Related papers: Self-similar solutions for the Muskat equation
In the present paper, we study the existence and uniqueness of solutions to some nonlocal singular elliptic problem under Dirichlet boundary condition. Problem is settled in Musielak-Sobolev spaces.
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
Existence of distributional solutions of a modified Surface Quasi-Geostrophic equation (mSQG) is proven for $\mu$-almost every initial condition, where $\mu$ is a suitable Gaussian measure. The result is the by-product of existence of a…
We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega,…
We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.
We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x = \eta_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are in divergence form with real, bounded, measurable coefficients that vary in the…
This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
We prove that for a given smooth initial value, if the finite element solution of the three-dimensional Navier-Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier-Stokes equations…
This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions…
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish…
Building upon the pioneering work [Merle, Rapha\"el, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] we construct exact, smooth self-similar imploding…
It is a well known fact that the eikonal equation is well posed in complete length spaces. Among the studied notions of solutions in the literature, there is one that can be defined in any metric space using the local (descent) slope and…
We consider the Swift-Hohenberg equation on manifolds with conical singularities and show existence, uniqueness and maximal regularity of the short time solution in terms of Mellin-Sobolev spaces. Moreover, we give a necessary and…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to…
This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $\Sigma$ with smooth boundaries $\partial\Sigma$. We establish the existence of solutions when the parameters are not…
This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak…
The existence of a positive solution to a class of Choquard equations with potential going at a positive limit at infinity possibly from above or oscillating is proved. Our results include the physical case and do not require any symmetry…
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the…