Related papers: Quasilinear parabolic stochastic partial different…
In this paper we consider stochastic differential equations with discontinuous diffusion coefficient of varying sign, for which weak existence and uniqueness holds but strong uniqueness fails. We introduce the notion of $\varphi $-strong…
In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the…
In this paper, we study the existence, stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise. We first prove the existence and uniqueness of tempered random…
The solutions to a large class of semi-linear parabolic PDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of…
In this paper, we study the existence and uniqueness of periodic solutions of the differential equation of the form . Here, we obtain some sufficient conditions which guarantee the existence of periodic solutions. This equation is a quite…
We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved…
We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on…
We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of…
We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.
Doubly nonlinear stochastic evolution equations are considered. Upon assuming the additive noise to be rough enough, we prove the existence of probabilistically weak solutions of Friedrichs type and study their uniqueness in law. This…
In this paper we analyze the existence of large positive radial solutions to some quasilinear elliptic systems. Also, a non-radially symmetric solution is obtained by using a lower and upper solution method. The equations are coupled by…
We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the…
In this paper, we study the solvability problem for one kind of fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es). With the help of the necessary and sufficient condition for the solvability of the linear…
We prove a maximum principle for local solutions of quasilinear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of It\^o's formula and estimates for the positive part of a local solution which is…
We prove some existence, uniqueness and non-existence results of stochastic strong solutions for a class of stochastic transport equations with a $q$-integrable (in time), bounded and $\alpha$-H\"{o}lder continuous (in space) drift…
We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative…
We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under…
In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial…
For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset \mathbb {R}^d$ and driven by an $L^2(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new…
This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence…