Related papers: A priori bounds for the dynamic fractional $\Phi^4…

We derive a priori bounds for the $\Phi^4$ equation in the full sub-critical regime using Hairer's theory of regularity structures. The equation is formally given by \begin{equation} \label{e}(\partial_t-\Delta)\phi = -\phi^3 + \infty \phi…

We obtain (small-parameter) well-posedness for the (space-time periodic) $\Phi^4$ equation in the full subcritical regime in the context of regularity structures based on multi-indices. As opposed to Hairer's more extrinsic tree-based…

We prove an a priori bound for the dynamic $\Phi^4_3$ model on the torus wich is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform…

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not…

In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -\Delta)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. First we derive…

We study the fractional $\Phi^4_3$-measure (with order $\alpha > 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity,…

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator . Under certain assumptions of the indefinite nonlinearity and its weight, we prove that there is no positive bounded solution,…

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$,…

We prove a priori subelliptic estimates, near a non-characteristic boundary point, for the heat operators associated to a wide class of maximally subelliptic quadratic forms. This is the third paper in a series devoted to studying general…

We show a priori bounds for solutions to $(\partial_t - \Delta) u = \sigma (u) \xi$ in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269--504, 2014]. We assume $\sigma \in C_b^2 (\mathbb{R})$ and that…

We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus, and Schauder's fixed…

We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus…

In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of…

We use the formalism of Hairer's regularity structures theory \cite{hai-14} to study a heat equation with non-linear perturbation driven by a space-time fractional noise. Different regimes are observed, depending on the global pathwise…

We establish boundary observability and control for the fractional heat equation over arbitrary time horizons $T > 0$, within the optimal range of fractional exponents $s \in (1/2, 1)$. Our approach introduces a novel synthesis of…

We investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove existence results by applying Schauder's fixed point technique. Moreover, we show fundamental…

The present work is devoted to the study of fractional powers of vector operators, with particular emphasis on the gradient operator with non-constant coefficients. Within the setting of Clifford algebra $\mathbb{R}_n$, this operator turns…

This paper consists of four parts. It begins by using the authors's generalized Schauder formula, [50], and the algebraic multiplicity, $\chi$, of Esquinas and L\'opez-G\'omez [18,17,40] to package and sharpening all existing results in…

In this paper, we establish gradient continuity for solutions to \[ (\partial_t - \operatorname{div}(A(x) \nabla u))^s =f,\ s \in (1/2, 1), \] when $f$ belongs to the scaling critical function space $L(\frac{n+2}{2s-1}, 1)$. Our main…