We study a mathematical model of two species competing in a chemostat for two internally stored essential nutrients, where the nutrients are added to the culture vessel by way of periodic forcing functions. Persistence of a single species happens if the nutrient supply is sufficient to allow it to acquire a threshold of average stored nutrient quota required for growth to balance dilution. More precisely, the population is washed out if a sub-threshold criterion holds, while there is a globally stable positive periodic solution, if a super-threshold criterion holds. When there is mutual invasibility of both semitrivial periodic solutions of the two-species model, both uniform persistence and the existence of periodic coexistence state are established.

We study the classification and evolution of bifurcation curves of positive solutions $u\in C^{2}(-L,L)\cap C[-L,L]$ for the one-dimensional prescribed mean curvature problem \begin{equation*} \left\{ \begin{array}{l} - \Big ( \dfrac{u^{\prime }(x)}{\sqrt{1+ ( {u^{\prime }(x)} ) ^{2}}} \Big ) ^{\prime }=\lambda [ \exp (\frac{au}{a+u})-1 ] , \ \ -L < x < L,\\ u(-L)=u(L)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is a bifurcation parameter, and $L,a>0$ are two evolution parameters. We prove that, on $(\lambda , \Vert u \Vert _{\infty })$ -plane, the bifurcation curve of this problem is (i) $\supset $-shaped when $ 0 < a < 2$; (ii) $\supset $-shaped or monotone decreasing when $a=2$; (iii) reversed $S$-like shaped or monotone decreasing when $a>2$. Moreover, for $ a>2,$ the bifurcation curve is reversed $S$-like shaped if $L>0$ is large enough and is monotone decreasing if $L>0$ is small enough.

By means of variational methods we prove the existence of a positive, homoclinic solution to an equation of the kind $u''=au-bu^p$, where $p>1$, and both coefficients $a(x)$, $b(x)$ are positive and asymptotically constant. Our main result requires a control from above on the ratios $M/\alpha$ and $\beta/\nu$, where $M=\sup a$, $\alpha=a(\infty)$, $\nu=\inf b$, $\beta=b(\infty)$.

We study asymptotic behavior of (all) positive solutions of the non\-oscillatory half-linear differential equation of the form $$ (r(t)|y'|^ {\alpha-1}\text{sgn}\, y')'=p(t)|y|^{\alpha-1}\text{sgn}\, y , $$ where $\alpha\in(1,\infty)$ and $r,p$ are positive continuous functions on $[a,\infty)$, with the help of the Karamata theory of regularly varying functions and the de Haan theory. We show that increasing resp. decreasing solutions belong to the de Haan class $\Gamma$ resp. $\Gamma_-$ under suitable assumptions. Further we study behavior of slowly varying solutions for which asymptotic formulas are established. Some of our results are new even in the linear case $\alpha=2$.

In this paper, we provide an example of a class of two reaction-diffusion-ODE equations with homogeneous Neumann boundary conditions, in which Turing-type instability not only destabilizes constant steady states but also induces blow-up of nonnegative spatially heterogeneous solutions. Solutions of this problem preserve nonnegativity and uniform boundedness of the total mass. Moreover, for the corresponding system with two non-zero diffusion coefficients, all nonnegative solutions are global in time. We prove that a removal of diffusion in one of the equations leads to a finite-time blow-up of some nonnegative spatially heterogeneous solutions.

In this paper, we consider a semi-linear, energy-critical, shifted wave equation on the hyperbolic space ${\mathbb H}^n$ with $3 \leq n \leq 5$: \[ \partial_t^2 u - (\Delta_{{\mathbb H}^n} + \rho^2) u = \zeta |u|^{4/(n-2)} u, \quad (x,t)\in {\mathbb H}^n \times {\mathbb R}. \] Here, $\zeta = \pm 1$ and $\rho = (n-1)/2$ are constants. We introduce a family of Strichartz estimates compatible with initial data in the energy space $H^{0,1} \times L^2 ({\mathbb H}^n)$ and then establish a local theory with these initial data. In addition, we prove a Morawetz-type inequality \[ \int_{-T_-}^{T_+} \int_{{\mathbb H}^n} \frac{\rho (\cosh |x|) |u(x,t)|^{2n/(n-2)}} {\sinh |x|} d\mu(x) dt \leq n {\mathcal E}, \] in the defocusing case $\zeta = -1$, where ${\mathcal E}$ is the energy. Moreover, if the initial data are also radial, we can prove the scattering of the corresponding solutions by combining the Morawetz-type inequality, the local theory and a pointwise estimate on radial $H^{0,1}({\mathbb H}^n)$ functions.

We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1,$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$, we obtain that the set of positive solutions is locally path connected. For a periodic potential $V$ this implies that the standard variational functional has discrete critical values in the low energy range and that a compact isolated set of positive solutions exists, under additional assumptions.

An initial-boundary value problem for the 3D Zakharov-Kuznetsov equation posed on bounded domains is considered. Existence and uniqueness of a global regular solution as well as exponential decay of the $H^2$-norm for small initial data are proven.

We will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will give an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.