85,161 research outputs found
Optimal existence classes and nonlinear-like dynamics in the heat equation in Rd
We analyse the behaviour of solutions of the linear heat equation in R d for initial data in the classes M ε (Rd) of Radon measures with ∫ R d e − ε | x | 2 d | u 0 | 0 M ε (Rd) consists precisely of those initial data for which the a solution of the heat equation can be given for all time using the heat kernel representation formula. After considering properties of existence, uniqueness, and regularity for such initial data, which can grow rapidly at infinity, we go on to show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t →∞ (in fact we show that this behaviour is generic), and that one can prescribe the behaviour of u (0 ,t ) to be any real-analytic function γ ( t ) on [0 , ∞ )
Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi Equations
We study the large time behavior of Lipschitz continuous, possibly unbounded,
viscosity solutions of Hamilton-Jacobi Equations in the whole space . The
associated ergodic problem has Lipschitz continuous solutions if the analogue
of the ergodic constant is larger than a minimal value . We
obtain various large-time convergence and Liouville type theorems, some of them
being of completely new type. We also provide examples showing that, in this
unbounded framework, the ergodic behavior may fail, and that the asymptotic
behavior may also be unstable with respect to the initial data
Data-driven and Model-based Verification: a Bayesian Identification Approach
This work develops a measurement-driven and model-based formal verification
approach, applicable to systems with partly unknown dynamics. We provide a
principled method, grounded on reachability analysis and on Bayesian inference,
to compute the confidence that a physical system driven by external inputs and
accessed under noisy measurements, verifies a temporal logic property. A case
study is discussed, where we investigate the bounded- and unbounded-time safety
of a partly unknown linear time invariant system
The Einstein-Vlasov system with a scalar field
We study the Einstein-Vlasov system coupled to a nonlinear scalar field with
a nonnegative potential in locally spatially homogeneous spacetime, as an
expanding cosmological model. It is shown that solutions of this system exist
globally in time. When the potential of the scalar field is of an exponential
form, the cosmological model corresponds to accelerated expansion. The
Einstein-Vlasov system coupled to a nonlinear scalar field whose potential is
of an exponential form shows the causal geodesic completeness of the spacetime
towards the future. The asymptotic behaviour of solutions of this system in the
future time is analysed in various aspects, which shows power-law expansion.Comment: 30 page
Robustness of time-dependent attractors in H1-norm for nonlocal problems
In this paper, the existence of regular pullback attractors as well as their upper semicontinuous behaviour in H1-norm are analysed for a parameterized family of non-autonomous nonlocal reaction-diffusion equations without uniqueness, improving previous results [Nonlinear Dyn. 84 (2016), 35–50].Ministerio de EconomÃa y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucÃ
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