Sharp profiles for periodic logistic equation with nonlocal dispersal

In this paper, we study the nonlocal dispersal logistic equation ut=J∗u-u+λu-[b(x)q(t)+δ]upinΩ¯×(0,∞),u(x,t)=0inRN\Ω¯×(0,∞),u(x,t)=u(x,t+T)inΩ¯×[0,∞),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=J*u-u+\lambda u-[b(x)q(t)+\delta ]u^p &{}\text {in}\,\bar{\Omega }\times (0,\infty ),\\ u(x,t)=0 &{}\text {in}\,{\mathbb {R}^N\setminus \bar{\Omega }}\times (0,\infty ),\\ u(x,t)=u(x,t+T) &{}\text {in}\,\bar{\Omega }\times [0,\infty ), \end{array}\right. } \end{aligned}$$\end{document}here Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^N$$\end{document} is a bounded domain, J is a nonnegative dispersal kernel, p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}, λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is a fixed parameter and δ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document}. The coefficients b, q are nonnegative and continuous functions, and q is periodic in t. We are concerned with the asymptotic profiles of positive solutions as δ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \rightarrow 0$$\end{document}. We obtain that the temporal degeneracy of q does not make a change of profiles, but the spatial degeneracy of b makes a large change. We find that the sharp profiles are different from the classical reaction–diffusion equations. The investigation in this paper shows that the periodic profile has two different blow-up speeds and the sharp profile is time periodic in domain without spatial degeneracy.