In this paper we survey the construction of Lyapunov functions (or functionals) for various ecological models which take the form of ODE systems (or Reaction-Diffusion PDE systems). First we consider the resources-consumers type ecological models which study the competition of $n$ microorgansims for a single limiting resource or two complementary resources in the chemostat. Next we consider the Gause-type predator-prey systems and the Lesile-type predator-prey systems. From the Lyapunov functions of the predator-prey system we construct new Lyapunov functions for three-level food chain models and one prey two predators models. Suppose a Lyapunov function is known for an ecological model which takes the form of an ODE system. Then we construct a Lyapunov functional for the corresponding reaction-diffusion PDE systems. Open problems are indicated when there is gap in the theory.

We obtain optimal lower estimates for the eigenvalue ratios $(\frac{\lambda_{m}}{\lambda_{n}})$ of Dirichlet and Neumann Schrödinger operators with nonpositive potentials and Dirichlet vibrating string problems with concave and positive densities. Our results supplement those of Ashbaugh-Benguria [2] and M. J. Huang [5].

We study the global and non-global existence of positive solutions of a nonlinear parabolic equation. For this, we consider the forward and backward self-similar solutions of this equation. We obtain a family of radial symmetric global solutions which tend to zero as the time tends infinity. Next, we show that there are initial data for which the corresponding solutions blow up in finite time. Finally, we also construct some self-similar single-point blow-up patterns with different oscillations.

We study the shape and the structure of the bifurcation curve $f_{a}(\rho)$ ($= \sqrt{\lambda}$) with $\rho: = \min_{x \in (0,1)} u(x)$ of (sign-changing and nonnegative) solutions of the boundary blow-up problem \[ \left\{ \begin{array}{l} -u''(x) = \lambda f(u(x)),\; 0 \lt x \lt 1, \\ \lim\limits_{x \to 0^+} u(x) = \infty = \lim\limits_{x \to x^-} u(x), \end{array} \right. \] where $\lambda$ is a positive bifurcation parameter and the Lipschitz continuous concave function \[ f = f_{a}(u) = \begin{cases} -|u|^p & \textrm{if } u \leq -a^{1/p}, \\ -a & \textrm{if } -a^{1/p} \lt u \lt a^{1/p}, \\ -|u|^p & \textrm{if } u \geq a^{1/p}, \end{cases} \] with constants $p \gt 1$ and $a \gt 0$. We mainly show that the bifurcation curve $G_{f_a}(\rho)$ satisfies $\lim_{\rho \rightarrow \pm \infty} G_{f_a}(\rho) = 0$ and $G_{f_a}(\rho)$ has a exactly one critical point, a maximum, on $(-\infty,\infty )$. Thus we are able to determine the exact number of (sign-changing and nonnegative) solutions of the problem for each $\lambda \gt 0$.

We discuss the homogenization process of second order differential equations involving highly oscillating coefficients in the time and space variables. It generate memory or nonlocal effect. For initial value problems, the memory kernels are described by Volterra integral equations; and for boundary value problems, they are characterized by Fredholm integral equations. When the equation is translation (in time or in space) invariant, the memory or nonlocal kernel can be represented explicitly in terms of the Young's measure.

The convergence of approximations to solutions of nonlinear elliptic equations is closely related to the structure of the equations. As examples, we examine certain quasilinear elliptic equations with quadratic growth in the gradient defined on bounded domains. $L^{\infty}$ and $H^1$ estimates on approximating solutions are performed to deduce the convergence to a solution in $H^1_0(\Omega) \cap L^{\infty}(\Omega)$. In some cases, $H^1$ a priori bound can be derived without referring to $L^{\infty}$ estimate. Furthermore, a $W^{2,p}(\mathbf{\Omega})$ bound is also established to deduce the existence of strong solutions in $W^{2,p}(\mathbf{\Omega}) \cap W^{1,p}_0(\mathbf{\Omega})$.

In this paper, we study a Palais-Smale condition in unbounded domains. Furthermore, we apply this result to prove that the semilinear elliptic equation in a Esteban-Lions domain with holes has multiple positive solutions.

In this paper we study the behavior of subRiemannian geodesics on a certain step $4$ subRiemannian manifold. We compute the length of the subRiemannian geodesics between the origin and any point on the $t-$axis, where the conjugate locus is. We characterize the number of subRiemannian geodesics between the origin and any other point.

Let $a \ge 0$. We give sufficient conditions on $\{c_{jk}\}$ to obtain Parseval's formula for the corresponding double Laguerre series $\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} c_{jk} \mathcal{L}^a_j(x) \mathcal{L}^a_k(y)$.

For a bounded linear injection $C$ on a Banach space $X$ and a (not necessarily densely defined) closed linear operator $A$ which commutes with $C$, we present various conditions for $A$ to generate a local $C$-semigroup. A Hille-Yosida type generation theorem in terms of the asymptotic $C$-resolvent of $A$ is proved, and various characterizations of a generator by means of existence of unique strong solutions of the associated abstract Cauchy problems are obtained.

We consider random fractals generated by random recursive constructions with overlaps. Our construction allows some overlaps among sets in the same generation. We introduce a certain ``limited overlaps condition''. Under this condition, we prove that the Hausdorff dimension of the generated fractal satisfies the expectation equation (upon non-extinction), which was studied previously by Falconer, Graf, Mauldin and Williams under open set condition. We also prove that the generated fractal is regular in the sense that its Hausdorff and upper box dimension are equal to a non-random constant (this result holds without assumption of limited overlaps condition).