This paper is concerned with the periodic-parabolic equation \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+\lambda u-[a(x)s(t)u^p+\varepsilon b(t)u^q]& \,\text { in }\,\Omega \times [0,T],\\ u(x,t)=0& \,\text { on }\,\partial \Omega \times [0,T],\\ u(x,0)=u(x,T)& \,\text { in }\,\Omega , \end{array}\right. } \end{aligned}\) where the parameter \(\varepsilon >0\) , \(\Omega \subset {\mathbb {R}}^N(N\ge 2)\) is a bounded domain with smooth boundary \(\partial \Omega \) , \(p>1\) , \(q>1\) , \(a\in C^\alpha (\bar{\Omega })\) \((0<\alpha <1)\) is nonnegative, the coefficients b(t) and s(t) are nonnegative and T-periodic in t. By analyzing the asymptotic behavior as \(\varepsilon \rightarrow 0\) , we show that the positive solution blows up in the region where a(x)s(t) degenerates. In sharp contrast to the result of Li et al. (Calc Var Partial Differ Equ 60:36, 2021), we prove that the positive solution always tends to zero as \(\varepsilon \rightarrow \infty \) . Our study reveals that the pattern of the positive solution undergoes a fundamental change between small and large values of \(\varepsilon \) .
