Positive profiles for the perturbed nonlocal dispersal equations

In this paper, we study the following perturbed nonlocal dispersal equation \(\begin{aligned} \int _\Omega J(x-y)u(y)\,\mathrm{{d}}y-u(x)+\lambda u(x)+\varepsilon b(x)u(x)-a(x)u^p(x)=0, \quad x\in {\bar{\Omega }}, \end{aligned}\) $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}J(x-y)u(y)\mathrm{d}y-u(x)+\lambda u(x)+\epsilon b(x)u(x)-a(x)u^p(x)=0,x\in \overline{\mathrm{\Omega }},\end{array}$$ where \(\Omega \subset {\mathbb {R}}^{N}\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, \(p>1\) $p>1$ , \(\lambda \) $\lambda$ and \(\varepsilon >0\) $\epsilon >0$ are parameters, the coefficient \(a(\cdot )\) $a(·)$ and the kernel function \(J(\cdot )\) $J(·)$ are nonnegative, while b(x) can be indefinite. We are interested in the asymptotic profiles and limiting behavior of patterns for the positive solutions. It is shown that the positive solution converges to the unique positive solution of nonlocal dispersal logistic equation as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ . However, we obtain that the new pattern appears when \(\varepsilon \) $\epsilon$ is large. Among others, we find that the positive solutions exhibit quenching or blow-up profiles. Our study reveals how the existence of new profiles of patterns is determined by the behavior of b(x).