602 research outputs found
Blow-up and superexponential growth in superlinear Volterra equations
This paper concerns the finite-time blow-up and asymptotic behaviour of
solutions to nonlinear Volterra integrodifferential equations. Our main
contribution is to determine sharp estimates on the growth rates of both
explosive and nonexplosive solutions for a class of equations with nonsingular
kernels under weak hypotheses on the nonlinearity. In this superlinear setting
we must be content with estimates of the form ,
where is the blow-up time if solutions are explosive or
if solutions are global. Our estimates improve on the sharpness of results in
the literature and we also recover well-known blow-up criteria via new methods.Comment: 24 page
Multiple positive solutions to elliptic boundary blow-up problems
We prove the existence of multiple positive radial solutions to the
sign-indefinite elliptic boundary blow-up problem where is a function superlinear at zero and at infinity,
and are the positive/negative part, respectively, of a sign-changing
function and is a large parameter. In particular, we show how the
number of solutions is affected by the nodal behavior of the weight function
. The proof is based on a careful shooting-type argument for the equivalent
singular ODE problem. As a further application of this technique, the existence
of multiple positive radial homoclinic solutions to is also considered
Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis
This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be
replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We
therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends
on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering
this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose
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