Abstract:
We study existence and uniqueness of bounded solutions to a fractional nonlinear porous medium equation with a variable density, in one space dimension.

Abstract:
We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfy a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e. solutions of the corresponding elliptic problem) and time periodic solutions.

Abstract:
We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate homogeneous Dirichlet boundary conditions. As $\mathcal{L}$ we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian $(-\Delta)^s$, $01$. In this paper we propose a suitable class of solutions of the equation, and cover the basic theory: we prove existence, uniqueness of such solutions, and we establish upper bounds of two forms (absolute bounds and smoothing effects), as well as weighted-$L^1$ estimates. The class of solutions is very well suited for that work. The standard Laplacian case $s=1$ is included and the linear case $m=1$ can be recovered in the limit. In a companion paper [12], we will complete the study with more advanced estimates, like the upper and lower boundary behaviour and Harnack inequalities, for which the results of this paper are needed.

Abstract:
We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.

Abstract:
We prove the existence of strong time-periodic solutions and their asymptotic stability with the total energy of the perturbations decaying to zero at an exponential decay rate as $t \rightarrow \infty$ for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domain. The mathematical model includes a mechanical dissipation and a periodic forcing function of period $T$. In the second part of the paper, we consider a magnetoelastic system in the form of a semilinear initial boundary value problem in a bounded, simply-connected two-dimensional domain. We use LaSalle invariance principle to obtain results on the asymptotic behavior of solutions. This second result was obtained for the system under the action of only one dissipation (the natural dissipation of the system).

Abstract:
In this work, we introduce and study the well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on bounded domains and on the torus (Briefly dD-FSNSE). We prove the existence of a martingale solution for the general regime. We establish the uniqueness in the case a martingale solution enjoys a condition of Serrin's type on the fractional Sobolev spaces. If an $L^2-$ local weak (strong in probability) solution exists and enjoys conditions of Beale-Kato-Majda type, this solution is global and unique. These conditions are automatically satisfied for the 2D-FSNSE on the torus if the initial data has $ H^1-$regularity and the diffusion term satisfies growth and Lipschitz conditions corresponding to $ H^1-$spaces. The case of 2D-FSNSE on the torus is studied separately. In particular, we established thresholds for the global existence, uniqueness, space and time regularities of the weak (strong in probability) solutions in the subcritical regime.

Abstract:
The existence and nonexistence of $\lambda$-harmonic functions in unbounded domains of $\mathbb{H}^n$ are investigated. We prove that if the $(n-1)/2$ Hausdorff measure of the asymptotic boundary of a domain $\Omega$ is zero, then there is no bounded $\lambda$-harmonic function of $\Omega$ for $\lambda \in [0,\lambda_1(\mathbb{H}^n)]$, where $\lambda_1(\mathbb{H}^n)=(n-1)^2/4$. For these domains, we have comparison principle and some maximum principle. Conversely, for any $s>(n-1)/2,$ we prove the existence of domains with asymptotic boundary of dimension $s$ for which there are bounded $\lambda_1$-harmonic functions that decay exponentially at infinity.

Abstract:
We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary.

Abstract:
We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system , , , , where , and , is the standard Riemann-Liouville derivative, is linear functionals given by Riemann-Stieltjes integrals, is a function of bounded variation, and can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder's fixed-point theorem and upper and lower solution method.