Related papers: Sharp energy estimates for nonlinear fractional di…
We consider the partial differential equation $$ u-f={\rm div}\left(u^m\frac{\nabla u}{|\nabla u|}\right) $$ with $f$ nonnegative and bounded and $m\in\mathbb{R}$. We prove existence and uniqueness of solutions for both the Dirichlet…
The survey is devoted to numerical solution of the fractional equation $A^\alpha u=f$, $0 < \alpha <1$, where $A$ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain…
We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-\Delta)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and…
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…
This paper provides sharp quantitative and constructive estimates of nonnegative solutions $u(t,x)\geq 0$ to the nonlinear fractional diffusion equation, $$\partial_t u +{\mathcal L} F(u)=0,$$ also known as filtration equation, posed in a…
We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the…
We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in dimension $d \in \{3,4,5\}$ and prove the exterior scattering of solutions if $3\leq d \leq 5$ and $1+6/d<p<1+4/(d-2)$. More…
We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-\Delta)^s u = \chi_{\{u>c\}} \] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free…
Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential…
A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law $\partial_t \alpha + \mathrm{div}(\boldsymbol{u}f(\alpha)) = 0$ in two and three spatial dimensions with an initial data of…
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the…
We consider the drift-diffusion equation $u_t-\epsilon\Delta u + \nabla \cdot(u\nabla K^*u)=0$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case…
We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\infty)\to\mathbb{R}$ is a locally…
In this paper, we study the defocusing energy-critical nonlinear Schr\"odinger equations $$ i\partial_t u + \Delta u = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in…
In this paper, the sharp threshold of scattering for the fractional nonlinear Schr\"{o}dinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\frac{4s}{N}<p<1+\frac{4s}{N-2s}$, and $$…
This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where…
In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-H\'enon equation $$ (-\Delta)^\sigma u = |x|^\alpha u^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin,…
We study vector minimizers u of the Allen-Cahn functional with potentials possessing N global minima defined on bounded domains, with certain geometrical features and Dirichlet conditions on the boundary. We derive a sharp lower bound for…
Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…
This paper studies the inhomogeneous fractional Sch\"odinger equation $$i\dot u-(-\Delta)^s u=\pm(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to…