Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations

Non-existence of local solutions of semilinear heatequations of Osgood type in bounded domains

We establish a local non-existence result for semilinear heat equations with Dirichlet boundary conditions and initial data in L^q when the source term f is non-decreasing. We construct a locally Lipschitz f satisfying the Osgood condition (which ensures global existence for bounded initial data), such that for every q there is an initial condition in L^q for which the corresponding semilinear problem has no local-in-time solution

Gaussian lower bounds on the Dirichlet heat kernel and non-existence of local solutions for semilinear heat equations of Osgood type

We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditionsand initial data in $L^q(\Omega)$ when the source term $f$ is non-decreasing and $\limsup_{s\to\infty}s^{-\gamma}f(s)=\infty$ for some $\gamma>q(1+2/n)$.This allows us to construct a locally Lipschitz $f$ satisfying the Osgood condition \int_{1}^{\infty}1/f(s)\ \,\d s =\infty, which ensures global existence for bounded initial data, such that for every $q$ with $1\le

A blow-up dichotomy for semilinear fractional heat equations

We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation

Prediction, management and control of odour from landfill sites.

Thesis (M.Sc.Eng.)-University of Natal,Durban, 2002.Due to the spread of urbanisation and increased environmental awareness, odour has become a major problem in communities surrounding landfills. The aim of this research was to investigate odour emissions from landfills and develop a management tool that operators could use to assist in minimising the impacts of odour. The management tool would be in the form of real-time predictions of odour concentrations in the vicinity of a source. The Bisasar Road landfill in Springfield, Durban was a case study site for the research. The methodologies used in this project can be divided into three broad categories. Firstly, flow visualisation experiments were conducted on the case study site to investigate the effects of complex terrain and the results compared to predictions from a dispersion model. Secondly, source characterisation was done on-site. Sources of odour were identified using a portable odour monitor (Electronic nose). Sources of odour were then sampled using sorbent tubes and analysis done using Gas Chromatography - Mass Spectrometry. Thirdly, numerical dispersion modelling was done. Five available dispersion models were assessed and compared against one another in order to select the most suitable model for this application. A software management tool or 'Odour Management System' (OMS), was designed and implemented on a computer at the Bisasar Road landfill. Qualitative results of the flow visualisation experiments show that terrain does have an effect on a dispersing plume path for short-range predictions. Comparisons between the flow experiments and model predictions are qualitatively consistent. Quantitative results were not obtained for the emission flow rate and emission concentration of landfill gas. The chemical composition of the fresh waste gas was determined. ADMSTM(Advanced Dispersion Modelling System) was found to be the most suitable dispersion model for this application. The OMS has been installed on-site to produce odour concentration graphics every ten minutes. A fence line odour control misting system has been installed along approximately 600 metres of the landfill border based on work done as part of this project. Weather conditions and information provided by the OMS, assist in running the odour control system economically

Solvability of Superlinear Fractional Parabolic Equations

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure

Cruciform structures are a common DNA feature important for regulating biological processes

DNA cruciforms play an important role in the regulation of natural processes involving DNA. These structures are formed by inverted repeats, and their stability is enhanced by DNA supercoiling. Cruciform structures are fundamentally important for a wide range of biological processes, including replication, regulation of gene expression, nucleosome structure and recombination. They also have been implicated in the evolution and development of diseases including cancer, Werner's syndrome and others

A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations

© 2017 Elsevier Inc. In their (1968) paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) and the nonlinearity f satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, similar, non-uniqueness phenomena in parabolic equations. In this paper we provide an elementary proof of non-uniqueness for the PDE without any such concavity assumption on f. An important consequence of our result is that uniqueness of the trivial solution of the PDE is equivalent to uniqueness of the trivial solution of the corresponding ODE, which in turn is known to be equivalent to an Osgood-type integral condition on f

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

We consider the scalar semilinear heat equation ut−Δu=f(u), where f:[0,∞)→[0,∞) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in Lq(Ω) for all non-negative initial data u0∈Lq(Ω), when Ω⊂Rd is a bounded domain with Dirichlet boundary conditions. For q∈(1,∞) this holds if and only if limsups→∞s−(1+2q/d)f(s

Convergence to equilibrium in degenerate parabolic equations with delay

© 2012 Elsevier Ltd In [11], Busenberg & Huang (1996) showed that small positive equilibria can undergo supercritical Hopf bifurcation in a delay-logistic reaction–diffusion equation with Dirichlet boundary conditions. Consequently, stable spatially inhomogeneous time-periodic solutions exist. Previously in [12] Badii, Diaz & Tesei (1987) considered a similar logistic-type delay-diffusion equation, but differing in two important respects: firstly by the inclusion of nonlinear degenerate diffusion of so-called porous medium type, and secondly by the inclusion of an additional ‘dominating instantaneous negative feedback’ (where terms local in time majorize the delay terms, in some sense). Sufficient conditions were given ensuring convergence of non-negative solutions to a unique positive equilibrium. A natural question to ask, and one which motivated the present work, is: can one still ensure convergence to equilibrium in delay-logistic diffusion equations in the presence of nonlinear degenerate diffusion, but in the absence of dominating instantaneous negative feedback? The present paper considers this question and provides sufficient conditions to answer in the affirmative. In fact the results are much stronger, establishing global convergence for a much wider class of problems which generalize the porous medium diffusion and delay-logistic terms to larger classes of nonlinearities. Furthermore the results obtained are independent of the size of the delay