In this paper, we establish and analyze a mosquito population suppression model with seasonal switching. Under the assumption that the ratio of sexually active Wolbachia-infected males and wild mosquitoes is kept at a constant level during the favorable seasons, we give a rather complete description on the dynamics including the stability of the origin, existence, stability and semi-stability of a unique, exact two periodic solutions, and so on. We obtain sufficient conditions for the origin to be globally asymptotically stable, for the model to have a unique semi-stable periodic solution, and to have exactly two periodic solutions among which one is stable and the other is unstable, respectively. Numerical examples to support our theoretical results and brief discussions are also provided.

The chemotaxis-Navier-Stokes system $$ \begin{cases} n_t+u\cdot\nabla n = \Delta \big(n\phi(c)\big),\\ c_t+ u\cdot\nabla c = \Delta c -c+n,\\ u_t + (u\cdot\nabla) u =\Delta u+\nabla P + n\nabla\Phi, \qquad \nabla\cdot u=0, \end{cases} $$ is considered along with homogeneous boundary conditions of no-flux type for $n$ and $c$, and of Dirichlet type for $u$, in a smoothly bounded planar domain $\Omega$, where $\Phi \in W^{2,\infty}(\Omega)$ is given. A concept of generalized solvability is introduced, and in this framework a statement on global existence is derived for all reasonably regular initial data $(n_0,c_0,u_0)$, provided that $\phi\in C^3((0,\infty))$ is positive and bounded on $(\xi_0,\infty)$ for each $\xi_0>0$, and such that $$ \xi\phi(\xi) \to +\infty \qquad \mbox{and} \qquad \frac{\xi\phi'^2(\xi)}{\phi(\xi)} \to 0 \qquad \mbox{as } \xi\to\infty. $$ These solutions particularly satisfy $\int_0^T \Omega n\ln (n+1) < \infty$ for all $T > 0$ as well as $\int_\Omega n(\cdot,t)=\int_\Omega n_0$ for a.e. $t > 0$.

In this paper, we consider a general model of an abstract evolution equation with history memory and time delay with a nonlinear source term in the Hilbert space. First, we consider a linear evolution system with history memory and time delay. Under appropriate assumptions, we prove the well-posedness by using semigroup arguments and the stability results of the system by defining a suitable Lyapunov functional under a suitable condition on the time delay. We obtain the unique dissipation, given by history memory, is strong enough to exponentially stabilize the system. Second, we consider a semilinear evolution equation where the nonlinear source term and the nonlinear dissipative term verify some Lipschitz continuity conditions. In this later, we establish the well-posedness and the exponential decay results by a direct proof using Duhamel's formula and a perturbation method. In both cases, some examples are given to illustrate our abstract results. Our systems generalize the earlier problems in the literature.

We investigate multi-bump positive and nodal solutions to the logarithmic scalar field equation $$ -\Delta u + V(y) u = u\log |u|,\quad u\in H^1(\mathbb R^N), $$ where $N\geq2$ and the potential $V$ is bounded radially symmetric, which is a class of important Schrödinger equations in mathematical physics. The main difficulties to apply Lyapunov-Schmidt reduction to logarithmic scalar equations are caused by the non-smooth property and sub-linear growth of the logarithmic nonlinearity. To overcome these difficulties, we develop a new approach to carry out the Lyapunov-Schmidt reduction, which can be used to construct not only positive but also nodal solutions to the logarithmic equations. Finally, both infinitely many positive and infinitely many nodal solutions with an arbitrarily large number of bumps are constructed for scalar field equations when the potential satisfies different proper decay conditions.