Realization of prescribed patterns in the competition model

We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D₁,…,D_m of a given spatial domain Ω in R^N, if d₁,d₂,a₁,a₂,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D?D₁∪?∪D_m and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model

     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v₀ is initially assumed to be in H¹ and the velocity variable u₀ to be in L² on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L²([0,1]) and v approaches the set of stationary solutions in H¹([0,1]).     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

A ground state alternative for singular Schrödinger operators

Let a be a quadratic form associated with a Schrödinger operator L=-∇·(A∇)+V on a domain Ω⊂R^d. If a is nonnegative on , then either there is W>0 such that for all , or there is a sequence and a function ?>0 satisfying L?=0 such that a[?_k]→0, ?_k→? locally uniformly in Ω?{x₀}. This dichotomy is equivalent to the dichotomy between L being subcritical resp. critical in Ω. In the latter case, one has an inequality of Poincaré type: there exists W>0 such that for every satisfying there exists a constant C>0 such that for all .     

Kinematical conservation laws in a space of arbitrary dimensions

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ω_t in d-dimensional x-space, where x = (x ₁, x ₂,..., x _d) ∈ R^d. The KCL are derived in a specially defined ray coordinates (ξ = (ξ₁, ξ₂,..., ξ_d?1), t), where ξ₁, ξ₂,..., ξ_d?1 are surface coordinates on Ω_t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω_t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω_t when Ω_t is a curve in R² and is a curve on Ω_t when it is a surface in R³. Across a kink the normal n to Ω_t and normal velocity m on Ω_t are discontinuous.     

Cauchy integral decomposition for harmonic forms

Let ann-dimensional differential form Ω be defined at points of aC ¹-smooth boundary π of a domainG ? ? ⁿ . Under what condition can Ω be represented as Ω = Ω₊ + Ω₊ + Ω_-, where Ω_± are forms insideG and outsideG, harmonic in the sense of Hodge? A necessary condition is that both restrictions Ω{inπ and *Ω{inπ be closed in the sense of currents. This condition, with an additional smoothness assumption, turns out to be sufficient as well. This is an analogue of the Cauchy integral decomposition of functions in the plane.     

Asymptotic expansions for distributions of latent roots in multivariate analysis

Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S₁(S₁ + S₂)^?1, where S₁ is $\text{W}{}_{\mathrm{<mtext>m</mtext>}}\text{(n}{}_1\text{, Σ, Ω)}$ and S₂ is W_m(n₂, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n^?1S, where S in W_m(n, Σ), for large n, and S₁S₂^?1, where S₁ is W_m(n₁, Σ) and S₂ is W_m(n₂, Σ), for large n₁ + n₂.     

Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance

We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses m₁, m₂ satisfying the resonance relation m₂=2m₁>0. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate O(|t|⁻¹) as t→±∞ in L^∞. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of J.-M. Delort, D. Fang and R. Xue [J.-M. Delort, D. Fang, R. Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004) 288-323].     

Energy method in the partial Fourier space and application to stability problems in the half space

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x^′∈Rn−1. For the variable x₁∈R₊ in the normal direction, we use L² space or weighted L² space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].     

Scalar curvature type problem on the three dimensional bounded domain

In this paper we prove an existence result for the nonlinear elliptic problem:-△u = Ku~5,u 0 in Ω,u = 0 on?Ω,where Ω is a smooth bounded domain of R~3 and K is a positive function in Ω.Our method relies on studying its corresponding subcritical approximation problem and then using a topological argument.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

On the solvability of the jump problem in clifford analysis

Let Ω be a bounded open and oriented connected subset of ? ⁿ which has a compact topological boundary Γ, let C be the Dirac operator in ? ⁿ , and let ?_0,n be the Clifford algebra constructed over the quadratic space ? ⁿ . An ?_0,n -valued smooth function f : Ω → ?_0,n in Ω is called monogenic in Ω if Df = 0 in Ω. The aim of this paper is to present the most general condition on Γ obtained so far for which a Hölder continuous function f can be decomposed as F ⁺ ? F ^? = f on Γ, where the components F ^± are extendable to monogenic functions in Ω^± with Ω⁺ := Ω, and Ω^? := ? ⁿ \ (Ω ? Γ), respectively.     

Existence and uniqueness of solutions for a non-uniformly parabolic equation

In this paper, we study a generalized Burgers equation u_t+(u²)_x=tu_xx, which is a non-uniformly parabolic equation for t>0. We show the existence and uniqueness of classical solutions to the initial-value problem of the generalized Burgers equation with rough initial data belonging to .     

Remote points in large products

A point p ∈ βX\X is a remote point of X if p? cl_βXD for any nowhere dense D ? X. Van Douwen, and independently Chae and Smith, have shown that each non-pseudocompact space of countable π-weight has a remote point. Van Mill showed that many spaces of π-weight ω₁, such as ω×2^ω₁ also have remote points.We show that arbitrarily large products of spaces with countable π-weight which are not pseudocompact have remote points. In particular, ω×2^? for any infinite cardinal ?.     

On some questions concerning permanents of (1,−1)-matrices

Let Ω _n denote the set of alln×n-(1,?1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω _n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2 ^k ?1,k≧2, even for singularA∈Ω _n . Moreover, we prove that perA≠0 ifA∈Ω _n ,n=2 ^k ?1, and we derive new results concerning the divisibility of the permanent in Ω _n by powers of 2.     

The behavior of the best Sobolev trace constant and extremals in thin domains

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω)?Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω),α)?Lq(P(Ω),β).For the special case p=q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case.