pinecone/core-2020-05-10-deduplication huggingface dataset:
Reaction–Diffusion in Irregular Domains

AbstractWe consider the Cauchy–Dirichlet and Dirichlet problems for the nonlinear parabolic equationut−a(um)xx+buβ=0,where a>0, b∈R1, m>0, and β>0. The problems are considered in noncylindrical domains with nonsmooth boundaries. Existence, uniqueness, and comparison results are established. Constructed solutions are continuous up to the nonsmooth boundary if at each interior point the left modulus of the lower (respectively upper) semicontinuity of the left (respectively right) boundary curve satisfies an upper (respectively lower) Hölder condition near zero with Hölder exponent ν>12. The value 12 is critical as in the classical theory of the heat equation ut=uxx
On Periodic Solutions inside Isolating Chains

AbstractWe develop a geometric approach to problems concerning the existence of T-periodic solutions of a non-autonomous time-T periodic ordinary differential equation. We consider isolating segments, subsets of the extended phase space of the equation, which in some ways resemble isolating blocks from the theory of isolated invariant sets. The union of several contiguous isolating segments is called an isolating chain. Isolating segments determine some homomorphisms in reduced singular homologies. The main theorem asserts that the Lefschetz number of the composition of the homomorphisms determined by segments such that their union is a periodic isolating chain is equal to the fixed point index of the Poincaré map of the equation in the set of initial values of T-periodic solutions contained inside the chain. We give some applications of the theorem to planar polynomial equations. In particular, we prove that the equation ż=z5+sin2(φt)z has four nonzero (π/φ)-periodic solutions provided 0<φ⩽π/336
Critical Curve for p-q Systems of Nonlinear Wave Equations in Three Space Dimensions

AbstractThe existence of the critical curve for p-q systems for nonlinear wave equations was already established by D. Del Santo, V. Georgiev, and E. Mitidieri [1997, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, in “Geometric Optics and Related Topics” (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117–139, Birkhäuser, Basel] except for the critical case. Our main purpose is to prove a blow-up theorem for which the nonlinearity (p, q) is just on the critical curve in three space dimensions. Moreover, the lower and upper bounds of the lifespan of solutions are precisely estimated, including the sub-critical case
Global Solutions of Reaction–Diffusion Systems with a Balance Law and Nonlinearities of Exponential Growth

AbstractWe consider the initial boundary value problem of the form ut−aΔu=−f(u, v), vt−cΔu−dΔv=+f(u, v), x∈Ω∈RN, N⩾1, t∈R+ where f(u, v)⩾0, f(0, v)=0, v∈R; f(u, v)⩽Kφ(u)eσv, K and σ are positive constants, φ(.) is any continuous, nonnegative, locally Lipschitzian function on R such that φ(0)=0, d>a, and c<d−a with bounded continuous nonegative initial data. This system contains in particular the Frank–Kamenetskii approximation to an nth-order exothermic chemical reaction of Arrhenius type and non-systemically autocatalysed reaction–diffusion systems. We prove the existence of global classical solutions and study their large time behaviour. Our main tools are estimates of the Neumann function for the heat equation and local Lp a priori estimates independent of time
Decay of Mass for the Equation ut=Δu−a(x)up|∇u|q

AbstractConsider the quasilinear Cauchy problemut=Δu−a(x)up|∇u|q,x∈Rd,t>0u(x, 0)=φ(x)⩾0,x∈Rd,where a>0, p and q satisfyp⩾0andq⩾1orp>1andq=0,and0≨φ∈L1(Rd)∩C3, αb(Rd).This paper proves that the above equation possesses a unique positive classical solution and then investigates whether or not γ≡limt→∞∫Rdu(x, t)dx=0. In particular, it is shown that if a is on the order |x|m for large |x|, then γ=0 if dp+(d+1)q⩽d+2+m. Under the assumption that for compactly supported φ,‖∇uεφ(·, t)‖∞⩽εcφ(t+1)β,forsomeβ⩾0,andforε∈(0, 1], (*)where uεφ denotes the solution to the above equation with initial condition εφ, it is shown that γ>0 if dp+2βq>d+2+max(m, −d). For a certain range of the parameters d, p, q, m, it is proved that (*) holds with β=(d+1)/2, and for many other parameter values it is proved that (*) holds with β=d/2. Note that if β=(d+1)/2, then the above condition for γ>0 becomes dp+(d+1)q>d+2+max(m, −d), which in light of the parameter restrictions is equivalent to dp+(d+1)q>d+2+m
The Effect of Varying Coefficients on the Dynamics of a Class of Superlinear Indefinite Reaction–Diffusion Equations

AbstractIn this paper we analyze how the dynamics of a class of superlinear indefinite reaction–diffusion equations varies as the nodal behavior of a coefficient changes. To perform this analysis we use both theoretical and numerical tools. The analysis aids the numerical study, and the numerical study confirms and completes the analysis. The numerics in addition provides us with some further results for which-at-first glance analytical tools are not available yet. Our main analytical result shows that the problem possesses a unique positive solution which is linearly asymptotically stable if the trivial state is linearly unstable and the model admits some positive solution. This result is a relevant feature for superlinear indefinite problems, since our numerical computations show how these models can have an arbitrarily large number of positive solutions if the trivial state is unstable
Iteration Inequalities of the Maslov-Type Index Theory with Applications

AbstractIn this paper, by using the ω-index theory introduced by Y. Long in (1999, Pacific J. Math.187, 113–149), in particular, the splitting numbers, Maslov-type mean index, and the homotopy component of symplectic matrix, we establish various inequalities of the Maslov-type index theory for iterations of symplectic paths starting from the identity. As an application, these results are used to study Rabinowitz' conjecture on the prescribed minimal period solution problem of nonlinear Hamiltonian systems
Local Bifurcations and a Survey of Bounded Quadratic Systems

AbstractThis paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems. It is shown that the only finite-codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle-node and the Hopf–Takens bifurcations of codimensions 1 and 2 and the Bogdanov–Takens bifurcations of codimensions 2 and 3; furthermore, it is shown that whenever a bounded quadratic system has one of these critical points, then a full generic unfolding of the critical point exists in the class of bounded quadratic systems. Finally, we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system
Exchange Lemmas for Singular Perturbation Problems with Certain Turning Points

AbstractIn this work, singular perturbation problems with a certain type of turning point are studied from a geometric point of view. We first describe the delay phenomenon, initially studied by Pontryagin, of dynamics near the slow manifold possessing the turning points. Based on the local properties of such turning points and the center manifold theory for general invariant manifolds developed by Chow, Liu, and Yi, we extend the well-known exchange lemma, first formulated by Jones and Kopell for problems with normally hyperbolic slow manifolds, to problems with this type of turning point. Applications to singular boundary value problems with turning points are discussed
Generic Hopf Bifurcation from Lines of Equilibria without Parameters I. Theory

AbstractMotivated by decoupling effects in coupled oscillators, by viscous shock profiles in systems of nonlinear hyperbolic balance laws, and by binary oscillation effects in discretizations of systems of hyperbolic balance laws, we consider vector fields with a one-dimensional line of equilibria, even in the absence of any parameters. Besides a trivial eigenvalue zero we assume that the linearization at these equilibria possesses a simple pair of nonzero eigenvalues which cross the imaginary axis transversely as we move along the equilibrium line. In normal form and under a suitable nondegeneracy condition, we distinguish two cases of this Hopf-type loss of stability, hyperbolic and elliptic. Going beyond normal forms we present a rigorous analysis of both cases. In particular, α- and ω-limit sets of nearby trajectories consist entirely of equilibria on the line
The Relaxation of the Hydrodynamic Model for Semiconductors to the Drift–Diffusion Equations

AbstractWe establish the convergence and consistency of approximate solutions derived by the modified Godunov scheme for the initial-boundary value problem to a simplified one-dimensional hydrodynamic model for semiconductors using the compensated compactness method. The traces of weak solutions are introduced and then the weak solutions are proved to satisfy the natural boundary conditions. The zero relaxation limit of the hydrodynamic model to the drift–diffusion model is proved when the momentum relaxation time tends to zero
Global Travelling Waves in Reaction–Convection–Diffusion Equations

AbstractWe study the existence and properties of solutions in travelling wave form, u(x, t)=φ(x−st), defined for every z=x−st∈R, for the reaction-convection- diffusion equationut=a(um)xx+b(un)x+kupforx∈R,t>0,with a, m, n>0; b, k, p∈R. In the reaction case k>0 we prove that there exist travelling waves vanishing for z→∞ if and only if b>0 and min{1, n}⩽m+p2⩽max{1, n},p⩽max{1, n}.Moreover, if m+p≠2n, there exists a minimal velocity s*(a, b, k, m, n, p)>0, for which there are travelling waves only with s⩾s*, while in the case m+p=2n there are travelling waves only when 4amk⩽b2n and for every velocity s⩾0. Some properties of the function s* are established. All the waves are decreasing in their support and waves having bounded support from the right exist if and only if m>min{1, n}. Also, the absorption case k<0 is treated, where we find that, for different values of the parameters, there exists a unique travelling wave for every velocity s∈R, but for some case where only negative velocities exist. The cases b=0 or k=0 are well known in the literature
On the Integrability of Non-polynomial Scalar Evolution Equations

AbstractWe show the existence of infinitely many symmetries for λ-homogeneous equations when λ=0. If the equation has one generalized symmetry, we prove that it has infinitely many and these can be produced by recursion operators. Identifying equations under homogeneous transformations, we find that the only integrable equations in this class are the Potential Burgers, Potential Modified Korteweg–de Vries, and Potential Kupershmidt Equations. We can draw some conclusions from these results for the case λ=−1 which, although theoretically incomplete, seem to cover the known integrable systems for this case
Further Reductions of Normal Forms for Dynamical Systems

AbstractWe propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields, with concrete and interesting applications. We use lower order nonlinear terms in the normal form for the simplifications of higher order terms. Our approach is applicable for both the non nilpotent and the nilpotent cases. For dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal (or unique) with respect to equivalence by formal near identity transformations. We can compute at the same time a formal diffeormorphism that realizes the normalization. Comparisons with other methods are given for several examples
The Gevrey Asymptotics in the Case of Singular Perturbations

AbstractIn this paper, we present results in the Gevrey asymptotics which correspond to some existing results concerning asymptotic solutions in the Poincaré asymptotics of singularly perturbed ordinary differential equations. The main idea is based on a characterization of the Gevrey flat functions and a characterizaton of the Gevrey asymptotic expansions as it had been already exhibited by Y. Sibuya (1990, Lecture Notes in Pure and Applied Math., Vol. 124, pp. 393–401, Dekker, New York) and P. F. Hsieh and Y. Sibuya (1999, “Basic Theory of Ordinary Differential Equations,” Chap. XI, Sect. XI-2 and Chap. XII, Sect. XII-5, Springer-Verlag, Berlin)
On Similarity Solutions of a Heat Equation with a Nonhomogeneous Nonlinearity

AbstractWe are interested in positive radially symmetric solutions of the semilinear equationΔw−y·∇w2−Aw+|y|lwp=0,inRn,n⩾3, where p>1, l>−2 and A≡l+22(p−1). This equation is satisfied by self-similar solutions of a semilinear heat equation. We prove existence and non existence of solutions for various values of the parameters l and p. When solutions exist we study their asymptotic behavior and discuss their uniqueness. Our proofs are based on various continuity, comparison and Pohozaev type arguments
Spatially Extended Differential Equations in a Potential: On the Borders of the Basins of Attraction Related to Local Minima

AbstractWe consider a class of non-linear parabolic or hyperbolic partial differential equations on unbounded domains. These equations can be viewed as spatial extensions of dissipative oscillators in a potential. We prove that the basins of attraction of the homogeneous stationary solutions corresponding to local minima of the potential are open, and we describe the asymptotic behavior of a class of solutions which belong to the borders of these basins. We also study the basin of attraction of a homogeneous stationary solution corresponding to a global minimum of the potential
Nonlinear Standing and Rotating Waves on the Sphere

AbstractWe establish the existence of time-periodic solutions of semi-linear wave equations on the unit sphere in R3. The problem has been studied previously in (1982, V. Benci and Fortunato, Ann. Mat. Pura Appl.132, 215–242) using variational techniques. Our results here are much sharper: We employ delicate methods of bifurcation theory (1979, H. Kielhöfer, J. Math. Anal. Appl.68, 408–420; 1987, H. Kielhöfer and P. Kötzner, J. Appl. Math. Phys.38, 201–212) combined with well-known group-theoretic ideas to find branches of small-amplitude solutions. Precise spatio-temporal patterns of solutions are uncovered as a by-product of the analysis. Moreover, in certain cases we prove the existence of global solution branches (in the sense of P. Rabinowitz)
Classification of the Solutions of Semilinear Elliptic Problems in a Ball

AbstractIn this paper we fully describe the set of the positive and nodal (regular and singular) radial solutions of the superlinear elliptic PDE. −Δu=λu+|u|p−1uinB1,u=0on∂B1,p>1, (1)without restriction on the range of λ∈R. Here, B1 is the unit ball in RN. More precisely, in all subcritical, critical and supercritical cases, we analyze the possible singularities of radial solutions at the origin and the number of bounded and unbounded solutions. The solutions will be of three different types: bounded with a finite number of zeroes in (0, 1), singular at the origin, still with a finite number of zeroes and singular with sign changing oscillations at the origin
Asymptotic Behaviors of Solutions of Quasilinear Hyperbolic Equations with Linear Damping II

AbstractThis paper, as a follow-up to the recent work of K. Nishihara (1997, J. Differential Equations133, 384–395), is concerned with the asymptotic behaviors of the solution of quasilinear hyperbolic equation with linear damping Vtt−a(Vx)x+Vt=0,a′(v)>0forallv∈R,which satisfies the following prescribed initial condition:(V(t, x), Vt(t, x))|t=0=(V0(x), V1(x)),x∈R.Compared with the results obtained by K. Nishihara, the main novelties of our present paper lie in the following: First, we recover all the results obtained by K. Nishihara under some other smallness assumptions on the initial data (V0(x), V1(x)) which are much weaker than those needed in K. Nishihara's arguments. Second, under some additional assumptions on the nonlinear function a(v) and on the initial data (V0(x), V1(x)), when δ0=0, δ1≠0, we succeed in proving that φ(t, x) is still an asymptotic profile of V(t, x) and this answers partially the open problem left by K. Nishihara
Large Time Behavior for Convection-Diffusion Equations in RN with Periodic Coefficients

AbstractWe describe the large time behavior of solutions of the convection-diffusion equationut−div(a(x)∇u)=d·∇(|u|q−1u)in(0, ∞)×RNwhere d∈RN and a=a(x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that a∈W1, ∞(RN). First, we consider the linear problem (d=0) and prove that the large time behavior of solutions is given by the fundamental solution of the diffusion equation with a≡ah where ah is the homogenized matrix. In the nonlinear case, when q=1+1N, we prove that the large time behavior of solutions with initial data in L1(RN) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant homogenized diffusion a≡ah. When q>1+1N, we prove that the large time behavior of solutions is given by the fundamental solution of the linear-diffusion equation with a≡ah
Stability of Steady State Solutions for an Isentropic Hydrodynamic Model of Semiconductors of Two Species

AbstractWe discuss the stability of the steady state solutions for an isentropic hydrodynamic model of semiconductors of two species. We consider not only the mass and momentum equations for electrons, but also those two equations for holes. The recombination effects between electrons and holes are taken into consideration as well. We study the case where the doping profile is close to zero and depends on the spacial variable x. We shall show that steady state solutions are asymptotically stable
On Solutions to Nonlinear Reaction–Diffusion–Convection Equations with Degenerate Diffusion

AbstractThis paper consists of three parts. In Section 2, the Cauchy problem for general reaction-convection equations with a special diffusion term G(u)=um in multidimensional space is studied and Hölder estimates of weak solutions with explicit Hölder exponents are obtained by applying the maximum principle. In Section 3, for any nondecreasing smooth function G, the sharp regularity estimate G(u)∈C(1) up to the boundaries for the radial solution u of the general equation of Newtonian filtration is obtained by applying the maximum principle with the Minty's device. A direct by-product is the sharp regularity estimate of the temperature to the classical two-phase Stefan model. In Section 4, the Hölder continuity of weak solutions of the initial-boundary value problem for general nonlinear reaction–diffusion–convection equations is considered. Under the critical condition on the diffusion function G:meas{u:G′(u)=g(u)=0}=0, we obtain a Hölder continuous solution u and the sharp regularity estimate G(u)∈C(1) up to the boundaries. Our proof is based on the maximum principle
Resonance Tongues in Hill's Equations: A Geometric Approach

AbstractThe geometry of resonance tongues is considered in, mainly reversible, versions of Hill's equation, close to the classical Mathieu case. Hill's map assigns to each value of the multiparameter the corresponding Poincaré matrix. By an averaging method, the geometry of Hill's map locally can be understood in terms of cuspoid Whitney singularities. This adds robustness to the result. The algorithmic nature of the averaging method enables a pull-back to the resonance tongues of the original system
Stiff Well-Posedness and Asymptotic Convergence for a Class of Linear Relaxation Systems in a Quarter Plane

AbstractIn this paper we study the asymptotic equivalence of a general linear system of 1-dimensional conservation laws and the corresponding relaxation model proposed by S. Jin and Z. Xin (1995, Comm. Pure Appl. Math.48, 235–277) in the limit of small relaxation rate. The main interest is this asymptotic equivalence in the presence of physical boundaries. We identify and rigorously justify a necessary and sufficient condition (which we call the Stiff Kreiss Condition, or SKC in short) on the boundary condition to guarantee the uniform well-posedness of the initial boundary value problem for the relaxation system independent of the rate of relaxation. The SKC is derived and simplified by using a normal mode analysis and a conformal mapping theorem. The asymptotic convergence and boundary layer behavior are studied by the Laplace transform and a matched asymptotic analysis. An optimal rate of convergence is obtained
Viability for Nonautonomous Semilinear Differential Equations

AbstractLet X be a reflexive Banach space, A:D(A)⊂X→X the infinitesimal generator of a compact C0-semigroup S(t):X→X, t⩾0, D a locally closed subset in X and (t, x)↦f(t, x) a function on [a, b)×D which is locally integrably bounded, measurable with respect to t and continuous with respect to x. The main result of the paper is: Theorem.Under the general assumptions above a necessary and sufficient condition in order that for each (τ, ξ)∈[a, b)×Dthere exists at least one mild solutionu:[τ, T]→Dofdudt(t)=Au(t)+f(t, u(t))satisfyingu(τ)=ξis the tangency condition below: There is a negligible subset Z of [a, b) such that for eacht∈[a, b)\Zand eachx∈D,liminfh↓01hd(S(h)x+hf(t, x), D)=0
Topological Equivalence of Planar Vector Fields and Their Generalised Principal Part

AbstractLet X(R2) be the space of C∞ planar vector fields. We consider the space V⊂X(R2) of vector fields with an isolated singularity and a fixed Newton diagram. We define the generalised principal part XΔ of the vector field X∈V and give the nondegeneracy condition on XΔ, using the Newton diagram. We prove that X∈V is locally topologically equivalent to its minimal generalised principal part XΔ, if XΔ is nondegenerate and X is not a monodromic vector field. In the proof we use the normal form method and the blowing-up method
Simplicial Models for the Global Dynamics of Attractors

AbstractGiven an unknown attractor A in a continuous dynamical system, how we can discover the topology and dynamics of A? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from A onto a model system M whose topology and dynamics are known. The complexity of M then provides a lower bound for the complexity of A. In this paper, we use the techniques of the Conley index to construct a simplicial model and a surjective semi-conjugacy for a large class of attractors. The essential features of this construction are that the model M can be explicitly described and that the finite amount of information needed to construct it is computable
On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow

AbstractWe consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities. It is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and amplitude of the imperfections tend to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem, and the tangential drag force and the effective mass flow are determined up to order O(ε3/2). The value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of O(ε)