Global asymptotic stability of bifurcating, positive equilibria of p-Laplacian boundary value problems with p-concave nonlinearities

We consider the parabolic, initial value problem $$v_t =\Delta_p(v)+\lambda g(x,v)\phi_p(v), \quad \text{in $\Omega \times (0,\infty),$}$$ $$v =0, \text{in $\partial\Omega \times (0,\infty),$}\tag{IVP} v =v_0\ge0, \text{in $\Omega \times \{0\},$}$$ where $\Omega$ is a bounded domain in ${\mathbb R}^N$, for some integer $N\ge1$, with smooth boundary $\partial\Omega$, $\phi_p(s):=|s|^{p-1} {\rm sgn}s$, $s\in{\mathbb R}$, $\Delta_p$ denotes the $p$-Laplacian, with $p>\max\{2,N\}$, $v_0\in C^0(\overline{\Omega})$, and $\lambda>0$. The function $g:\overline{\Omega } \times [0,\infty)\to(0,\infty)$ is $C^0$ and, for each $x\in\overline{\Omega }$, the function $g(x,\cdot):[0,\infty)\to(0,\infty)$ is Lipschitz continuous and strictly decreasing. Clearly, (IVP) has the trivial solution $v\equiv0$, for all $\lambda>0$. In addition, there exists $0<\lambda_{\rm min}(g)<\lambda_{\rm max}(g)$ ($\lambda_{\rm max}(g)$ may be $\infty$) such that: $(a)$ if $\lambda\not\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP) has no non-trivial, positive equilibrium; $(b)$ if $\lambda\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP) has a unique, non-trivial, positive equilibrium $e_\lambda\in W_0^{1,p}(\Omega)$. We prove the following results on the positive solutions of (IVP): $(a)$ if $0<\lambda<\lambda_{\rm min}(g)$ then the trivial solution is globally asymptotically stable; $(b)$ if $\lambda_{\rm min}(g)<\lambda<\lambda_{\rm max}(g)$ then $e_\lambda$ is globally asymptotically stable; $(c)$ if $\lambda_{\rm max}(g)<\lambda$ then any non-trivial solution blows up in finite time

Post Natal Impact of Maternal Tryptophan Deficiency on Central CO2/PH Chemosensitivity

Cells and mechanisms underlying central chemosensitivity, are poorly understood and can be controversial. Our overarching hypothesis is that brainstem 5-HT and/or GABA neurons contribute to detection and response to changes in pH/CO2. Our experiments are designed to provide insight into respiratory physiology, and pathologies thought to result from chemosensory dysfunction such as the Sudden Infant Death Syndrome (SIDS). A deficiency of 5-HT resulting from maternal dietary restriction could enhance vulnerability to SIDS. It was recently shown that rat pups born to dams fed a tryptophan deficient diet have a reduced number of central 5-HT neurons and reduced ventilatory sensitivity to CO2 (Nattie et al. 2011). Unknown are the relative contributions of central vs peripheral chemoreceptors to this observation, or the residual contributions of 5-HT in the face of this deficiency. In the present study we are extending this initial description using a perfused in situ brainstem model to determine the degree of central chemosensory deficit imparted by maternal tryptophan restriction. We also repeat these studies with pharmacological blockade of a population of 5-HT receptors to illustrate remaining 5-HT and non-5-HT contributions to chemosensitivity. This work reveals important interactions between nutrition and ventilatory control that may aid in the understanding of SIDS

Digital Image Access & Retrieval

The 33th Annual Clinic on Library Applications of Data Processing, held at the University of Illinois at Urbana-Champaign in March of 1996, addressed the theme of "Digital Image Access & Retrieval." The papers from this conference cover a wide range of topics concerning digital imaging technology for visual resource collections. Papers covered three general areas: (1) systems, planning, and implementation; (2) automatic and semi-automatic indexing; and (3) preservation with the bulk of the conference focusing on indexing and retrieval.published or submitted for publicatio

Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Landesman-Lazer conditions at half-eigenvalues of the p-Laplacian

We study the existence of solutions of the Dirichlet problem {gather} -\phi_p(u')' -a_+ \phi_p(u^+) + a_- \phi_p(u^-) -\lambda \phi_p(u) = f(x,u), \quad x \in (0,1), \label{pb.eq} \tag{1} u(0)=u(1)=0,\label{pb_bc.eq} \tag{2} {gather} where $p>1$, \phi_p(s):=|s|^{p-1}\sgn s for $s \in \mathbb{R}$, the coefficients $a_\pm \in C^0[0,1]$, $\lambda \in \mathbb{R}$, and $u^\pm := \max\{\pm u,0\}$. We suppose that $f\in C^1([0,1]\times\mathbb{R})$ and that there exists $f_\pm \in C^0[0,1]$ such that $\lim_{\xi\to\pm\infty} f(x,\xi) = f_\pm(x)$, for all $x \in [0,1]$. With these conditions the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to have a `jumping nonlinearity'. We also suppose that the problem {gather} -\phi_p(u')' = a_+ \phi_p(u^+) - a_- \phi_p(u^-) + \lambda \phi_p(u) \quad\text{on} \ (0,1), \tag{3} \label{heval_pb.eq} {gather} together with \eqref{pb_bc.eq}, has a non-trivial solution $u$. That is, $\lambda$ is a `half-eigenvalue' of \eqref{pb_bc.eq}-\eqref{heval_pb.eq}, and the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem \eqref{pb.eq}-\eqref{pb_bc.eq} has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients $a_\pm$ are constants, and the resonance has been at a point in the `Fucik spectrum'. Even in this constant coefficient case our result extends previous results. In particular, previous variational approaches have required strong conditions on the location of the resonant point, whereas our result applies to any point in the Fucik spectrum.Comment: 14 page

Tackling the Global NCD Crisis: Innovations in Law and Governance

35 million people die annually of non-communicable diseases (NCDs), 80% of them in low- and middle-income countries—representing a marked epidemiological transition from infectious to chronic diseases and from richer to poorer countries. The total number of NCDs is projected to rise by 17% over the coming decade, absent significant interventions. The NCD epidemic poses unique governance challenges: the causes are multifactorial, the affected populations diffuse, and effective responses require sustained multi-sectorial cooperation. The authors propose a range of regulatory options available at the domestic level, including stricter food labeling laws, regulation of food advertisements, tax incentives for healthy lifestyle choices, changes to the built environment, and direct regulation of food and drink producers. Given the realities of globalization, such interventions require global cooperation. In 2011, the UN General Assembly held a High-level meeting on NCDs, setting a global target of a 25% reduction in premature mortality from NCDs by 2025. Yet concrete plans and resource commitments for reaching this goal are not yet in the offing, and the window is rapidly closing for achieving these targets through prevention--as opposed to treatment, which is more costly. Innovative global governance for health is urgently needed to engage private industry and civil society in the global response to the NCD crisis

Subsurface Irrigation Research in Arkansas

A pilot study conducted in 1963 indicated that (1) cotton yields could be increased by subirrigation, (2) drilled orifices were unsatisfactory because of internal plugging caused by burrs and drilling particles, and (3) operating pressures of 5 and 10 psi were excessive, and 5 psi probably should be considered as an upper-limit pressure. A greenhouse study of a subirrigation system indicated that 2 psi should, in general, for in-wall orifices, be considered as a lower-limit pressure when orifices are built into the pipe wall. The greenhouse study also indicated that a placement depth of 12 inches was preferable to 18 inches in terms of the amount of water required. Twelve inches was used as the depth of placement for the field subirrigation system. A device was developed for forming orifices with a hot needle. This method has advantages over the drilled orifices used in the pilot study in that the forming process does not produce loose particles to fall inside the pipe or burrs which remain attached to the inner edge of the orifice and later cause stoppage problems. This method also seems preferable to punched orifices which tend to become smaller with time due to rebound of the plastic