4 research outputs found
A generalization of Osgood's test and a comparison criterion for integral equations with noise
In this work, we prove a generalization of Osgood's test for the explosion of
the solutions of initial-value problems. We also establish a comparison
criterion for the solution of integral equations with noise, and provide
estimations of the time of explosion of problems arising in the investigation
of crack failures where the noise is the absolute value of the Brownian motionComment: 10 page
Feedback exponential stabilization of the semilinear heat equation with nonlocal initial conditions
The present paper is devoted to the problem of stabilization of the one-dimensional semilinear heat equation with nonlocal initial conditions. The control is with boundary actuation. It is linear, of finite-dimensional structure, given in an explicit form. It allows to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel, then to apply a fixed point argument in a convenient space
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
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