All Outputs (18)

Local solvability and dilation-critical singularities of supercritical fractional heat equations (2024)
Journal Article

We consider the Cauchy problem for fractional semilinear heat equations with supercritical nonlinearities and establish both necessary conditions and sufficient conditions for local-in-time solvability. We introduce the notion of a dilation-critical... Read More about Local solvability and dilation-critical singularities of supercritical fractional heat equations.

Solvability of superlinear fractional parabolic equations (2022)
Journal Article

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 o... Read More about Solvability of superlinear fractional parabolic equations.

A blow-up dichotomy for semilinear fractional heat equations (2020)
Journal Article

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... Read More about A blow-up dichotomy for semilinear fractional heat equations.

Well-posedness of semilinear heat equations in L1 (2019)
Journal Article

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.... Read More about Well-posedness of semilinear heat equations in L1.

A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations (2017)
Journal Article

© 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 i... Read More about A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations.

A low frequency persistent reservoir of a genomic island in a pathogen population ensures island survival and improves pathogen fitness in a susceptible host (2016)
Journal Article

The co-evolution of bacterial plant pathogens and their hosts is a complex and dynamic process. Host resistance imposes stress on invading pathogens that can lead to changes in the bacterial genome enabling the pathogen to escape host resistance. We... Read More about A low frequency persistent reservoir of a genomic island in a pathogen population ensures island survival and improves pathogen fitness in a susceptible host.

A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces (2015)
Journal Article

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... Read More about A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces.

Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains (2014)
Journal Article

We establish a local non-existence result for the equation ut-δu=f(u) with Dirichlet boundary conditions on a smooth bounded domain Ω⊂Rn and initial data in Lq(Ω) when the source term f is non-decreasing and limsups→∞s-γf(s)=∞ for some exponent γ>q(1... Read More about Non-existence of local solutions of semilinear heat equations of Osgood type in bounded domains.

Non-existence of local solutions for semilinear heat equations of Osgood type (2013)
Journal Article

We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, non-negative source term f. Global (in time) solutions of the scalar ODE v;=f(v) exist for v(0)>0 if and... Read More about Non-existence of local solutions for semilinear heat equations of Osgood type.

Gaussian lower bounds on the Dirichlet heat kernel and non-existence of local solutions for semilinear heat equations of Osgood type (2013)
Journal Article

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 conditions
and initial data in $L^q... Read More about Gaussian lower bounds on the Dirichlet heat kernel and non-existence of local solutions for semilinear heat equations of Osgood type.

Convergence to equilibrium in degenerate parabolic equations with delay (2013)
Journal Article

© 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... Read More about Convergence to equilibrium in degenerate parabolic equations with delay.

Finite time extinction in nonlinear diffusion equations (2004)
Journal Article

We consider a class of degenerate diffusion equations where the nonlinearity is assumed to be singular (non-Lipschitz) at zero. It is shown that solutions with compactly supported initial data become identically zero in finite time. Such extinction f... Read More about Finite time extinction in nonlinear diffusion equations.

Trajectories of a DAE near a pseudo-equilibrium (2004)
Journal Article

We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that th... Read More about Trajectories of a DAE near a pseudo-equilibrium.

Sequential and continuum bifurcations in degenerate elliptic equations (2003)
Journal Article

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 bifurc... Read More about Sequential and continuum bifurcations in degenerate elliptic equations.

The flow of a DAE near a singular equilibrium (2003)
Journal Article

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 quasilin... Read More about The flow of a DAE near a singular equilibrium.

Transversality and separation of zeros in second order differential equations (2003)
Journal Article

Sufficient conditions on the non-linearity f are given which ensure that non-trivial solutions of second order differential equations of the form Lu = f(t, u) have a finite number of transverse zeros in a given finite time interval. We also obtain a... Read More about Transversality and separation of zeros in second order differential equations.

Global asymptotic behaviour in some functional parabolic equations (2002)
Journal Article

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. Resul... Read More about Global asymptotic behaviour in some functional parabolic equations.

Instability of equilibria in some delay reaction-diffusion systems (2000)
Journal Article

A new result is derived which extends a known instability result for a class of reaction-diffusion equations to a corresponding system incorporating time delay effects. For a significant class of nonlinear equations it is shown that an unstable equil... Read More about Instability of equilibria in some delay reaction-diffusion systems.