Showing papers in "Journal of Differential Equations in 1994"

TL;DR: In this article, the authors studied the nonlinear wave equation involving the damping term u t | u t| m −1 and a source term of type u | u | p −1.

TL;DR: For simple models of hyperbolic systems of conservation laws, a delta-shock wave as discussed by the authors is a Dirac delta function supported on a shock, and it is shown that delta-shocks are w*-limits in L 1 of solutions to some reasonable viscous perturbations as the viscosity vanishes.

TL;DR: In this paper, a simple algebraic proof of the center conditions, λ = 0 and B = 0, A = − 1 2, B = 1 or A = Ā, B = ǫ, C = C or Aǫ = 2, BÁ = 1, |C| = 1.

TL;DR: In this article, the existence of an exponential dichotomy for a linear evolutionary system on a Banach space was shown to hold for linear evolutionary systems which are uniformly α-contracting and applied to the study of the linearization of nonlinear equations of the following type: (a) parabolic PDEs, including systems of reaction diffusion equations and the Navier-Stokes equations, including the nonlinear wave equation with dissipation; (b) retarded differential equations; and (c) certain neutral differential delay equations.

TL;DR: In this article, the existence of positive radial solutions of Δu+g(|x|) ƒ(u) = 0 in annuli with Dirichlet (Dirichlet/Neumann) boundary conditions was proved.

TL;DR: In this article, a technique for finding homoclinic orbits in singularly perturbed systems was developed, where the desired orbit lies near a singular solution that has both fast and slow time scales.

TL;DR: In this paper, a uniqueness theorem for the potential of the derived Sturm Liouville problem is proved when the data are the given spectra and partial knowledge of the potential is given.

TL;DR: In this paper, it was shown that the convergence of the trajectories holds good in the sense of weak topology, and in fact this convergence holds true in the strong topology.

TL;DR: In this article, the Riemann problem for a two-dimensional 2 × 2 hyperbolic system of nonlinear conservation laws has been solved analytically; 57 exact solutions and corresponding criteria have been obtained.

TL;DR: In this paper, a systematic approach towards a positive answer to Hilbert's 16th problem for quadratic systems, namely the existence of a uniform bound for the number of limit cycles, is discussed.

TL;DR: In this paper, a unified approach to both the cooperative and non-cooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques.

TL;DR: In this article, the authors studied the dynamics of the one-dimensional Cahn-Hilliard equation in a neighborhood of an equilibrium having N + 1 transition layers, and proved that a solution of the Cahnhilliard equation starts outside but close to the slow channel around the approximate manifold, then it will approach the channel at an exponentially large speed.

TL;DR: In this paper, the authors consider initial boundary value problems, including boundary damping, for the equations of motion of a viscous, heat-conducting, one-dimensional real gas and prove the existence of globally defined smooth solutions.

TL;DR: In this paper, it was shown that the wave is stable in polynomially weighted L ∞ spaces with respect to the growth rate of the polynomial near infinity.

TL;DR: In this paper, it was shown that invariant tori still exist in perturbed integrable Hamiltonian systems with n degrees of freedom H(p, q) = N(p) + P(p, q) if the hypothesis of nondegeneracy is replaced by a much weaker one that the rank of the mapping ∂N/∂p can be any positive integer smaller than n provided the image of any open set under the mapping includes a Cn + 2 curved one-dimensional submanifold.

TL;DR: In this paper, it was shown that for a dissipative, three dimensional, competitive, and irreducible system of ordinary differential equations having a unique equilibrium point, at which point the Jacobian matrix has negative determinant, either the equilibrium point is stable or there exists an orbitally stable periodic orbit.

TL;DR: In this paper, the authors apply the Picard-Vessiot differential Galois theory to the integrability of a Hamiltonian system and obtain results for the case of a finite arbitrary number of degrees of freedom.

TL;DR: In this article, it was shown that there are 28 non-equivalent topologic phase portraits of quadratic Hamiltonian vector fields (QH/H/X, − ∆H/∆x, ∆X/x) for a cubic polynomial in two variables over R.

TL;DR: In this paper, necessary and sufficient conditions on the resolvent of the generator of a strongly continuous semigroup were presented in order that the corresponding semigroup is hyperbolic.

TL;DR: In this paper, the authors solved the 2D Riemann problem with the initial data projecting some contact discontinuities and rarefaction waves, and revealed a variety of geometric structures for the interaction of elementary waves.

TL;DR: In this article, the problem of determining sufficient geometric conditions for some of the periodic solutions on this invariant manifold to survive after perturbation is addressed. And the main idea is to use a Lyapunov-Schmidt reduction for an appropriate displacement function in order to obtain the bifurcation function for the problem in a form which can be recognized as a generalization of the subharmonic Melnikov function.

TL;DR: In this article, necessary and sufficient conditions for disconjugacy, disfocality, or oscillation of the second order linear difference equation are given explicitly in terms of the coefficients in the equation.

TL;DR: The O (ϵ) part of solutions obtained via Glimm′s scheme of strictly hyperbolic conservation laws with O(ϵ)-periodic BV initial data is shown to tend as ϵ → 0 to the unique entropy solution of the modulation equations of weakly nonlinear geometric optics derived by Majda and Rosales.