2011 | OriginalPaper | Chapter
Stability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process
Authors : S. Chakraborty, E. T. Kolkovska, J. A. López-Mimbela
Published in: Stochastic Analysis with Financial Applications
Publisher: Springer Basel
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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.