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

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

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

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

Transitions to food democracy through multilevel governance

status: publishe

Global asymptotic behaviour in some functional parabolic equations

The global asymptotic behavior in some functional parabolic equations was presented. The sufficient condition for the global convergence to a spatially uniform equilibrium in the system of functional partial differential equations was provided. Results were observed to be valid for all non-negative spatio-temporal kernels, which also satisfied a uniform normailzation condition

Well-posedness of semilinear heat equations in L1

The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term f has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in L 1 , a necessary and sufficient integral condition on f emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with L 1 initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We also establish sufficient conditions for the global-in-time continuation of solutions for small initial data in L 1

The flow of a DAE near a singular equilibrium

We extend the differential-algebraic equation (DAE) taxonomy by assuming that the linearization of a DAE about a singular equilibrium has a particular index-2 Kronecker normal form. A Lyapunov-Schmidt procedure is used to reduce the DAE to a quasilinear normal form which is shown to posses quasi-invariant manifolds which intersect the singularity. In turn, this provides solutions of the DAE which pass through the singularity