In this paper the existence of integral manifolds for impulsive differential systems with time-varying delay and with impulsive effect at fixed moments are investigated. The main results are obtained by using of piecewise continuous Lyapunov’s functions and Razumikhin’s technique.

In this paper, we investigate a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and transmission conditions at an interior point. A self-adjoint linear operator $A$ is defined in a suitable Hilbert space $H $such that the eigenvalues of such a problem coincide with those of $A$. We show that the operator $A$ has only point spectrum, the eigenvalues of the problem are algebraically simple, and the eigenfunctions of $A$ are complete in $H$.

We study the solutions of a quasilinear Schrödinger equation which has been derived in many areas of physical modeling. Using the Morawetz Radial Identity, we show that the local energy of a solution is integrable in time and the local $L^2$ norm of the solution approaches zero as time approaches the infinity.

We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection.

In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved in Dirac concentrations in "Lotka-Volterra parabolic PDEs", Indiana Univ. Math. J., 57:7 (2008), pp. 3275–3301, for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

We formally derive the simplest possible periodic wave structure consistent with time-periodic sound wave propagation in the $3 × 3$ nonlinear compressible Euler equations. The construction is based on identifying the simplest periodic pattern with the property that compression is counter-balanced by rarefaction along every characteristic. Our derivation leads to an explicit description of shock-free waves that propagate through an oscillating entropy field without breaking or dissipating, indicating a new mechanism for dissipation free transmission of sound waves in a nonlinear problem. The waves propagate at a new speed, (different from a shock or sound speed), and sound waves move through periods at speeds that can be commensurate or incommensurate with the period. The period determines the speed of the wave crests, (a sort of observable group velocity), but the sound waves move at a faster speed, the usual speed of sound, and this is like a phase velocity. It has been unknown since the time of Euler whether or not time-periodic solutions of the compressible Euler equations, which propagate like sound waves, are physically possible, due mainly to the ubiquitous formation of shock waves. A complete mathematical proof that waves with the structure derived here actually solve the Euler equations exactly, would resolve this long standing open problem.

We consider second order parabolic and elliptic systems with leading coefficients having the property of vanishing mean oscillation (VMO) in the spatial variables. An $L_q − L_p$ theory is established for systems both in divergence and non-divergence form.

A class of fully-discrete high-resolution schemes using flux limiters was constructed by P. K. Sweby, which amounted to add a limited anti-diffusive flux to a first order scheme. This technique has been very successful in obtaining high-resolution, second order, oscillation free, explicit difference schemes. However, the entropy convergence of such schemes has been open. For the scalar convex conservation laws, we use one of Yang’s convergence criteria to show the entropy convergence of the schemes with van Leer’s flux limiter when the building block of the schemes is the Godunov or the Engquish-Osher. The entropy convergence of the corresponding problems in semi-discrete case, for convex conservation laws with or without a source term, has been settled by Jiang and Yang.