Related papers: Energy Estimates for Fractional Evolution Equation…
We study a time fractional heat equation in a noncylindrical domain. The problem is one-dimensional. We prove existence of properly defined weak solutions by means of the Galerkin approximation.
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…
We consider an evolution equation involving the fractional powers, of order $s \in (0,1)$, of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order $\gamma \in (1,2]$. Since it has been…
Using the Galerkin method, we obtain the unique existence of the weak solution to a time fractional wave problem, and establish some regularity estimates which reveal the singularity structure of the weak solution in time.
We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As the main…
In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time…
We will give some regularity results about fractional diffusion-wave equations.
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity.…
We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…
Space-time fractional evolution equations are a powerful tool to model diffusion displaying space-time heterogeneity. We prove existence, uniqueness and stochastic representation of classical solutions for an extension of Caputo evolution…
In this article, we investigate the existence and properties of time-periodic solutions for damped evolutionary partial differential equations subject to periodic forcing. Particular emphasis is placed on configurations where the energy…
We introduce a notion of weak solution for abstract fractional differential equations, motivated by the definition of Caputo derivative. We prove existence results for weak and strong solutions. We also give two examples as application of…
This manuscript is dedicated to prove a new inequality that involves an important case of Leibniz rule regarding Riemann-Liouville and Caputo fractional derivatives of order $\alpha\in(0,1)$. In the context of partial differential…
In this work, we introduce a novel variational framework for the study of the unsteady Stokes equations in a bounded open Lipschitz domain in R^n, involving a Caputo fractional derivative in time. The nonlocal nature of the fractional…
A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…
Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology…
We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a~priori bounded but the heat flux provides merely \mbox{$L^1$-coercivity}.…
Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving…
We consider an evolution equation whose time-diffusion is of fractional type and we provide decay estimates in time for the $L^s$-norm of the solutions in a bounded domain. The spatial operator that we take into account is very general and…
A class of linear evolutionary equations with material laws involving fractional time-derivatives is considered. The main result is well-posedness and causality for this problem class. The approach is illustrated with two examples: a…