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[Consider the nonlinear equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{\partial}{\partial t}u(x,t)=\Delta_\alpha u (x, t) + a(x) \sum\limits_{k=2}^{\infty} pk^{{u^k}} (x, t)+(p0 + p1 u(x, t))\phi(x), x\in \mathbb{R}^d,$$ \end{document} 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.]
Published: Jul 12, 2011
Keywords: Semi-linear partial differential equation; positive solutions; global solutions
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