Growth Rate Of Solutions Research Articles

Nonlinear Volterra Integrodifferential Equations from above on Unbounded Time Scales

The paper is devoted to studying the existence, uniqueness and certain growth rates of solutions with certain implicit Volterra-type integrodifferential equations on unbounded from above time scales. We consider the case where the integrand is estimated by the Lipschitz type function with respect to the unknown variable. Lipschitz coefficient is an unbounded rd-function and the Banach fixed-point theorem at a functional space endowed with a suitable Bielecki-type norm.

Growth Rates of Sublinear Functional and Volterra Differential Equations

This paper considers the growth rates of positive solutions of scalar nonlinear functional and Volterra differential equations. The equations are assumed to be autonomous (or asymptotically so), and the nonlinear dependence grows less rapidly than any linear function. We impose extra regularity properties on a function asymptotic to this nonlinear function, rather than on the nonlinearity itself. The main result of the paper demonstrates that the growth rate of the solution can be found by determining the rate of growth of a trivial functional differential equation (FDE) with the same nonlinearity and all its associated measure concentrated at zero; the trivial FDE is nothing other than an autonomous nonlinear ODE. We also supply direct asymptotic information about the solution of the FDE under additional conditions on the nonlinearity, and exploit the theory of regular variation to sharpen and extend the results.

A proof of the [formula omitted]-regularity conjecture in the plane

We establish a new oscillation estimate for solutions of nonlinear partial differential equations of elliptic, degenerate type. This new tool yields a precise control on the growth rate of solutions near their set of critical points, where ellipticity degenerates. As a consequence, we are able to prove the planar counterpart of the longstanding conjecture that solutions of the degenerate p-Poisson equation with a bounded source are locally of class Cp′=C1,1p−1; this regularity is optimal.

Porosity of free boundaries in the obstacle problem for quasilinear elliptic equations

In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp et al. (J. Diff. Equ.164 (2000) 110–117) for p-Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ.40(2) (2012) 11–13, 18) for p-Laplacian type equations.

Kinetic anomalies of crystal growth: Development of methodical approaches and interpretations

Crystal growth rate in solutions is characterized by the so-called anomalies in the form of maxima and minima alternating with a change in temperature. This phenomenon was discovered by Sipyagin and Chernov in 1967. To date, it has been found for all thoroughly investigated materials. In this paper, the kinetic anomalies of electrochemical deposition of metals (Cu and Hg) from CuSO4 and Hg(NO3)2 solutions are described for the first time. The data on the growth of CuSO4 · 5H2O crystals in a solution are reported. On the basis of comparison of the characteristics of anomalies for Cu-CuSO4 · 5H2O and Cu-Hg pairs, with due regard for the expected general and different features of the structure of solutions and adsorption layers, it is concluded that this phenomenon is caused by the structural alterations in the bulk of the solution with a significant influence of the adsorption layer.

Almost-sure asymptotic stability of a general four-dimensional system driven by real noise

In the first part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the rotation numbers for a general four-dimensional dynamical system driven by a small-intensity real noise process. Stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. This work is an extension of the work of Sri Namachchivaya and Van Roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered.

The blowup property of solutions to some diffusion equations with localized nonlinear reactions

In this paper we investigate the blowup property of solutions to the equation u t = Δu + ƒ(u(x 0, t)) , where x 0 is a fixed point in the domain. We show that under certain conditions the solution blows up in finite time. Moreover, we prove that the set of all blowup points is the whole region. Furthermore, the growth rate of solutions near the blowup time is also derived. Finally, the results are generalized to the following nonlocal reaction-diffusion equation u t = Δu + ∝ Ω ƒ(u) dx .

Investigation of the Solution Status of TGS and ATGSP Crystals

AbstractThis paper reports on the solubility curves of TGS and ATGSP crystals with different pH's. The relationship of (010) growth rate in solutions with different pH varied with the supersaturation, has been studied. The influence of pH on the metastable region of solution have been measured. The experimential results are discussed with respect to the structure and viscosity of solution and the solubility of the crystals. Finally the optimum conditions for the rapid growth of TGS and ATGSP crystals is given.

The effect of pH on the urinary inhibition of calcium oxalate crystal growth.

Urinary inhibition of calcium oxalate crystal growth was measured in metastable solutions of sodium oxalate and calcium chloride at different pH. Inhibition of calcium oxalate crystal growth in urine from patients with calcium oxalate stone disease increased with increasing pH. The increase was most pronounced between pH 5.5 and 7.0. Sodium pyrophosphate in a similar way inhibited calcium oxalate crystal growth in this pH interval, and between pH 6.0 and 7.5 there was also a slight increase in the inhibiting activity by chondroitin sulphate and citrate. Approximative correction factors were calculated in order to obtain a more appropriate value of the inhibition index when urine pH was different from 6.0, and a reasonably good correlation was obtained between inhibition indices calculated by means of the correction factor and inhibition indices derived from direct measurements of the crystal growth rate in solutions with different pH.