Related papers: Discrepancy principle for DSM
The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum…
We consider the recovery of an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation that can be written in the form $\mathcal{L} u_f=c(f,u_f)$, for a differential operator $\mathcal{L}$ that…
In this paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space $\mathcal{H}$. We find necessary and sufficient conditions on a bounded linear operator $A\in\mathcal{B}(\mathcal{H})$ and a set of…
We consider systems of linear differential and difference equations \begin{eqnarray*} \partial Y(x) =A(x)Y(x), \sigma Y(x) =B(x)Y(x) \end{eqnarray*} with $\partial = \frac{d}{dx}$, $\sigma$ a shift operator $\sigma(x) = x+a$, $q$-dilation…
It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and…
This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$…
Comparison principles for Volterra equations play a role analogous to maximum principles in PDEs: they provide positivity and stability information on the solution and allow one to control the output of bounded inputs. In the continuous…
We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_q)^s u &= \frac{\lambda}{u^{\delta}} + u^r \text{ in } \Omega\newline u > 0 \text{ in }…
Inspired by a recent pointwise differential inequality for positive bounded solutions of the fourth-order H\'enon equation $\Delta^2 u = |x|^a u^p$ in ${\mathbb R}^n$ with $a \geqslant 0$, $p > 1$, $n \geqslant 5$ due to Fazly, Wei, and Xu…
The celebrated Heisenberg Uncertainty Principle \Delta x \Delta p\ge \hbar/2 can allow measurement accuracies less than \Delta x or \Delta p. Classical analog of this is known as sub-Fourier sensitivity. We illustrate this phenomenon in a…
Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…
The purpose of this paper is to present an example of a C1 (in the Fr\'echet sense) discrete dynamical system in a infinite-dimensional separable Hilbert space for which the origin is an exponentially asymptotically stable fixed point, but…
This paper proposes a new approach to describe the stability of linear time-invariant systems via the torsion $\tau(t)$ of the state trajectory. For a system $\dot{r}(t)=Ar(t)$ where $A$ is invertible, we show that (1) if there exists a…
This paper considers a local and non-local problem characterized by singular nonlinearity and a source term. Specifically, we focus on the following problem: \begin{equation}\label{A}\tag{P} -\Delta_{p} u + (-\Delta)^{s}_{q} u = f(x)…
Dynamical spontaneous breaking of some discrete symmetries including special parities and time reversal and their restoration at finite temperature T are researched in 3D Gross-Neveu model by means of Schwinger-Dyson equation in the…
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…
In a Hilbert space $\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\infty$, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features. Given $\mathcal H$ a general Hilbert…
The uncertainty principle is fundamentally rooted in the algebraic asymmetry between observables. We introduce a new class of uncertainty relations grounded in the resource theory of asymmetry, where incompatibility is quantified by an…
In this paper the uncertainty principle is found via characteristics of continuous and nowhere differentiable functions. We prove that any physical system that has a continuous and nowhere differentiable position function is subject to an…
This work aims to study the initial-boundary value problem of the reaction-diffusion equation $\pa_{t}u-\Delta u=f(u)+g(u(t-\tau(t,u_t)))+h(t,x)$ in a bounded domain with state-dependent delay and supercritical nonlinearities. We establish…