Hamilton-Jacobi equations having only action functions as solutions

Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution of the Hamilton-Jacobi equation

The Brezis-Nirenberg problem near criticality in dimension 3

We consider the problem of finding positive solutions of Δu+λu+u^q=0 in a bounded, smooth domain Ω in , under zero Dirichlet boundary conditions. Here q is a number close to the critical exponent 5 and 0<λ<λ₁. We analyze the role of Green's function of Δ+λ in the presence of solutions exhibiting single and multiple bubbling behavior at one point of the domain when either q or λ are regarded as parameters. As a special case of our results, we find that if , where λ∗ is the Brezis-Nirenberg number, i.e., the smallest value of λ for which least energy solutions for q=5 exist, then this problem is solvable if q>5 and q−5 is sufficiently small.     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

Godunov-type approximation for a general resonant balance law with large data

We consider the Cauchy problem for the 2×2 nonstrictly hyperbolic system

     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

Regularity of weak solution to Maxwell's equations and applications to microwave heating

In this paper we first study the regularity of weak solution for time-harmonic Maxwell's equations in a bounded anisotropic medium Ω. It is shown that the weak solution to the linear degenerate system, , is Hölder continuous under the minimum regularity assumptions on the complex coefficients γ(x) and ξ(x). We then study a coupled system modeling a microwave heating process. The dynamic interaction between electric and temperature fields is governed by Maxwell's equations coupled with an equation of heat conduction. The electric permittivity, electric conductivity and magnetic permeability are assumed to be dependent of temperature. It is shown that under certain conditions the coupled system has a weak solution. Moreover, regularity of weak solution is studied. Finally, existence of a global classical solution is established for a special case where the electric wave is assumed to be propagating in one direction.     

On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem

We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem.     

Finitely generated varieties of distributive double <Emphasis Type="Italic">p</Emphasis>-algebras universal modulo a group

For any monoid M, any universal variety contains arbitrarily large algebras whose endomorphism monoid is isomorphic to M. A variety universal modulo a group G contains arbitrarily large algebras whose endomorphism monoid is isomorphic to the direct product M x G. One of the results of this paper structurally characterizes all finitely generated varieties of distributive double p-algebras universal modulo a group, and shows that any unavoidable direct factor G is a Boolean group with at most eight elements.