on the half-line. This is done in a separate paper
Asymptotic Behavior of Solutions of Quasilinear Hyperbolic Equations with Linear Damping

AbstractWe consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingVtt−a(Vx)x+αVt=0,(x,t)∈R×(0,∞), ((*))subsequent to [K. Nishihara,J. Differential Equations131(1996), 171–188]. In that article, the system with dampingvt−ux=0,ut+p(v)x=−αu,p′(v)<0(v>0) was treated, and the convergence rates to the diffusion wave by [L. Hsiao and T.-P. Liu,Comm. Math. Phys.143(1992), 599–605 and L. Hsiao and T.-P. Liu,Chinese Ann. Math. Ser. B14(1993), 465–480] were improved. For initial data (v0,u0)(x) satisfying (v0,u0)(±∞)=(v̄0,0) and ∫∞−∞(v0(x)−v̄0)dx=0 this system is reduced to the quasilinear hyperbolic equation with linear dampingVtt+(p(v0+Vx)−p(v0))x+αVt=0, and the decay rates ‖V(t)‖L∞=O(t−1/2), ‖Vx(t)‖L∞=O(t−1), ‖Vt(t)‖L∞=O(t−3/2), were obtained provided that (V,Vt) |t=0∈(H3×H2) is small and inL1×L1. From these decay rates,Vis expected to behave as a solution to the parabolic equation. The aims of this paper are to seek the asymptotic profileφ(x,t) satisfying ‖φ(t)‖L∞=O(t−1/2), ‖φx(t)‖L∞=O(t−1), ‖φt(t)‖L∞=O(t−3/2), and to show that the solution of (*) satisfies ‖(V−φ)(t)‖L∞=O(t−1), ‖(V−φ)x(t)‖L∞=O(t−3/2), ‖(V−φ)t(t)‖L∞=O(t−2). The proof is based on the elementary energy method and the Green function method for the parabolic equation
Spectral Decomposition and Invariant Manifolds for Some Functional Partial Differential Equations

AbstractWe study the integrodifferential convolution equationddt(x+μ∗x)−Ax−ν∗x=fon [0, +∞),x=φon (−∞, 0],as well as a nonlinear perturbation of the corresponding homogeneous equation. HereAis the generator of an analytic semigroup on a Hilbert spaceH, andμandνare operator-valued dominated measures with values inL(H) andL(D(A), H) respectively. Under the assumption that the operator given by the Laplace transform of the left-hand side of the equation is boundedly invertible on some right half-plane and on a line in the left half-plane, parallel to the imaginary axis, we decompose the solutions into components with different exponential growth rates. We construct projectors onto the stable and unstable subspaces, which are then used for the construction of stable and unstable manifolds for the nonlinear equation, which can have a fully nonlinear character. The results are applied to two equations of parabolic type. Moreover, the spectrum of the generator of the translation semigroup in various weighted spaces is determined, including the stable and unstable subspaces of our problem
A Difference Differential Equation of Euler–Cauchy Type

AbstractWe study a class of advanced argument linear difference differential equations analogous to Euler–Cauchy ordinary differential equations. Solutions of two equations of this type have arisen as adjoint functions in sieve theory, and they are also of use in control theory. Here we study the problem in a general setting. Subject to mild assumptions, each of our equations is shown to have a unique solution which is analytic in the right half-plane. In some cases the solution is a polynomial, and in others it has an asymptotic expansion. Finally, the solution is shown to have a representation as an exponential of a Hellinger type integro-differential operator acting on a monomial
Uniqueness Theorems for Positive Radial Solutions of Quasilinear Elliptic Equations in a Ball

AbstractWe establish a new Pohozaev-type identity and use it to prove a theorem on the uniqueness of positive radial solutions to the quasilinear elliptic problem div(|∇u|m−2emsp14;∇u)+f(u)=0 inB, andu=0 on ∂B, whereBis a finite ball in Rn,n⩾3 and 1<m⩽n. Applying this main uniqueness result we can prove that the semilinear problemΔu+μup+uq=0 inB, andu=0 on ∂B, whereμ>0 and 1⩽p<q⩽(n+2)/(n−2), has a unique positive solution whenn⩾6. This gives a complete answer to an open problem raised by Brezis and Nirenberg in 1983 in the casen⩾6. We shall also derive some partial results to the open problem in the casesn=3, 4, 5
Order Structures and the Heat Equation

AbstractIn this paper we shall introduce the notion of order structure and use it to study properties of a solutionu(x,t) of a scalar linear parabolic equation. It is well known that in one space dimension the number of zeros ofu(·,t) is nonincreasing witht. This zero number is an example of an order structure and the zero number property is very useful in unfolding the structure of the global attractor of the semiflow generated by a scalar parabolic equation. We shall prove that in one space dimension the zero number is the only order structure preserved by linear parabolic equations. In two dimensions the only order structure preserved by linear parabolic equations is the order structure induced by the comparison principle for second order equations. Consequently, in two dimensions there does not exist a fine decomposition into equivalence classes as in the one dimensional case
Global Solutions to a Game-Theoretic Riccati Equation of Stochastic Control

AbstractIn this paper the existence of global solutions to game-theoretic Riccati equations associated to a controlled stochastic differential equation with state dependent white noise is investigated. One proves that if a stabilizing and attenuating feed- back exists then a matrix differential game-theoretic Riccati equation of stochastic control has a unique global bounded stabilizing positive semidefinite solution; this solution is periodic if the coefficients are periodic functions and it solves an algebraic Riccati equation if the coefficients are constant. Conversly, if such a solution exists it allows an explicit construction of a stabilizing and disturbance attenuating state feedback
Unfolding of a Quadratic Integrable System with Two Centers and Two Unbounded Heteroclinic Loops

AbstractIn this paper we present a complete study of quadratic 3-parameter unfoldings of some integrable system belonging to the classQR3, and having two centers and two unbounded heteroclinic loops. We restrict to unfoldings that are transverse toQR3, obtain a versal bifurcation diagram and all global phase portraits, including the precise number and configuration of the limit cycles. It is proved that 3 is the maximal number of limit cycles surrounding a single focus, and only the (1,1)-configuration can occur in case of simultaneous nests of limit cycles. Essentially the proof relies on a careful analysis of a related non-conservative Abelian integral
On the Application of KAM Theory to Discontinuous Dynamical Systems

AbstractSo far the application of Kolmogorov–Arnold–Moser (KAM) theory has been restricted to smooth dynamical systems. Since there are many situations which can be modeled only by differential equations containing discontinuous terms such as state-dependent jumps (e.g., in control theory or nonlinear oscillators), it is shown by a series of transformations how KAM theory can be used to analyze the dynamical behaviour of such discontinuous systems as well. The analysis is carried out for the examplex+x+asgn(x)=p(t)withp∈C6being periodic. It is known that all solutions are unbounded for smalla>0. We prove that all solutions are bounded fora>0 sufficiently large, and that there are infinitely many periodic and quasiperiodic solutions in this case
Asymptotic Stability for a Class of Delay Differential Inclusions

AbstractIn this work we provide sufficient conditions for the asymptotic stability of an equilibrium for a class of differential inclusions of finite retarded type where the right-hand side is a set-valued mapping having a bounded, closed, convex graph. In particular our result includes “linear differential inclusions” in which the right-hand side is a closed, convex process. This result partially generalizes an earlier result of Leizarowitz (1985,SIAM J. Control Optim.23, 514–522) in the non-delay case
Exponential Dichotomies for Solitary-Wave Solutions of Semilinear Elliptic Equations on Infinite Cylinders

AbstractIn applications, solitary-wave solutions of semilinear elliptic equationsΔu+g(u,∇u)=0(x,y)∈R×Ωin infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov–Schmidt reduction near solitary waves can be applied. As an example, existence of shift dynamics near solitary waves is shown if a perturbationμh(x,u,∇u) periodic inxis adde
Some Qualitative Properties of the Riemann Problem in Gas Dynamical Combustion

AbstractWe study the Chapman–Jouguet (CJ) model and the selfsimilar Zeldovich–von Neumann–Döring (SZND) model in chemically reacting gas flows. We discover some basic relationships among ignition temperatureTi, total chemical binding energyQ, and the adibatic exponentγof polytropic gas. From these relations, we can determine when temperatures along the SZND burning solutions are higher than the ignition temperatureTi. We also study the all possible selfsimilar solutions for the SZND-model. From these results, we can determine when selfsimilar solutions for the CJ-model are the limits of selfsimilar solutions of the SZND model when the reaction rate tends to infinity
Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev Exponent

AbstractConsider the equation−Δpu=λg(x)|u|p−2u+f(x)|u|p*−2u ((1))in RN, where 1<p<Nandp*=Np/(N−p) is the critical Sobolev exponent. Letλ+1>0 be the principal eigenvalue of−Δpu=λg(x)|u|p−2uinRN,∫RNg(x)|u|p>0, ((2))withu+1>0 the associated eigenfunction. We prove: (i) Equation (1) has at least one positive solution ifλ∈(0,λ+1). (ii) Suppose ∫RNf(x)(u+1)p*<0. Then there existsλ0>λ+1such that (1) has at least one positive solution forλ∈[λ+1,λ0). Moreover, ifp⩾2, there exists λ0>λ+1such that (1) has at least two positive solutions forλ∈(λ+1,λ0)
Global Regularity of Solution for General Degenerate Parabolic Equations in 1-D

AbstractThis paper considers the Cauchy problem for the general degenerate parabolic equations (1.1) with initial data (1.2). In the critical condition meas{u:g(u)=0{=0 we obtain the regular estimateG(u)∈C(1), whereG(u)=∫u0g(s)ds. A new maximum principle is introduced to obtain the estimate and is applied to some special equations such as prous media equation, an infiltration equation to obtain the optimal estimate |(um−1)x|⩽M. Finally an interesting equation related to the Broadwell model (whereg(u) has two zero points) is studied and a uniquely regular solutionu∈C(1)is obtained. Moreover the estimatesux⩽ρ(f(u)−u2)/g(u) andρ⩾infxρ0(x)/(1+4t(infxρ0(x))) are proved for the solution of the Navier–Stokes equations corresponding to the Broadwell model
Global Existence of Solutions of an Extended Landau Equation

AbstractThe existence of solutions of a certain integro-differential equation, which contains a small parameter, is studied. This equation describes the amplitude of a disturbance of a boundary layer as a function of time. The present paper deals with the case of a weakly unstable boundary-layer flow. In this case, the amplitude equation is shown to have solutions, which are defined for all times, if the parameter of the equation is chosen sufficiently small. These solutions are determined uniquely by their initial values and tend to a common limit as time goes to infinity
Structural Stability of Flows under Numerics

AbstractIn our previous paper (SIAM J. Math. Anal.28(1997), 381–388), we showed that the qualitative properties of a Morse–Smale gradient-like flow are preserved by its discretization mapping obtained via numerical methods. In this paper, we extend the result to flows which satisfy Axiom A and the strong transversality condition. We prove that ifp⩾2,Φtis aCp+1 flow on a compact manifold satisfying Axiom A and the strong transversality condition, andNhis a numerical method of step sizehand orderp, then for all sufficiently smallh, there are a homeomorphismHhand a continuous real-valued functionτhonMsuch thatHh∘Φh+hτh(x)(x)=Nh∘Hh(x) andHhisO(hp)-close to the identity map onM
Connecting Orbit Structure of Monotone Solutions in the Shadow System

AbstractThe shadow system \begin{align}u_t= & \varepsilon ^2u_{xx}+f(u)-\xi ,\\ \xi = & \int^{}_{I} g(u, \xi) dx,\end{align}\quad I=[0, 1] is a scalar reaction diffusion equation coupled with an ODE. The extra freedom coming from the ODE drastically influences the solution structure and dynamics as compared to that of a single scalar reaction diffusion system. In fact, it causes secondary bifurcations and coexistence of multiple stable equilibria. Our long term goal is a complete description of the global dynamics on its global attractor A as a function ofε,f, andg. Since this is still far beyond our capabilities, we focus on describing the dynamics of solutions to the shadow system which are monotone inx, and classify the global connecting orbit structures in the monotone solution space up to the semi-conjugacy. The maximum principle and hence the lap number arguments, which have played a central role in the analysis of one dimensional scalar reaction diffusion equations, cannot be directly applied to the shadow system, although there is a Lyapunov function in an appropriate parameter regime. In order to overcome this difficulty, we resort to the Conley index theory. This method is topological in nature, and allows us to reduce the connection problem to a series of algebraic computations. The semi-conjugacy property can be obtained once the connection problem is solved. The shadow system turns out to exhibit minimal dynamics which displays the mechanism of basic pattern formation, namely it explains the dynamic relation among the trivial rest states (constant solutions) and the event patterns (large amplitude inhomogeneous solutions)
Dimension Estimate of Almost Periodic Attractors by Simultaneous Diophantine Approximation

AbstractIn this paper we estimate fractal dimensions of almost periodic trajectories for a semilinear parabolic partial differential equation: ∂u/∂t−d(t) Δu+g(t)u∋f(t), where we assume periodicity:d(t+α1)=d(t),g(t+α2)=g(t) for irrational numbersα1,α2, which are linearly independent over the rationals, andf(t+1)equals;f(t). By using simultaneous Diophantine approximation, we can show that the dimension of the almost periodic attractor is majorized by 1/γ1+2/γ2, whereγ1is the minimum number of the exponents of Hölder's conditions on the periodic functions andγ2is the secondary minimum
Stability in a Semilinear Boundary Value Problem via Invariant Conefields

AbstractWe give a geometric proof of stability for spatially nonhomogeneous equilibria in the singular perturbation problemut=ε2uxx+f(x,u),t∈R+, −1⩽u⩽1, with the Neumann boundary conditions onx∈[0,1]. The nonlinearity is of the formf(x,u)≔(1−u2)(u−c(x)), wherec(x) is merely continuous with a finite number of zeros. The strength of the method is in dealing with non-transversal zeros ofc, the case escaping the existing techniques of singular perturbations. The approach is also used for showing existence of unstable equilibria with one transition layer
Existence of Two Boundary Blow-Up Solutions for Semilinear Elliptic Equations

AbstractIn this paper we consider the boundary blow-up problemΔu=f(u)inΩ,u(x)→∞asx→∂Ω,and its non-autonomous version in a bounded, convexC2-domainΩof RN. We give growth conditions onfat ±∞ which imply the existence of two distinct blow-up solutions. The cases, (a)fhas a zero, and (b) minf>0, are fundamentally different. In case (a) we have a positive and a sign-changing blow-up solution. In case (b) we introduce a bifurcation parameterλinto the equationΔu=λf(u) and show that for 0<λ<λcritthere are blow-up solutions and forλ>λcritthere is no blow-up solution
Dynamics of Spiral Waves on Unbounded Domains Using Center-Manifold Reductions

AbstractAn equivariant center-manifold reduction near relative equilibria ofG-equivariant semiflows on Banach spaces is presented. In contrast to previous results, the Lie groupGinduces a strongly continuous group action on the underlying function space. In fact,Gmay act discontinuously. The results are applied to bifurcations of stable patterns arising in reaction–diffusion systems on the plane or in three-space modeling chemical systems such as catalysis on platinum surfaces and Belousov–Zhabotinsky reactions. These systems are equivariant under the Euclidean symmetry group. Hopf bifurcations from rigidly-rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated
Strong Uniqueness for Second Order Differential Operators

AbstractWe prove a strong unique continuation result for differential inequalities of the form |P(x,D)u|⩽C1|x|−2|u|+C2|x|−1|∇u|, whereP(x,D)=∑nj,k=1ajk(x)DjDkis an elliptic second order differential operator with Lipschitz coefficients such thatajk(0) is real.C1andC2are positive constants such thatC2is sufficiently small. Our assumption on the constantC2is justified by counterexamples due to Alinhac and Baouendi [2] and Wolff [6] showing that the strong unique continuation fails ifC2is not small