# Positive solutions for nonlocal dispersal equation with spatial degeneracy

Positive solutions for nonlocal dispersal equation with spatial degeneracy In this paper, we consider the positive solutions of the nonlocal dispersal equation \begin{aligned} \int \limits _{\Omega }J(x,y)[u(y)-u(x)]\mathrm{d}y=-\lambda m(x)u(x)+[c(x)+\varepsilon ]u^p(x) \quad \text { in }\bar{\Omega }, \end{aligned} ∫ Ω J ( x , y ) [ u ( y ) - u ( x ) ] d y = - λ m ( x ) u ( x ) + [ c ( x ) + ε ] u p ( x ) in Ω ¯ , where $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N is a bounded domain, $$\lambda ,\varepsilon$$ λ , ε and $$p>1$$ p > 1 are positive constants. The dispersal kernel J and the coefficient c(x) are nonnegative, but c(x) has a degeneracy in some subdomain of $$\Omega$$ Ω . In order to study the influence of heterogeneous environment on the nonlocal system, we study the sharp spatial patterns of positive solutions as $$\varepsilon \rightarrow 0$$ ε → 0 . We obtain that the positive solutions always have blow-up asymptotic profiles in $$\bar{\Omega }$$ Ω ¯ . Meanwhile, we find that the profiles in degeneracy domain are different from the domain without degeneracy. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Zeitschrift für angewandte Mathematik und Physik Springer Journals

# Positive solutions for nonlocal dispersal equation with spatial degeneracy

, Volume 69 (1) – Dec 18, 2017
9 pages

/lp/springer_journal/positive-solutions-for-nonlocal-dispersal-equation-with-spatial-KczGejplMS
Publisher
Springer International Publishing
Copyright © 2017 by Springer International Publishing AG, part of Springer Nature
Subject
Engineering; Theoretical and Applied Mechanics; Mathematical Methods in Physics
ISSN
0044-2275
eISSN
1420-9039
D.O.I.
10.1007/s00033-017-0903-8
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we consider the positive solutions of the nonlocal dispersal equation \begin{aligned} \int \limits _{\Omega }J(x,y)[u(y)-u(x)]\mathrm{d}y=-\lambda m(x)u(x)+[c(x)+\varepsilon ]u^p(x) \quad \text { in }\bar{\Omega }, \end{aligned} ∫ Ω J ( x , y ) [ u ( y ) - u ( x ) ] d y = - λ m ( x ) u ( x ) + [ c ( x ) + ε ] u p ( x ) in Ω ¯ , where $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N is a bounded domain, $$\lambda ,\varepsilon$$ λ , ε and $$p>1$$ p > 1 are positive constants. The dispersal kernel J and the coefficient c(x) are nonnegative, but c(x) has a degeneracy in some subdomain of $$\Omega$$ Ω . In order to study the influence of heterogeneous environment on the nonlocal system, we study the sharp spatial patterns of positive solutions as $$\varepsilon \rightarrow 0$$ ε → 0 . We obtain that the positive solutions always have blow-up asymptotic profiles in $$\bar{\Omega }$$ Ω ¯ . Meanwhile, we find that the profiles in degeneracy domain are different from the domain without degeneracy.

### Journal

Zeitschrift für angewandte Mathematik und PhysikSpringer Journals

Published: Dec 18, 2017

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