We address local- and global-in-time well-posedness of the Cauchy problem for nonlinear heat equations without imposing growth rate restrictions on the nonlinearity a priori. Our results constitute a non-trivial expansion of the classical L^q-theory for nonlinearities dominated by polynomial growth and the exponential-Orlicz space theory for nonlinearities of exponential growth, to one dealing with nonlinearities of arbitrarily large growth rate. A key ingredient is a new smoothing estimate for the action of the heat semigroup between two arbitrary Orlicz spaces, and in particular into L^∞. For nonlinearities growing at least exponentially we are able to identify explicitly a critical space for local well-posedness and for small initial data global well-posedness.

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 singularity (DCS) of the initial data and show that such singularities always exist for a large class of supercritical nonlinearities. Moreover, we provide exact formulae for such singularities.

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.

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 L1, a necessary and sufficient integral condition on f emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with L1 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 L1.

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.

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.

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 have observed this phenomenon with the plant pathogen Pseudomonas syringae pv. phaseolicola where isolates that have lost the genomic island PPHGI-1 carrying the effector gene avrPphB from its chromosome are infective against previously resistant plant hosts. However, we have never observed island extinction from the pathogen population within a host suggesting the island is maintained. Here, we present a mathematical model which predicts different possible fates for the island in the population; one outcome indicated that PPHGI-1 would be maintained at low frequency in the population long term, if it confers a fitness benefit. We empirically tested this prediction and determined that PPHGI-1 frequency in the bacterial population drops to a low but consistently detectable level during host resistance. Once PPHGI-1-carrying cells encounter a susceptible host, they rapidly increase in the population in a negative frequency-dependent manner. Importantly, our data show that mobile genetic elements can persist within the bacterial population and increase in frequency under favourable conditions.

We consider the scalar semilinear heat equation u(t) - Delta u = f (u), where f : [0, infinity) -> [0, infinity) 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 L-q (Omega) for all non-negative initial data u(0) epsilon L-q (Omega), when Omega subset of R-d is a bounded domain with Dirichlet boundary conditions. For q epsilon (1, infinity) this holds if and only if lim sup(s ->infinity)s(-(1+ 2q/d)) f (s) < infinity; and for q =1 if and only if integral(infinity)(1) s(-(1+2/d)) F(s) ds < infinity, where F(s) = sup1 <= t <= s f (t)/t. This shows for the first time that the model nonlinearity f (u) = u(1+2q/d) is truly the 'boundary case' when q epsilon (1, infinity), but that this is not true for q = 1. The same characterisations hold for the equation posed on the whole space R-d provided that lim sup(s -> 0) f (s)/s < infinity. (C) 2015 Elsevier Masson SAS. All rights reserved.

We establish a local non-existence result for the equation ut - Delta u = f (u) with Dirichlet boundary conditions on a smooth bounded domain Omega subset of R-n and initial data in L-q(Omega) when the source term f is non-decreasing and lim sups, S f (s) = co for some exponent y > q(1 + 2/n). This allows us to construct a locally Lipschitz f satisfying the Osgood condition fr 1/f (s)ds =co, which ensures global existence for initial data in L (52), such that for every q with 1 < q < co there is a non-negative initial condition u0 E L5 (Q) for which the corresponding semilinear problem has no local-in-time solution ('immediate blow-up'). (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

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. (C) 2012 Elsevier Ltd. All rights reserved.

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) over dot = f (v) exist for v(0) > 0 if and only if the Osgood-type condition integral(infinity)(1) = ds/f(s) = infinity holds; by comparison this ensures the existence of global classical solutions of u(t) = Delta u + f(u) for bounded initial data u(0) is an element of L-infinity (R-n). It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in L-q(R-n) for some 1 <= q < infinity. Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in L-loc(1)(R-n) for t > 0. (C) 2013 Elsevier Inc. All rights reserved.

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(Ω) when the source term f is non-decreasing and lim sup_s→∞s^-γf(s)=∞ for some γ>q(1+2/n). This allows us to construct a locally Lipschitz f satisfying the Osgood condition ∫_1^∞1/f(s) ṣ =∞, which ensures global existence for bounded initial data, such that for every q with 1≤ q<∞ there is an initial condition u_0∈ L^q() for which the corresponding semilinear problem has no local-in-time solution.

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.

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 follows by comparison with newly constructed finite travelling waves connecting two stable equilibria.

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 the constraint manifold has a quadratic fold along its singularity.From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general.Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE.

We consider a bistable degenerate diffusion equation where the nonlinearity () is assumed to be singular (non-differentiable) at = 0. It is shown that the finite travelling wave (FTW) connecting the two stable equilibria = 1 and = 0, which is known to exist when is differentiable, persists to the singular case considered here. More interestingly however, such a wavefront can also exist with negative velocity, a situation which does not occur in the non-singular case. Furthermore, there also exist FTWs connecting = 0 to the unstable equilibrium . We also construct composite travelling waves consisting of two FTWs moving in the same direction but with different speeds, which we term .The implications for the full Cauchy problem are considered, in particular with regard to finite propagation and finite-time extinction of solutions. Finally, generalisations to multi-stable diffusion equations are discussed.

In this paper, we consider a reaction-diffusion equation with continuous delay and spatial variable coefficients which models the evolution of a single species. We establish a sharp threshold dynamic result: there exists a critical value λ(a) such that if λ(a)<0 the positive steady state solution of the equation is globally attractive, while if λ(a)≥0 the trivial steady state is globally attractive. To this end, we analyze the ω-limit set of the equation and prove that it is a singleton. Moreover, we apply our method to obtain global attractivity of the positive steady state of a spatially nonlocal diffusive logistic model.

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 priori lower bounds on the separation of zeros of solutions. In particular our results apply to non-Lipschitz non-linearities. Applications to non-linear porous medium equations are considered, yielding information on the existence and strict positivity of equilibrium solutions in some important classes of equations.