Related papers: Space Regularity of Evolution Equations Driven by …

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…

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\ge 0, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need…

We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending…

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy…

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…

Given the abstract evolution equation \[ y'(t)=Ay(t),\ t\in \mathbb{R}, \] with scalar type spectral operator $A$ in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a…

We consider the Cauchy problem for a second-order evolution equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the…

We study the convergence of semilinear parabolic stochastic evolution equations, posed on a sequence of Banach spaces approximating a limiting space and driven by additive white noise projected onto the former spaces. Under appropriate…

We study conditions for the well-posedness of nonautonomous perturbation of evolution equations of the form \[ u'(t)=(A+B(t))u(t), \quad t \in [a,b], \] where $A$ generates a $\mathrm{C}_0$-semigroup $\left (T(t)\right )_{t\ge 0}$ with $\|…

This paper revisits the H\"{o}lder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces $L^p(\mathcal{O}),$ with $p\geq 2$ and $\mathcal{O}\subset\mathbb{R}^d$ a bounded domain. We find conditions on $p,…

We prove a new linearization principle for the nonlinear stability of solutions to semilinear evolution equations of parabolic type. We assume that the set of equilibria forms a finite dimensional manifold of normally stable and normally…

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 study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -\Delta + (-\Delta)^{\alpha/2} \), where \( 0 < \alpha < 2 \). The Cauchy problem under…

In this study, we analyze a semilinear damped evolution equation under different damping conditions, including the undamped $(\theta=0)$, effectively damped $(0<2\theta<\sigma)$, critically damped $(2\theta=\sigma)$, and non-effectively…

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…

I prove the bistability of linear evolution equations $x' = A(t)x$ in a Banach space $E$, where the operator-valued function $A$ is of the form $A(t) = f'(t)G(t,f(t))$ for a binary operator-valued function $G$ and a scalar function $f$. The…

This paper addresses the problem of wellposedness of non-autonomous linear evolution equations $\dot x = A(t)x$ in uniformly convex Banach spaces. We assume that $A(t):D \subset X\to X$, for each $t$ is the generator of a quasi-contractive…

We study a stochastic linear evolution equation $dX+A(t)Xdt=F(t)dt+ G(t)dw_t$ in a Banach space of M-type 2. We construct unique strict solutions to the equation on the basis of the theory of deterministic linear evolution equations. The…

We prove $L^2$-maximal regularity of linear non-autonomous evolutionary Cauchy problem \begin{equation}\label{eq00}\nonumber \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator…

This paper is devoted to studying stochastic parabolic evolution equations with additive noise in Banach spaces of M-type 2. We construct both strict and mild solutions possessing very strong regularities. First, we consider the linear…