Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

A maximum principle for evolution Hamilton-Jacobi equations on Riemannian manifolds

We establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form ut+F(t,dxu)=0, u(0,x)=u₀(x), where is a bounded uniformly continuous function, M is a Riemannian manifold, and . This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations.     

Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications

A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three u_tt−Δ_xu+cu_t+λu=f(t,x), when c>0, λ∈(0,c²/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u_tt−Δ_xu+cu_t=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given.     

On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity

It is well known that a scalar differential equation , where f(t,x) is continuous, T-periodic in t and weakly convex or concave in x has no, one or two T-periodic solutions or a connected band of T-periodic solutions. The last possibility can be excluded if f(t,x) is strictly convex or concave for some t in the period interval. In this paper we investigate how the actual number of T-periodic solutions for a given equation of this type in principle can be determined, if f(t,x) is also assumed to have a continuous derivative . It turns out that there are three cases. In each of these cases we indicate the monotonicity properties and the domain of values for the function P(ξ)=S(ξ)−ξ, where S(ξ) is the Poincaré successor function. From these informations the actual number of periodic solutions can be determined, since a zero of P(ξ) represents a periodic solution.     

Interval criteria for oscillation of second-order half-linear differential equations

By employing a generalized Riccati technique and an integral averaging method, interval oscillation criteria are established for the second-order half-linear differential equation [r(t)|x′(t)|^α−1x′(t)]′+q(t)|x(t)|^α−1x(t)=0. These criteria are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. They also extend, improve, and complement a number of existing results, and can be applied to extreme cases such as . In particular, several interesting examples that illustrate the importance of our results are included.     

On the optimal control problem for Schrödinger equation with complex potential

The existence and uniqueness for the solution of the problem of determining the v(x,t) potential in the Schrödinger equation from the measured final data ψ(x,T)=y(x) is investigated. For the objective functional , it is proven that the problem has at least one solution for α?0, and has a unique solution for α>0. The necessary condition for solvability the problem is stated as the variational principle.     

On the existence and multiplicity of positive periodic solutions for first-order vector differential equation

In this paper, we consider the existence and multiplicity of positive periodic solutions for first-order vector differential equation x^′(t)+f(t,x(t))=0, a.e. t∈[0,ω] under the periodic boundary value condition x(0)=x(ω). Here ω is a positive constant, and is a Carathéodory function. Some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones.     

Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.     

Hardy type inequality and application to the stability of degenerate stationary waves

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t−α/4 as t→∞, provided that the initial perturbation is in the weighted space for α<αc(q):=3+2/q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in for α>αc(q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

A generalization of Barbashin-Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,b∈K, such that for some c∈K. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,T]×BH.     

Stability switches in a system of linear differential equations with diagonal delay

This paper deals with the stability problem of a delay differential system of the form x^′(t)=-ax(t-τ)-by(t), y^′(t)=-cx(t)-ay(t-τ), where a, b, and c are real numbers and τ is a positive number. We establish some necessary and sufficient conditions for the zero solution of the system to be asymptotically stable. In particular, as τ increases monotonously from 0, the zero solution of the system switches finite times from stability to instability to stability if ; and from instability to stability to instability if . As an application, we investigate the local asymptotic stability of a positive equilibrium of delayed Lotka-Volterra systems.     

On α-times integrated C-cosine functions and abstract Cauchy problem I

In this paper we first investigate some basic properties concerning nondegenerate α-times integrated C-cosine functions on a Banach space X, and then characterize their generator A in terms of the unique existence of strong solutions of the following abstract Cauchy problem: for t>0, u(0)=x, u^′(0)=y.     

Fractional derivatives of products of Airy functions

Fractional derivatives of the products of Airy functions are investigated, and Dα{Ai(x)×Bi(x)}, where Ai(x) and Bi(x) are the Airy functions of the first and second type, respectively. They turn out to be linear combinations of Dα{Ai(x)} and Dα{Gi(x)}, where Gi(x) is the Scorer function. It is also proved that the Wronskian W(x) of the system of half integrals {D−1/2Ai(x),D−1/2Gi(x)} and its Hilbert transform can be considered special functions in their own right since they are expressed in terms of and Ai(x)Bi(x), respectively. Various integral relations are established. Integral representations for Dα{Ai(x−a)Ai(x+a)} and its Hilbert transform −HDα{Ai(x−a)Ai(x+a)} are derived.     

Homoclinic solutions for a class of nonperiodic and noneven second-order Hamiltonian systems

The existence of homoclinic solutions is obtained for second-order Hamiltonian systems , as the limit of the solutions of a sequence of nil-boundary-value problems which are obtained by the Mountain Pass theorem, when L(t) and W(t,x) are neither periodic nor even with respect to t.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X∼f(∥x-θ∥²) and let δ_π(X) be the generalized Bayes estimator of θ with respect to a spherically symmetric prior, π(∥θ∥²), for loss ∥δ-θ∥². We show that if π(t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator δ₀(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e^-αtβ and e^-αt+βφ(t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .