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Journal articles on the topic 'Solutions with exponential growth'

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Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Fattorini, H. O. "On the growth of solutions to second order differential equations in Banach spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 101, no. 3-4 (1985): 237–52. http://dx.doi.org/10.1017/s0308210500020801.

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SynopsisWe obtain estimates for the exponential growth of the solutions to u″(t) = (A + ζ2I)u(t) in terms of the exponential growth of the solutions to u″(t) = Au(t), where ζ is an arbitrary complex number. Estimates in exponentially weighted L2 norms are also considered in Hilbert space.

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2

Hang, Fengbo, and Fanghua Lin. "Exponential growth solutions of elliptic equations." Acta Mathematica Sinica, English Series 15, no. 4 (October 1999): 525–34. http://dx.doi.org/10.1007/s10114-999-0084-2.

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3

Popivanov, N., T. Popov, and R. Scherer. "Singular solutions with exponential growth to Protter’s problems." Siberian Advances in Mathematics 23, no. 3 (July 2013): 219–26. http://dx.doi.org/10.3103/s1055134413030073.

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4

Lubinsky, Doron S., and Paul Nevai. "Sub-Exponential Growth of Solutions of Difference Equations." Journal of the London Mathematical Society s2-46, no. 1 (August 1992): 149–60. http://dx.doi.org/10.1112/jlms/s2-46.1.149.

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5

SCHEUTZOW, MICHAEL. "EXPONENTIAL GROWTH RATES FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS." Stochastics and Dynamics 05, no. 02 (June 2005): 163–74. http://dx.doi.org/10.1142/s0219493705001468.

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In this survey, we provide some tools to obtain estimates for the almost sure exponential growth rate of a stochastic delay differential equation (sdde) which fixes zero. In particular, we are interested in determining whether the solutions of a given sdde are exponentially stable (i.e. have a negative exponential growth rate) or not. We focus on equations without drift, which are a good testground to assess if a method is powerful enough to discriminate between stability and instability when a certain parameter (e.g. noise intensity) varies. The most powerful tool we provide is the method of Lyapunov functionals which is used to obtain upper bounds for p-th moment exponents for small (positive and negative) values of p.

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6

ZAIDI, A. A., and B. VAN BRUNT. "ASYMMETRICAL CELL DIVISION WITH EXPONENTIAL GROWTH." ANZIAM Journal 63, no. 1 (January 2021): 70–83. http://dx.doi.org/10.1017/s1446181121000109.

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AbstractAn advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to “exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal.

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7

Zaidi, Ali, and Bruce Van Brunt. "Asymmetrical cell division with exponential growth." ANZIAM Journal 63 (July 30, 2021): 70–83. http://dx.doi.org/10.21914/anziamj.v63.16116.

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An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to "exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal. doi:10.1017/S1446181121000109

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8

ZHANG, ZHITAO, MARTA CALANCHI, and BERNHARD RUF. "ELLIPTIC EQUATIONS IN ℝ2 WITH ONE-SIDED EXPONENTIAL GROWTH." Communications in Contemporary Mathematics 06, no. 06 (December 2004): 947–71. http://dx.doi.org/10.1142/s0219199704001549.

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We consider elliptic equations in bounded domains Ω⊂ℝ2 with nonlinearities which have exponential growth at +∞ (subcritical and critical growth, respectively) and linear growth λ at -∞, with λ>λ1, the first eigen value of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical groups and Morse index of these solutions are given. Also the case λ<λ1 is considered.

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9

Benameur, Jamel, and Mongi Blel. "Asymptotic Study of the 2D-DQGE Solutions." Journal of Function Spaces 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/538374.

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We study the regularity of the solutions of the surface quasi-geostrophic equation with subcritical exponent1/2<α≤1. We prove that if the initial data is small enough in the critical spaceH˙2-2α(R2), then the regularity of the solution is of exponential growth type with respect to time and itsH˙2-2α(R2)norm decays exponentially fast. It becomes then infinitely differentiable with respect to time and has value in all homogeneous Sobolev spacesH˙s(R2)fors≥2-2α. Moreover, we give some general properties of the global solutions.

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10

Alves, Claudianor O., and Sérgio H. M. Soares. "Nodal solutions for singularly perturbed equations with critical exponential growth." Journal of Differential Equations 234, no. 2 (March 2007): 464–84. http://dx.doi.org/10.1016/j.jde.2006.12.006.

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11

TAN, ZHONG, and FEI FANG. "NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH." Analysis and Applications 11, no. 03 (May 2013): 1350005. http://dx.doi.org/10.1142/s021953051350005x.

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Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

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12

Souza, Manassés, Uberlandio Batista Severo, and Thiago Luiz do Rêgo. "Nodal solutions for fractional elliptic equations involving exponential critical growth." Mathematical Methods in the Applied Sciences 43, no. 6 (January 10, 2020): 3650–72. http://dx.doi.org/10.1002/mma.6145.

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13

Binyamini, Gal. "Bezout-type theorems for differential fields." Compositio Mathematica 153, no. 4 (March 13, 2017): 867–88. http://dx.doi.org/10.1112/s0010437x17007035.

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We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first $l$ derivatives of an $n$-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on $n$ and $l$) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.

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14

Liu, Jian-Guo, and Robert Strain. "Global stability for solutions to the exponential PDE describing epitaxial growth." Interfaces and Free Boundaries 21, no. 1 (May 9, 2019): 61–86. http://dx.doi.org/10.4171/ifb/417.

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15

Price, Brock C., and Xiangsheng Xu. "Strong solutions to a fourth order exponential PDE describing epitaxial growth." Journal of Differential Equations 306 (January 2022): 220–50. http://dx.doi.org/10.1016/j.jde.2021.10.034.

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16

Tarsi, Cristina. "Uniqueness of positive solutions of nonlinear elliptic equations with exponential growth." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 6 (December 2003): 1409–20. http://dx.doi.org/10.1017/s0308210500003012.

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By combining a technique inspired to the theory of sublinear elliptic equations with the Emden-Fowler inversion technique of Atkinson and Peletier, we obtain uniqueness of positive solutions of the following equation where B ⊂ Rn is the ball of radius one, λ > 0 and 1 < ϑ ≤ 2.

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17

Squassina, Marco, and Cristina Tarsi. "Multiple solutions for quasilinear elliptic problems¶in ℝ2 with exponential growth." manuscripta mathematica 106, no. 3 (November 2001): 315–37. http://dx.doi.org/10.1007/pl00005886.

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18

Chi, H., H. Poorkarimi, J. Wiener, and S. M. Shah. "On the exponential growth of solutions to non-linear hyperbolic equations." International Journal of Mathematics and Mathematical Sciences 12, no. 3 (1989): 539–45. http://dx.doi.org/10.1155/s0161171289000670.

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Existence-uniqueness theorems are proved for continuous solutions of some classes of non-linear hyperbolic equations in bounded and unbounded regions. In case of unbounded region, certain conditions ensure that the solution cannot grow to infinity faster than exponentially.

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19

Liu, Yanjun, and Chungen Liu. "Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term." Communications on Pure & Applied Analysis 19, no. 5 (2020): 2819–38. http://dx.doi.org/10.3934/cpaa.2020123.

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20

KAREV, GEORGY P. "DYNAMICS OF INHOMOGENEOUS POPULATIONS AND GLOBAL DEMOGRAPHY MODELS." Journal of Biological Systems 13, no. 01 (March 2005): 83–104. http://dx.doi.org/10.1142/s0218339005001410.

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The dynamic theory of inhomogeneous populations developed during the last decade predicts several essential new dynamic regimes applicable even to the well-known, simple population models. We show that, in an inhomogeneous population with a distributed reproduction coefficient, the entire initial distribution of the coefficient should be used to investigate real population dynamics. In the general case, neither the average rate of growth nor the variance or any finite number of moments of the initial distribution is sufficient to predict the overall population growth. We developed methods for solving the heterogeneous models and explored the dynamics of the total population size together with the reproduction coefficient distribution. We show that, typically, there exists a phase of "hyper-exponential" growth that precedes the well-known exponential phase of population growth in a free regime. The developed formalism is applied to models of global demography and the problem of "population explosion" predicted by the known hyperbolic formula of world population growth. We prove here that the hyperbolic formula presents an exact solution to the Malthus model with an exponentially distributed reproduction coefficient and that "population explosion" is a corollary of certain implicit unrealistic assumptions. Alternative models of world population growth are derived; they show a notable phenomenon, a transition from protracted hyperbolical growth (the phase of "hyper-exponential" development) to the brief transitional phase of exponential growth and, subsequently, to stabilization. The model solutions are consistent with real data and produce relatively accurate forecasts.

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21

Zhou, Shaobo. "Almost Surely Exponential Stability of Numerical Solutions for Stochastic Pantograph Equations." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/751209.

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Our effort is to develop a criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stable analysis of the exact solution. We will prove that the Euler-Maruyama (EM) method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. And the backward EM method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. A highly nonlinear example is provided to illustrate the main theory.

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22

Abou-Senna, Amr, and Boping Tian. "Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps." Mathematics 10, no. 17 (September 1, 2022): 3137. http://dx.doi.org/10.3390/math10173137.

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The stability analysis of the numerical solutions of stochastic models has gained great interest, but there is not much research about the stability of stochastic pantograph differential equations. This paper deals with the almost sure exponential stability of numerical solutions for stochastic pantograph differential equations interspersed with the Poisson jumps by using the discrete semimartingale convergence theorem. It is shown that the Euler–Maruyama method can reproduce the almost sure exponential stability under the linear growth condition. It is also shown that the backward Euler method can reproduce the almost sure exponential stability of the exact solution under the polynomial growth condition and the one-sided Lipschitz condition. Additionally, numerical examples are performed to validate our theoretical result.

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23

Ma, Weijun, Wei Liu, Quanxin Zhu, and Kaibo Shi. "Dynamics of the Exponential Population Growth System with Mixed Fractional Brownian Motion." Complexity 2021 (December 30, 2021): 1–18. http://dx.doi.org/10.1155/2021/5079147.

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This paper examines the dynamics of the exponential population growth system with mixed fractional Brownian motion. First, we establish some useful lemmas that provide powerful tools for studying the stochastic differential equations with mixed fractional Brownian motion. We offer some explicit expressions and numerical characteristics such as mathematical expectation and variance of the solutions of the exponential population growth system with mixed fractional Brownian motion. Second, we propose two sufficient and necessary conditions for the almost sure exponential stability and the k th moment exponential stability of the solution of the constant coefficient exponential population growth system with mixed fractional Brownian motion. Furthermore, we conduct some large deviation analysis of this mixed fractional population growth system. To the best of the authors’ knowledge, this is the first paper to investigate how the Hurst index affects the exponential stability and large deviations in the biological population system. It is interesting that the phenomenon of large deviations always occurs for addressed system when 1 / 2 < H < 1 . Moreover, several numerical simulations are reported to show the effectiveness of the proposed approach.

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24

Georgiades, Evripides, Michael J. S. Lowe, and Richard V. Craster. "Leaky wave characterisation using spectral methods." Journal of the Acoustical Society of America 152, no. 3 (September 2022): 1487–97. http://dx.doi.org/10.1121/10.0013897.

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Leaky waves are an important class of waves, particularly for guiding waves along structures embedded within another medium; a mismatch in wavespeeds often leads to leakage of energy from the waveguide, or interface, into the medium, which consequently attenuates the guided wave. The accurate and efficient identification of theoretical solutions for leaky waves is a key requirement for the choices of modes and frequencies required for non-destructive evaluation inspection techniques. We choose a typical situation to study: an elastic waveguide with a fluid on either side. Historically, leaky waves are identified via root-finding methods that have issues with conditioning, or numerical methods that struggle with the exponential growth of solutions at infinity. By building upon a spectral collocation method, we show how it can be adjusted to find exponentially growing solutions, i.e., leaky waves, leading to an accurate, fast, and efficient identification of their dispersion properties. The key concept required is a mapping, in the fluid region, that allows for exponential growth of the physical solution at infinity, whilst the mapped numerical setting decays. We illustrate this by studying leaky Lamb waves in an elastic waveguide immersed between two different fluids and verify this using the commercially available software disperse.

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25

Omaba, McSylvester Ejighikeme. "Moment bounds for a class of stochastic nonlinear fractional Volterra integral equations of the second kind." International Journal of ADVANCED AND APPLIED SCIENCES 9, no. 8 (August 2022): 152–57. http://dx.doi.org/10.21833/ijaas.2022.08.019.

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This paper studies and compares the second moment (Energy growth) bounds for solutions to a class of stochastic fractional Volterra integral equations of the second kind, under some Lipschitz continuity conditions on the parameters. The result shows that both solutions exhibit exponential growth but at different rates. The existence and uniqueness of the mild solutions are established via the Banach fixed point theorem.

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26

Bartłomiejczyk, Agnieszka, and Henryk Leszczyński. "Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations." Annales Polonici Mathematici 111, no. 3 (2014): 309–26. http://dx.doi.org/10.4064/ap111-3-7.

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27

Figueiredo, Giovany M., and Vicenţiu D. Rădulescu. "Positive solutions of the prescribed mean curvature equation with exponential critical growth." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 5 (March 1, 2021): 2213–33. http://dx.doi.org/10.1007/s10231-021-01077-7.

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AbstractIn this paper, we are concerned with the problem $$\begin{aligned} -\text{ div } \left( \displaystyle \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = f(u) \ \text{ in } \ \Omega , \ \ u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}$$ - div ∇ u 1 + | ∇ u | 2 = f ( u ) in Ω , u = 0 on ∂ Ω , where $$\Omega \subset {\mathbb {R}}^{2}$$ Ω ⊂ R 2 is a bounded smooth domain and $$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : R → R is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.

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28

Soave, Nicola, and Alessandro Zilio. "Entire solutions with exponential growth for an elliptic system modelling phase separation." Nonlinearity 27, no. 2 (January 17, 2014): 305–42. http://dx.doi.org/10.1088/0951-7715/27/2/305.

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29

Rebiai, Belgacem, and Saïd Benachour. "Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth." Journal of Evolution Equations 10, no. 3 (March 5, 2010): 511–27. http://dx.doi.org/10.1007/s00028-010-0059-x.

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30

Figueiredo, Giovany M., and Vicenţiu D. Rădulescu. "Nonhomogeneous equations with critical exponential growth and lack of compactness." Opuscula Mathematica 40, no. 1 (2020): 71–92. http://dx.doi.org/10.7494/opmath.2020.40.1.71.

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We study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth.

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31

Han, Haegyeong, and Hwajoon Kim. "The Solution of Exponential Growth and Exponential Decay by Using Laplace Transform." International Journal of Difference Equations 15, no. 2 (December 30, 2020): 191–95. http://dx.doi.org/10.37622/ijde/15.2.2020.191-195.

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32

Shvets, Vadim, and Boris Zeide. "Investigating parameters of growth equations." Canadian Journal of Forest Research 26, no. 11 (November 1, 1996): 1980–90. http://dx.doi.org/10.1139/x26-224.

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Two differential forms of growth equations, called the power decline, or PD form, and the exponential decline, or ED form, generate classic growth equations (such as the logistic, Chapman–Richards, Korf) and many other integral forms. Having a full range of these integral solutions allows us to classify them, establish requirements to their parameters, and relate these parameters and initial values (starting age and tree size). Comparisons with data confirm theoretical results. Some applications of the results are discussed.

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33

Deng, Shengbing, and Junwei Yu. "On a class of singular Hamiltonian Choquard-type elliptic systems with critical exponential growth." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 121501. http://dx.doi.org/10.1063/5.0110352.

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In this paper, using the Moser functions and linking theorem, we study the existence of solutions for a class of Hamiltonian Choquard-type elliptic systems in the plane with exponential growth involving singular weights.

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34

sci, Chong Wang. "Existence of Nontrivial Weak Solutions to Quasi-linear Elliptic Equations with Exponential Growth." Journal of Partial Differential Equations 26, no. 1 (June 2013): 25–38. http://dx.doi.org/10.4208/jpde.v26.n1.3.

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35

Lam, Nguyen, and Guozhen Lu. "Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth." Discrete & Continuous Dynamical Systems - A 32, no. 6 (2012): 2187–205. http://dx.doi.org/10.3934/dcds.2012.32.2187.

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36

Liao, Fangfang, and Xiaoping Wang. "Ground state solutions for Schrödinger–Poisson system with critical exponential growth in R2." Applied Mathematics Letters 120 (October 2021): 107340. http://dx.doi.org/10.1016/j.aml.2021.107340.

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37

Zennir, Khaled. "Exponential growth of solutions with Lp -norm of a nonlinear viscoelastic hyperbolic equation." Journal of Nonlinear Sciences and Applications 06, no. 04 (November 10, 2013): 252–62. http://dx.doi.org/10.22436/jnsa.006.04.03.

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38

de Freitas, Luciana R. "Multiplicity of solutions for a class of quasilinear equations with exponential critical growth." Nonlinear Analysis: Theory, Methods & Applications 95 (January 2014): 607–24. http://dx.doi.org/10.1016/j.na.2013.10.010.

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39

Li, Qin, and Zuodong Yang. "Multiple solutions for N-Kirchhoff type problems with critical exponential growth in RN." Nonlinear Analysis: Theory, Methods & Applications 117 (April 2015): 159–68. http://dx.doi.org/10.1016/j.na.2015.01.005.

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40

Song, Hongxue, Caisheng Chen, and Qinglun Yan. "Infinitely many solutions for quasilinear Schrödinger equation with critical exponential growth in RN." Journal of Mathematical Analysis and Applications 439, no. 2 (July 2016): 575–93. http://dx.doi.org/10.1016/j.jmaa.2016.03.002.

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41

Sugimura, Kunihiko. "Existence of infinitely many solutions for a perturbed elliptic equation with exponential growth." Nonlinear Analysis: Theory, Methods & Applications 22, no. 3 (February 1994): 277–93. http://dx.doi.org/10.1016/0362-546x(94)90020-5.

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42

Chen, Caisheng, and Hongxue Song. "Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in ℝ N." Applications of Mathematics 61, no. 3 (May 18, 2016): 317–37. http://dx.doi.org/10.1007/s10492-016-0134-x.

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43

Li, Shuoshuo, Zifei Shen, and Minbo Yang. "Multiplicity of solutions for a nonlocal nonhomogeneous elliptic equation with critical exponential growth." Journal of Mathematical Analysis and Applications 475, no. 2 (July 2019): 1685–713. http://dx.doi.org/10.1016/j.jmaa.2019.03.039.

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44

Chen, Sitong, and Xianhua Tang. "Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth." Journal of Differential Equations 269, no. 11 (November 2020): 9144–74. http://dx.doi.org/10.1016/j.jde.2020.06.043.

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45

Pucci, Patrizia, and Letizia Temperini. "(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group." Open Mathematics 18, no. 1 (November 27, 2020): 1423–39. http://dx.doi.org/10.1515/math-2020-0108.

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Abstract The paper deals with the existence of solutions for (p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in {{\mathbb{H}}}^{n} .

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46

Hall, A. J., and G. C. Wake. "Functional differential equations determining steady size distributions for populations of cells growing exponentially." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 31, no. 4 (April 1990): 434–53. http://dx.doi.org/10.1017/s0334270000006779.

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AbstractA population of cells growing and dividing often goes through a phase of exponential growth of numbers, during which the size distribution remains steady. In this paper we study the function differential equation governing this steady size distribution in the particular case where the individual cells themselves are growing exponentially in size. A series solution is obtained for the case where the probability of cell division is proportional to any positive power of the cell size, and a method for finding closed-form solutions for a more general class of cell division functions is developed.

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47

Abdel-Rehim, E. A. "The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process." Electronic Research Archive 30, no. 7 (2022): 2487–509. http://dx.doi.org/10.3934/era.2022127.

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<abstract><p>The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The approximate solutions corresponding to different values of the birth and death rates are also compared.</p></abstract>

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48

Fang, Fei, and Chao Ji. "The cone Moser–Trudinger inequalities and applications." Asymptotic Analysis 120, no. 3-4 (October 30, 2020): 273–99. http://dx.doi.org/10.3233/asy-191588.

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In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in x ∈ int ( B ) , u = 0 , on ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0, and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

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HUUSKO, JUHA-MATTI. "LOCALISATION OF LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC BY A CONFORMAL MAP." Bulletin of the Australian Mathematical Society 93, no. 2 (October 15, 2015): 260–71. http://dx.doi.org/10.1017/s0004972715001070.

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We obtain lower bounds for the growth of solutions of higher order linear differential equations, with coefficients analytic in the unit disc of the complex plane, by localising the equations via conformal maps and applying known results for the unit disc. As an example, we study equations in which the coefficients have a certain explicit exponential growth at one point on the boundary of the unit disc and consider the iterated $M$-order of solutions.

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50

Chaves, M., J. L. Vazquez, and M. Walias. "Optimal existence and uniqueness in a nonlinear diffusion–absorption equation with critical exponents." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 2 (1997): 217–42. http://dx.doi.org/10.1017/s0308210500023623.

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We study the existence and uniqueness of non-negative solutions of the nonlinear parabolic equationposed in Q = RN × (0, ∞) with general initial data u(x, 0) = u0(x) ≧ 0. We find optimal exponential growth conditions for existence of solutions. Similar conditions apply for uniqueness, but the growth rate is different. Such conditions strongly depart from the linear case m = 1, ut = Δu – u, and also from the purely diffusive case ut = Δum.

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