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Stability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process

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Stochastic Analysis with Financial Applications

Part of the book series: Progress in Probability ((PRPR,volume 65))

Abstract

Consider the nonlinear equation

$$\frac{\partial}{\partial t}u(x,t)=\Delta_\alpha u (x, t) + a(x) \sum\limits_{k=2}^{\infty} pk^{{u^k}} (x, t)+(p0 + p1 u(x, t))\phi(x), x\in \mathbb{R}^d,$$

where α ∈ (0, 2], u (x, 0) is nonnegative, {pk, k = 0, 1,...} is a probability distribution on ℤ+, and a and φ are positive functions satisfying certain growth conditions. We prove existence of non-trivial positive global solutions when p0, p1 and u(x, 0) are small.

Mathematics Subject Classification (2000).

Primary: 35K58, 35B44. Secondary:35B09, 35B51.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas Pan American, Edinburg, Texas, USA

    S. Chakraborty

  2. Department of Probability and Statistics, Centro de Investigación en Matemáticas, Guanajuato, Mexico

    E. T. Kolkovska & J. A. López-Mimbela

Authors
  1. S. Chakraborty
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  2. E. T. Kolkovska
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  3. J. A. López-Mimbela
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Corresponding author

Correspondence to S. Chakraborty .

Editor information

Editors and Affiliations

  1. Japan Science and Technology Agency, Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan

    Arturo Kohatsu-Higa

  2. Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-05-43, 21 Nanyang Link, Singapore, 637371, Singapore

    Nicolas Privault

  3. Department of Mathematics, National Central University, No.300, Zhongda Rd., Zhongli City, 320, Taiwan R.O.C.

    Shuenn-Jyi Sheu

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Chakraborty, S., Kolkovska, E.T., López-Mimbela, J.A. (2011). Stability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process. In: Kohatsu-Higa, A., Privault, N., Sheu, SJ. (eds) Stochastic Analysis with Financial Applications. Progress in Probability, vol 65. Springer, Basel. https://doi.org/10.1007/978-3-0348-0097-6_2

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