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DCDS

In this article, we investigate the periodic-parabolic logistic
equation on the entire space $\mathbb{R}^N\ (N\geq1)$:
$$
\begin{equation}
\left\{\begin{array}{ll}
\partial_t u-\Delta u=a(x,t)u-b(x,t)u^p\ \ \ \
& {\rm in}\ \mathbb{R}^N\times(0,T),\\
u(x,0)=u(x,T) \ & {\rm in}\ \mathbb{R}^N,
\end{array}
\right.
\end{equation}
$$
where the constants $T>0$ and $p>1$, and the functions $a,\ b$ with
$b>0$ are smooth in $\mathbb{R}^N\times\mathbb{R}$ and $T$-periodic in time. Under
the assumptions that $a(x,t)/{|x|^\gamma}$ and $b(x,t)/{|x|^\tau}$
are bounded away from $0$ and infinity for all large $|x|$, where
the constants $\gamma>-2$ and $\tau\in\mathbb{R}$, we study the existence
and uniqueness of positive $T$-periodic solutions. In particular, we
determine the asymptotic behavior of the unique positive
$T$-periodic solution as $|x|\to\infty$, which turns out to depend
on the sign of $\gamma$. Our investigation considerably generalizes
the existing results.

DCDS

This paper concerns a diffusive logistic equation with a free
boundary and seasonal succession, which is formulated to investigate
the spreading of a new or invasive species, where the free boundary
represents the expanding front and the time periodicity accounts for
the effect of the bad and good seasons. The condition to determine
whether the species spatially spreads to infinity or vanishes at a
finite space interval is derived, and when the spreading happens,
the asymptotic spreading speed of the species is also given. The
obtained results reveal the effect of seasonal succession on the
dynamical behavior of the spreading of the single species.

keywords:
free boundary
,
vanishing.
,
spreading
,
seasonal
succession
,
Diffusive logistic equation

DCDS-B

This paper continues the analysis on a bimolecular autocatalytic
reaction-diffusion model with saturation law. An improved result of
steady state bifurcation is derived and the effect of various
parameters on spatiotemporal patterns is discussed. Our analysis
provides a better understanding on the rich spatiotemporal patterns.
Some numerical simulations are performed to support the theoretical
conclusions.

DCDS

This paper is concerned with the stationary Gierer-Meinhardt system with singularity:

$\left\{\begin{array}{ll} d_1\Delta u-a_1 u+\frac{u^p}{v^q}+\rho_1(x)=0, \ \ & x\in\Omega, \\ d_2\Delta v-a_2 v+\frac{u^r}{v^s}+\rho_2(x)=0,\ \ & x\in\Omega,\\ u(x)>0,\ \ v(x)>0,\ \ & x\in \Omega,\\ \displaystyle u(x)=v(x)=0,\ \ & x\in\partial\Omega, \end{array}\right.$ |

where $-\infty < p < 1$, $-1 < s$, and $q, r, d_1, d_2$ are positive constants, $a_1, \, a_2$ are nonnegative constants, $\rho_1, \, \rho_2$ are smooth nonnegative functions and $\Omega\subset \mathbb{R}^d\, (d\geq1)$ is a bounded smooth domain. New sufficient conditions, some of which are necessary, on the existence of classical solutions are established. A uniqueness result of solutions in any space dimension is also derived. Previous results are substantially improved; moreover, a much simpler mathematical approach with potential application in other problems is developed.

keywords:
Gierer-Meinhardt system
,
singularity
,
stationary solution
,
Dirichlet boundary
,
existence
,
uniqueness

## Year of publication

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