The chaotic heat equation

The one dimensional heat equation ∂u ∂t = a ∂2u ∂x2 +b ∂u ∂x +cu for x ε ℝ, t ≥ 0 is governed by a chaotic semigroup for certain values of the coefficients (a,b,c) ε ℝ3 on certain weighted supremum norm spaces. © CSP - Cambridge, UK

Chaotic solution for the Black-Scholes equation

The Black-Scholes semigroup is studied on spaces of continuous functions on (0,∞) which may grow at both 0 and at ∞, which is important since the standard initial value is an unbounded function. We prove that in the Banach spaces with norm, the Black-Scholes semigroup is strongly continuous and chaotic for s \u3e 1, τ ≥ 0 with sν \u3e 1, where √2ν is the volatility. The proof relies on the Godefroy-Shapiro hypercyclicity criterion. © 2011 American Mathematical Society

Degenerate second order differential operators generating analytic semigroups in L\u3csup\u3ep\u3c/sup\u3e and W\u3csup\u3e1,p\u3c/sup\u3e

We deal with the problem of analyticity for the semigroup generated by the second order differential operator Au:= αu″ + βu′ (or by some restrictions of it) in the spaces Lp(0, 1), with or without weight, and in W1,p(0, 1), 1 \u3c p \u3c ∞. Here α and β are assumed real-valued and continuous in [0, 1], with α(x) \u3e 0 in (0, 1), and the domain of A is determined by the generalized Neumann boundary conditions and by Wentzell boundary conditions

Equipartition of energy for nonautonomous wave equations

Consider wave equations of the form u (t) + A2u(t) = 0 with A an injective selfadjoint operator on a complex Hilbert space H. The kinetic, potential, and total energies of a solution u are K(t) = ∥u\u27(t)∥2; P(t) = ∥Au(t)∥2; E(t) = K(t) + P(t): Finite energy solutions are those mild solutions for which E(t) is finite. For such solutions E(t) = E(0), that is, energy is conserved, and asymptotic equipartition of energy lim/t→±∞ K(t) = lim/t→±∞P(t) = E(0)/2 holds for all finite energy mild solutions iff eitA → 0 in the weak operator topology. In this paper we present the first extension of this result to the case where A is time dependent

Scaling and variants of Hardy’s inequality

The two related one space dimensional singular linear parabolic equations (1), (2) studied by H. Brezis et al. [Comm. Pure Appl. Math. 24 (1971), pp. 395–416] have different scaling properties. These scaling properties lead to new variants of the Hardy and Caffarelli-Kohn-Nirenberg inequalities. These results are proved, and they imply some non-wellposedness results when the constant in the singular potential term is large enough

Equipartition of energy for nonautonomous damped wave equations

The kinetic and potential energies for the damped wave equation u00 + 2Bu0 + A2u = 0 (DWE) are defined by K(t) = ku0(t)k2, P(t) = kAu(t)k2, where A, B are suitable commuting selfadjoint operators. Asymptotic equipartition of energy means tlim →∞KP((tt)) = 1 (AEE) for all (finite energy) non-zero solutions of (DWE). The main result of this paper is the proof of a result analogous to (AEE) for a nonautonomous version of (DWE)