Related papers: Regularity of solutions of abstract linear evoluti…
Since the pioneering works by Aronson & B\'enilan [C. R. Acad. Sci. Paris S\'er., 1979] and B\'enilan & Crandall [Johns Hopkins Univ. Press, 1981], it is well-known that first-order evolution problems governed by a nonlinear but homogeneous…
We prove existence of strongly continuous evolution systems in L^2 for Schroedinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations…
Our GBDT (generalised B\"acklund-Darboux transformation) approach is used to construct explicit solutions of the focusing nonlinear Schr\"odinger (NLS) equation in the case of the exponential seed $a \exp\{2 i (cx +dt)\}$. The corresponding…
We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We…
In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps $F$ between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames,…
We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form $u'=A(t)u+\epsilon H(t,u)+f(t)$, where $A(t)$ is, in general, an unbounded operator depending 1-periodically on $t$, $H$ is 1-periodic…
Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the…
In this paper, we study the regularity of solutions to a linear elliptic equation involving a mixed local-nonlocal operator of the form $$Lu - \operatorname{div}\big(a(x)\nabla u(x)\big)= f, \quad \text{in } \Omega \subset \mathbb{R}^n,$$…
Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with a scalar type spectral operator $A$ in a complex Banach space, we find conditions on $A$, formulated exclusively in terms of the location of its spectrum in the…
Existence and uniqueness of a strong solution in $H^{-1}(\mathbb R^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\rm div}(DX)dt-\Delta\beta(X)dt=X\,dW \mbox{ in }(0,T)\times\mathbb R^d,\ X(0)=x,$$ via a…
We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a…
We consider evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+ A(t)u(t)=0,\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form…
A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution…
In this paper, we deal with the existence of $\omega$-periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space $E$ $$u'(t)+Au(t)=F(t,u(t),u(t-\tau)),\ \ \ \ t\in\R,$$ where $A:D(A)\subset…
In the paper regularity of solutions to stochastic Volterra equations in a separable Hilbert space is studied. Sufficient conditions for the temporal and spatial regularity of stochastic convolutions corresponding to the equations under…
In an abstract Banach space we study conditions for the existence of piecewise continuous, almost periodic solutions for semi-linear impulsive differential equation with fixed and non-fixed moments of impulsive action
This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix $\tau > 0$ and let $\left(A(t)\right)_{t \in [0,\tau]} \subset \mathcal{L}(D,X)$, where $D$ and $X$ are two Banach spaces such that $D$…
In this paper we establish local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions. An analogous result is…
We study generalized solutions of an evolutionary equation related to a densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and find…
In this paper, we discuss the well-posedness of the Cauchy problem associated with the third-order evolution equation in time $$ u_{ttt} +A u + \eta A^{\frac13} u_{tt} +\eta A^{\frac23} u_t=f(u) $$ where $\eta>0$, $X$ is a separable Hilbert…