偏微分方程分析

We study the solvability of some linear nonhomogeneous elliptic problems and establish that under certain technical assumptions the convergence in $L^2$ of their right-hand sides yields the existence and the convergence in $H^4$ of the…

In this paper, we focus on a new type of non-linear kinetic Fokker-Planck equation where the non-linearity comes from a non-linear diffusion in the velocity variable. The existence of solutions in suitable Lebesgue spaces is proved,…

We mainly consider a Liouville-type problem for the three dimensional stationary fractional Navier-Stokes equations with arbitrary asymptotic state $u_\infty$ at infinity. When $u_\infty\neq 0$ and $\frac{1}{2}\leq s<1$, we prove a complete…

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in $\mathbb{R}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\geq 3$, we…

We study Ornstein--Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The…

This paper is concerned with the existence of non-radial positive classical solutions for the critical H\'enon equation \[ -\Delta u=|x|^\alpha u^{\frac{N+2+2\alpha}{N-2}} \qquad \text{in }\mathbb R^N, \] where \(\alpha>0\) and \(N\ge3\),…

We develop a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation $$\partial_tu-\Delta u=|u|^p+\mathbf{w}(x), \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0, \cdot)=u_0.$$ The…

In this paper, we establish a comparison principle for non-negative weak solutions to a class of doubly nonlinear parabolic fractional partial differential equations within a space-time cylinder…

This paper investigates the long-times behavior of the Fisher-KPP equation with slowly decaying initial data in an almost periodic medium. We mainly focus on two classes of initial data: exponentially decaying initial data and inital data…

In this paper, we prove estimates for the Stokes resolvent problem with no-slip boundary conditions on the half space in weighted $L^p$-spaces. The weights we consider are power weights both inside and outside the Muckenhoupt range. Our…

This paper focuses on the analysis of an initial-boundary value (direct) problem for the Hallaire-Luikov moisture transfer equation involving the $\psi$-Prabhakar integral-differential operator of fractional order. We establish the…

We provide an upper estimate \`a la Pleijel on the asymptotic number of nodal domains for eigenfunctions corresponding to the cogenus eigenvalues $\{\lambda_k(p;\Omega)\}$ of the $p$-Laplacian in a bounded domain $\Omega$, and identify…

We present a comparison principle for unbounded viscosity solutions to Hamilton-Jacobi equations on Wasserstein spaces of probability measures over $R^d$ . In addition to the use of standard techniques of viscosity solutions, our approach…

We consider the one-phase free boundary problem for the incompressible Navier-Stokes equations in $\mathbb{R}^d$ ($d\ge2$). The surface tension is taken into account. The initial domain, which is the outside of a bubble, is an exterior…

This paper establishes a mathematical framework for nonlinear subwavelength resonances and bound states in the continuum (BIC) in an acoustic metascreen with a cubic Kerr nonlinearity. We first use the quasiperiodic Dirichlet-to-Neumann…

In this article, we investigate the local well-posedness of the nonlinear Schr\"odinger equation on the two-dimensional sphere $\mathbb{S}^2$: \begin{align*} i\partial_tu+\Delta_{g}u=F(u). \end{align*} The nonlinearity $F(u)$ is assumed to…

In this paper, we construct several explicit examples of singular sets of Hausdorff dimension $(n-2)$ in $\mathbb{R}^n$ on free boundaries for an elliptic system modeling long range segregation. The system has been previously studied by…

We propose a variational approach to principal spectral values of non-selfadjoint operator pencils $\mathcal L u=\lambda\mathcal G u$, where the weight operator $\mathcal G$ may be singular. The aim is to obtain Rayleigh-type minimax…

This paper addresses the inverse problem of simultaneously recovering the fractional order $\alpha \in (0,1)\cup (1,2)$ and the time-dependent source factor $p(t)$ in the Cauchy problem for an evolution equation with a general self-adjoint…

We consider the $L^2$-critical nonlinear Hartree equation in $\mathbb{R}^{1+4}$ and multisoliton solutions for which the trajectories are approximated to leading order by an $m$-body law. We obtain soliton clusters asymptotically following…