Blow-up dynamics of self-attracting diffusive particles driven by competing convexities

In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is a consistent approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel in high dimensions

Asymptotic growth in nonlinear stochastic and deterministic functional differential equations

This thesis concerns the asymptotic growth of solutions to nonlinear functional differential equations, both random and deterministic. How quickly do solutions grow? How do growth rates of solutions depend on the memory and the nonlinearity of the system? What is the effect of randomness on the growth rates of solutions? We address these questions for classes of nonlinear functional differential equations, principally convolution Volterra equations of the second kind. We first study deterministic equations with sublinear nonlinearity and integrable kernels. For such systems, we prove that the growth rates of solutions are independent of the distribution of the memory. Hence we conjecture that stronger memory dependence is needed to generate growth rates which depend meaningfully on the delay structure. Using the theory of regular variation, we then demonstrate that solutions to a class of sublinear Volterra equations with non–integrable kernels grow at a memory dependent rate. We complete our treatment of sublinear equations by examining the impact of stochastic perturbations on our previous results; we consider the illustrative and important cases of Brownian and alpha–stable Lévy noise. In summary, if an appropriate functional of the forcing term has a limit L at infinity, solutions behave asymptotically like the underlying unforced equation when L = 0 and like the forcing term when L is infinite. Solutions inherit properties of both the forcing term and underlying unforced equation for finite and positive L. Similarly, we prove linear discrete Volterra equations with summable kernels inherit the behaviour of unbounded perturbations, random or deterministic. Finally, we consider Volterra integro–differential equations with superlinear nonlinearity and nonsingular kernels. We provide sharp estimates on the rate of blow–up if solutions are explosive, or unbounded growth if solutions are global. We also recover well–known necessary and sufficient conditions for finite–time blow–up via new methods

Boundary regularity for fully nonlinear integro-differential equations

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal L_*\subset \mathcal L_0$, which consists of infinitesimal generators of stable L\'evy processes belonging to the class $\mathcal L_0$ of Caffarelli-Silvestre. For fully nonlinear operators $I$ elliptic with respect to $\mathcal L_*$, we prove that solutions to $I u=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, satisfy $u/d^s\in C^{s+\gamma}(\overline\Omega)$, where $d$ is the distance to $\partial\Omega$ and $f\in C^\gamma$. We expect the class $\mathcal L_*$ to be the largest scale invariant subclass of $\mathcal L_0$ for which this result is true. In this direction, we show that the class $\mathcal L_0$ is too large for all solutions to behave like $d^s$. The constants in all the estimates in this paper remain bounded as the order of the equation approaches 2. Thus, in the limit $s\uparrow1$ we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.Comment: To appear in Duke Math.

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 $\lim_{t\to\tau}A(x(t),t) = 1$, where $\tau$ is the blow-up time if solutions are explosive or $\tau = \infty$ 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