Related papers: Schauder regularity results in separable Hilbert s…
The time dependent complex Schr\"odinger equation with cubic nonlinearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction to a coupled system of real equations enables to approach the space…
In this article we investigate the solvability of infinite-dimensional differential algebraic equations. Such equations often arise as partial differential-algebraic equations (PDAEs). A decomposition of the state-space that leads to an…
We study a class of quasi-linear parabolic equations defined on a separable Hilbert space, depending on a small parameter in front of the second order term. Through the nonlinear semigroup associated with such equation, we introduce the…
We study the Cauhcy problem for space-time fractional nonlinear Schr\"odinger equation with a general nonlinearity. We prove the local well-posedness of it in fractional Sobolev spaces based on the decay estimates and H\"older type…
We establish the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group $\mathbb{G}$. Despite its singularity we manage to invert…
In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of…
We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal space-time master equations $(\partial_t +L)^su = f$ in $\mathbb{R} \times \Omega$, where $L$ is an elliptic operator in divergence form, subject to…
We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -\Delta_\gamma u + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $\Delta_\gamma $ denotes the so-called…
In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans-Krylov theorem for nonlocal fully nonlinear translation…
We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such…
We study semilinear evolution equations $ \frac {{\rm d} U}{{\rm d} t}=AU+B(U)$ posed on a Hilbert space ${\cal Y}$, where $A$ is normal and generates a strongly continuous semigroup, $B$ is a smooth nonlinearity from ${\cal Y}_\ell =…
In this paper we study the convergence of a Lie-Trotter operator splitting for stochastic semi-linear evolution equations in a Hilbert space. The abstract Hilbert space setting allows for the consideration of convergence of the…
We establish existence, uniqueness, and Sobolev and H\"older regularity results for the stochastic partial differential equation $$ du=\left(\sum_{i,j=1}^d a^{ij}u_{x^ix^j}+f^0+\sum_{i=1}^d f^i_{x^i}\right)dt+\sum_{k=1}^{\infty}g^kdw^k_t,…
We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and H\"older continuity of weak solutions to this…
This paper investigates the existence, uniqueness, and regularity of solutions to evolution equations with time-measurable pseudo-differential operators in weighted mixed-norm Sobolev-Lipschitz spaces. We also explore trace embedding and…
Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator in divergence form in a bounded domain $\Omega$. We consider the fractional nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\…
In this paper, we prove the unique existence and investigate the $L^{p}$-regularity of solutions to stochastic partial differential equations in Hilbert spaces associated with pseudo-differential operators, driven by Hilbert space-valued…
In this paper we begin exploring a local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold $$ -\mathrm{div}(|y|^aA(x,y)\nabla…
We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded…
This work presents a more broadly applicable version of an energy inequality for weak solutions of evolution equations involving fractional time derivatives. Unlike the classical identity that relates the time derivative of the squared norm…