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Well-posedness of semilinear heat equations in L1 (2019)
Journal Article
Laister, R., & Sierżęga, M. (in press). Well-posedness of semilinear heat equations in L1. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, https://doi.org/10.1016/j.anihpc.2019.12.001

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
Sierżęga, M., Laister, R., Laister, R., Robinson, J. C., & Sierzega, M. (2017). A necessary and sufficient condition for uniqueness of the trivial solution in semilinear parabolic equations. Journal of Differential Equations, 262(10), 4979-4987. https://doi.org/10.1016/j.jde.2017.01.014

© 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
Neale, H., Laister, R., Payne, J., Preston, G., Jackson, R., & Arnold, D. L. (2016). A low frequency persistent reservoir of a genomic island in a pathogen population ensures island survival and improves pathogen fitness in a susceptible host. Environmental Microbiology, 18(11), 4144-4152. https://doi.org/10.1111/1462-2920.13482

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
Vidal-López, A., Sierżęga, M., Robinson, J. C., Laister, R., Laister, R., Robinson, J. C., …Vidal-Lopez, A. (2016). A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33(6), 1519-1538. https://doi.org/10.1016/j.anihpc.2015.06.005

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
equations of Osgood type in bounded domains. Comptes Rendus Mathématique, 352(7-8), 621-626. https://doi.org/10.1016/j.crma.2014.05.010

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 ens... 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
Sierz;E;ga, M., Robinson, J. C., Laister, R., Robinson, J. C., & Sierzega, M. (2013). Non-existence of local solutions for semilinear heat equations of Osgood type. Journal of Differential Equations, 255(10), 3020-3028. https://doi.org/10.1016/j.jde.2013.07.007

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
Laister, R., Robinson, J. C., & Sierzega, M. (2013). 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 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
Laister, R., & Laister, R. (2013). Convergence to equilibrium in degenerate parabolic equations with delay. Nonlinear Analysis: Theory, Methods and Applications, 81, 200-210. https://doi.org/10.1016/j.na.2012.10.019

© 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.

Sequential and continuum bifurcations in degenerate elliptic equations (2004)
Journal Article
Laister, R., Beardmore, R. E., & Laister, R. (2004). Sequential and continuum bifurcations in degenerate elliptic equations. Proceedings of the American Mathematical Society, 132(01), 165-174. https://doi.org/10.1090/S0002-9939-03-06979-X

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.

Finite time extinction in nonlinear diffusion equations (2004)
Journal Article
Beardmore, R. E., Peplow, A. T., & Laister, R. (2004). Finite time extinction in nonlinear diffusion equations. Applied Mathematics Letters, 17(5), 561-567. https://doi.org/10.1016/S0893-9659%2804%2990126-7

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
Laister, R., Beardmore, R. E., Laister, R., & Peplow, A. (2004). Trajectories of a DAE near a pseudo-equilibrium. Nonlinearity, 17(1), 253-279. https://doi.org/10.1088/0951-7715/17/1/015

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.

The flow of a DAE near a singular equilibrium (2003)
Journal Article
Laister, R., Beardmore, R. E., & Laister, R. (2003). The flow of a DAE near a singular equilibrium. SIAM Journal on Matrix Analysis and Applications, 24(1), 106-120. https://doi.org/10.1137/S0895479800378660

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
Laister, R., Laister, R., & Beardmore, R. E. (2003). Transversality and separation of zeros in second order differential equations. Proceedings of the American Mathematical Society, 131(1), 209-218. https://doi.org/10.1090/S0002-9939-02-06546-2

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
Laister, R. (2002). Global asymptotic behaviour in some functional parabolic equations. Nonlinear Analysis: Theory, Methods and Applications, 50(3), 347-361. https://doi.org/10.1016/S0362-546X%2801%2900766-0

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
Laister, R. (2000). Instability of equilibria in some delay reaction-diffusion systems. Journal of Mathematical Analysis and Applications, 247(2), 588-607. https://doi.org/10.1006/jmaa.2000.6883

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.