We consider the following shadow system of the Gierer-Meinhardt model: \[ \left\{\begin{array}{l} A_t= \epsilon^2 A_{xx} -A +\frac{A^p}{\xi^q},\, 0 <x <1,\, t>0,\\ \tau \xi_t= -\xi + \xi^{-s} \int_0^1 A^2 \,dx,\\ A>0,\, A_x (0,t)= A_x(1, t)=0, \end{array} \right. \] where $1 <p <+\infty,\, \frac{2q}{p-1} >s+1,\, s\geq 0$, and $\tau >0$. It is known that a nontrivial monotone steady-state solution exists if and only if $ \epsilon < \frac{\sqrt{p-1}}{\pi}.$ In this paper, we show that for any $\epsilon < \frac{\sqrt{p-1}}{\pi}$, and $p=2$ or $p=3$, there exists a unique $\tau_c>0$ such that for $\tau <\tau_c$ this steady state is linearly stable, while for $\tau>\tau_c$ it is linearly unstable. (This result is optimal.) The transversality of this Hopf bifurcation is proven. Other cases for the exponents as well as extensions to higher dimensions are also considered. Our proof makes use of functional analysis and the properties of Weierstrass functions and elliptic integrals.

Let $B$ be the unit ball of ${\mathbb{R}^n}$, $n\geq 5$, and $\rho:\mathbb{R}\rightarrow\mathbb{R}$ a smooth function. We consider the following critical problem: $$\left\{\begin{array}{ll} \Delta^2 u=|u|^{\frac{8}{n-4}}u+\rho(u) & \hbox{ in }B\\ u\not\equiv 0 & \\ u=\frac{\partial u}{\partial n}=0 & \hbox{ on }\partial B . \end{array}\right.$$ We give sufficient conditions for the existence of solutions to this problem. These conditions are close to being sharp, as we prove by considering the problem on arbitrary small balls.

We study the uniqueness problem for nonnegative solutions of $u_t=\Delta u^m$ in $\Omega \times [0,T)$, $-\frac{\partial u^m}{\partial \hat{n}}(x,t)=u^{\lambda}(x,t)$ on $\partial \Omega \times (0,T)$ and $u(x,0) \equiv 0$ on $\Omega$ where $m > 1$, $\lambda \ge 1$, and $\Omega$ is a bounded domain with smooth boundary in $\mathbf {R}^N$. We prove that the solution $u \equiv 0$ is unique if and only if $2\lambda \geq m+1$.

We study a class of higher-order elliptic operators in divergence form with nonsmooth, complex coefficients independent of time. In particular, we give some elliptic regularity results for weak solutions and establish upper bounds for higher derivatives of the heat kernel associated to this class of operators.

Let $\mathcal H$ denote the Hilbert transform and $\eta \ge 0$. We show that the initial-value problems $ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ and $ u_t + \frac{1}{2} (u_x)^2 + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \, u(\cdot , 0) = \phi (\cdot)$ are globally well-posed in $H^s(\mathbb{R})$, $s \ge 1$, $\eta>0$. We study the limiting behavior of the solutions of the first equation as $\eta$ tends to zero in $H^s(\mathbb{R})$ and $s \ge 2$. Moreover, we prove a unique continuation theorem for the first equation in $\mathcal F_{3,3}(\mathbb{R})=H^3(\mathbb{R}) \cap L^2_3(\mathbb{R})$, $\eta>0$.