Related papers: Monotonicity for the fractional semi-linear proble…

Regularization methods have been recently developed to construct stable approximate solutions to classical partial differential equations considered as final value problems. In this paper, we investigate the backward parabolic problem with…

A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations $\Delta u +f(u)=0$ in $\mathbb{R}^n$ must be radial and radially decreasing. In this paper, we consider…

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset \mathbb{R}^2$ a $C^{1,\alpha}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined…

A celebrated result by Gidas, Ni & Nirenberg asserts that classical positive solutions to semilinear equations $- \Delta u = f(u)$ in a ball vanishing at the boundary must be radial and radially decreasing. In this paper we consider small…

We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-\Delta u+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of…

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz…

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-\Delta)^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}$ with an isolated singularity, where $\sigma\in (0,1)$. We prove that…

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation $$(-\Delta)^s u + \mu u = (I_{\alpha}*F(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$, $N\geq 2$, $\alpha \in…

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…

In this paper, we establish the following Liouville theorem for fractional \emph{p}-harmonic functions. {\em Assume that $u$ is a bounded solution of $$(-\lap)^s_p u(x) = 0, \;\; x \in \mathbb{R}^n,$$ with $0<s<1$ and $p \geq 2$. Then $u$…

We establish global well-posedness for the mass subcritical nonlinear fractional Schr\"odinger equation $$iu_t - (-\Delta)^\frac{\beta}{2} u+F(u)=0$$ having radial initial data in modulation spaces $M^{p,\frac{p}{p-1}}(\mathbb R^n)$ for $n…

We develop a theory of existence, uniqueness and regularity for a porous medium equation with fractional diffusion, $\frac{\partial u}{\partial t} + (-\Delta)^{1/2} (|u|^{m-1}u)=0$ in $\mathbb{R}^N$, with $m>m_*=(N-1)/N$, $N\ge1$ and $f\in…

We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The variable coefficient…

We investigate symmetry properties of solutions to equations of the form $$ -\Delta u = \frac{a}{|x|^2} u + f(|x|, u)$$ in R^N for $N \geq 4$, with at most critical nonlinearities. By using geometric arguments, we prove that solutions with…

In this paper we define a new operator $J$ for the study of $$ \Delta u +f(u)=0,\quad x\in R ^N, N> 2. $$ Using $J$ we can easily see some qualitative properties of the solutions, for example we can determine how many times $u$ changes…

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads $\partial_t u-F(u) (-\Delta)^{s/2} u+(-\Delta)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. We are…

We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we…

We establish the optimal regularity of solutions to the Neumann problem for the fractional Laplacian, $(-\Delta)^s u=h$ in $\Omega$, with the external condition $\mathcal N^s u=0$ in $\Omega^c$. For this, a key point is to establish a 1D…

In this work we consider the boundary blow-up problem $$ \left\{ \begin{array}{ll} \Delta u = f(u) & \hbox{in } B\\ \ \ u=+\infty & \hbox{on }\partial B \end{array} \right. $$ where $B$ stands for the unit ball of $\mathbb{R}^N$ and $f$ is…

In this paper we study the following nonlinear Choquard equation $$ -\Delta u+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$…