Global Existence of Regular Solutions for the Vlasov–Poisson–Fokker–Planck System

AbstractWe study the global existence and uniqueness of regular solutions to the Cauchy problem for the Vlasov–Poisson–Fokker–Planck system. Two existence theorems for regular solutions are given under slightly different initial conditions. One of them completely includes the results of P. Degond (1986, Ann. Sci. Ecole Norm. Sup.19, 519–542)

Decay Rates of Solutions for Non-Degenerate Kirchhoff Type Dissipative Wave Equations

Consider the Cauchy problem for the non-degenerate Kirchhoff type dissipative wave equations with the initial data belonging to (H2(RN)∩L1(RN))×(H1(RN)∩L1(RN)). Using the Fourier transform method in the L2 ∩ L1-frame, we can improve the decay rates of the energies given by the energy method of the L2-frame

Decay Properties for Mildly Degenerate Kirchhoff Type

Under the assumption that the initial data belong to suitable Sobolev spaces, we derive the better decay estimate of the second order derivatives for the initial boundary value problem for degenerate dissipative wave equations of Kirchhoff type

Upper Decay Estimates for Non-Degenerate Kirchhoff Type Dissipative Wave Equations

We study on the Cauchy problem for non-degenerate Kirchhoff type dissipative wave equations ρu′′ + a (||A1/2u(t)||2) Au + u′ = 0 and (u(0), u′(0)) = (u0, u1), where u0 ≠ 0 and the nonlocal nonlinear term a(M) = 1+Mγ with γ > 0. Under the suitably smallness condition, we derive the upper decay estimates of the solution u(t) for the case of 0 < γ < 1 in addition to γ ≥ 1

Global Solvability for Mildly Degenerate Kirchhoff Type

Consider the initial boundary value problem for degenerate dissipative wave equations of Kirchhoff type. When the wave coefficient ρ > 0 or the initial energy E(0) is small, we show the global existence theorem

Lanchester Type Models with Time Dependent Coefficients

We consider an ordinary differential system which is a so-called Lanchester’s linear law model with time dependent coefficients. We study on asymptotic forms of solutions that decay to a point on the x-axis and y-axis

On Global Solvability for Some Degenerate Dissipative Nonlinear Kirchhoff Strings

Consider the initial boundary value problem for degenerate dissipative wave equations of Kirchhoff type with attractive force terms.We are interested in the case of 0 < γ < 1 for the degeneracy of nonlinear term Φ(r) = rγ. We prove the global solvability problem, provided that the initial data belong to the potential well and satisfy a suitable smallness condition. Moreover, we derive optimal decay estimates of the solutions

On Decay Properties of Solutions for the Vlasov-Poisson System

We study decay properties of solutions to the Cauchy problem for the collision-less Vlasov–Poisson system which appears Vlasov plasma physics and stems from Liouville’s equation coupled with Poisson’s equation for the determining the self-consistent electrostatics or gravitational forces

Lower Decay Estimates for Non-Degenerate Kirchhoff Type Dissipative Wave Equations

We consider the Cauchy problem for non-degenerate Kirchhoff type dissipative wave equations ρu′′+ a (∥A1/2u(t)∥2) Au + u′ = 0 and (u(0), u′(0)) = (u0, u1), where u0 ≠ 0. We derive the lower decay estimate ∥u(t)∥2 ≥ Ce−βt for t ≥ 0 with β > 0 for the solution u(t)