Title: On the binary-ternary Boltzmann equation
Abstract: The Boltzmann equation provides a statistical description of a dilute gas with predominantly binary particle interactions. When gas is dense enough, higher order interactions are more likely to happen. An equation that captures ternary particle interactions, as well as the binary ones, is called the binary-ternary Boltzmann equation. In this talk we present moment estimates and global well-posedness results for this equation. Moments estimates, in particular, show that the binary-ternary Boltzmann equation behaves better than the classical Boltzmann equation in the sense that it can generate exponential moments of higher order. The talk is based on a joint work with Ampatzoglou, Gamba and Pavlovic.

Title: An exact bifurcation diagram for a p-q Laplacian boundary value problem
Abstract: We study positive solutions to the $p$-$q$ Laplacian two-point boundary value problem: \begin{align*} \begin{cases} -\mu[(u’)^{p-1}]’ – [(u’)^{q-1}]’ = \lambda u(1-u) ; \hspace{.10in} (0,1) \\ u(0) = 0 = u(1) \end{cases} \end{align*} when $p = 4$ and $q=2$. Here $\lambda>0$ is a parameter and $\mu \geq 0$ is a weight parameter influencing the higher-order diffusion term. When $\mu = 0$ (the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, when $\lambda \leq \pi^2$ there are no positive solutions, and for $\lambda > \pi^2$ there exists a unique positive solution $u_{\lambda,\mu}$ such that $||u_{\lambda,\mu}||_{\infty} \rightarrow 0$ as $\lambda \rightarrow \pi^2$ and $||u_{\lambda,\mu}||_{\infty} \rightarrow 1$ as $\lambda \rightarrow \infty$. Here, we will prove that for all $\mu > 0$ similar bifurcation diagrams preserve, and they all bifurcate from $(\lambda,u) = (\pi^2,0)$. Our results are established via the method of sub-super solutions and a quadrature method. We also present computational evaluations of these bifurcation diagrams for various values of $\mu$ and illustrate how they evolve when $\mu$ varies.

Title: Σ-shaped bifurcation curves for classes of reaction diffusion equations with non-linear boundary conditions
Abstract: \noindent We analyse positive solutions of the steady state reaction diffusion equation of the form: \begin{equation*} \left\lbrace \begin{matrix}-\Delta u=\lambda f(u) ;~\Omega \\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda}g(u) u=0; ~ \partial\Omega \end{matrix} \right. \end{equation*} \noindent where $\lambda>0$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$; $N > 1$ with smooth boundary $\partial \Omega$ or $\Omega=(0,1)$, $\frac{\partial u}{\partial \eta}$ is the outward normal derivative of $u$, $f \in C^1([0,\infty),[0, \infty)) $ is an increasing function such that $f(0)=0$, $f'(0) > 0,$ and $ \lim\limits_{s \rightarrow \infty}\frac{f(s)}{s}=0$ (sublinear at infinity). Further, we assume that $g \in C([0,\infty), (0, \infty))$ is a non-increasing function such that $\lim\limits_{s \rightarrow \infty} g(s) = g_\infty > 0$. We discuss the existence of multiple positive solutions for certain ranges of $\lambda$ leading to the occurrence of $\Sigma$-shaped bifurcation diagrams. We establish our results via the method of sub-supersolutions.