Related papers: On a weak maximum principle for a class of fractio…

We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the…

In this paper we investigate maximum principles for functionals defined on solutions to special partial differential equations of elliptic type, extending results by Payne and Philippin. We apply such maximum principles to investigate one…

Doubly nonlinear stochastic evolution equations are considered. Upon assuming the additive noise to be rough enough, we prove the existence of probabilistically weak solutions of Friedrichs type and study their uniqueness in law. This…

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a…

In this article, we investigate the existence and properties of time-periodic solutions for damped evolutionary partial differential equations subject to periodic forcing. Particular emphasis is placed on configurations where the energy…

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of…

This article examines the existence and uniqueness of weak solutions to the d-dimensional micropolar equations ($d=2$ or $d=3$) with general fractional dissipation $(-\Delta)^{\alpha}u$ and $(-\Delta)^{\beta}w$. The micropolar equations…

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. For $s \in (1/2,1)$, we consider a problem of the form \[ \left\{\begin{aligned} (-\Delta)^s u & = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x)\,, & \quad \mbox{in}…

A general stochastic maximum principle is proved for optimal controls of semilinear stochastic evolution equations. Stochastic evolution operators, and the control with values in a general set enter into both drift and diffusion terms.

We develop a new, unified approach to the following two classical questions on elliptic PDE: the strong maximum principle for equations with non-Lipschitz nonlinearities, and the at most exponential decay of solutions in the whole space or…

We study vector valued solutions to non-linear elliptic partial differential equations with $p$-growth. Existence of a solution is shown in case the right hand side is the divergence of a function which is only $q$ integrable, where $q$ is…

In this paper we prove strong unique continuation principle and unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the…

We establish two Phragm\'en--Lindel\"{o}f theorems for a fully nonlinear elliptic equation. We consider a dynamic boundary condition that includes both spatial variable and time derivative terms. As a spatial term, we consider a non-linear…

We give an approach to exponential stability within the framework of evolutionary equations due to [R. Picard. A structural observation for linear material laws in classical mathematical physics. Math. Methods Appl. Sci.,…

This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination…

We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the Free-evolution approach, which…

We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: (-\Delta)^{\alpha/2}u=\lambda u^q+u^{\frac{N+\alpha}{N-\alpha}}, \quad…

Given a fractional differential equation of order $\alpha \in (0,1]$ with Caputo derivatives, we investigate in a quantitative sense how the associated solutions depend on their respective initial conditions. Specifically, we look at two…

We study the long-time behavior of solutions to a class of evolution equations arising from random-time changes driven by subordinators. Our focus is on fractional diffusion equations involving mixed local and nonlocal operators. By…

We provide a detailed (and fully rigorous) derivation of several fundamental properties of bounded weak solutions to initial-value problems for general conservative 2nd-order parabolic equations with p-Laplacian diffusion and (arbitrary)…