Related papers: A weak Harnack inequality for fractional evolution…

We study the weak finite element method solving convection-diffusion equations. A weak finite element scheme is presented based on a spacial variational form. We established a weak embedding inequality that is very useful in the weak finite…

In this paper, we deal with the inverse source problem of determining a source in a time fractional diffusion equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution does not depend continuously on the…

We consider fractional diffusion equations and study the stability of the inverse problem of determining the time-dependent parameter in a source term or a coefficient of zero-th order term from observations of the solution at one point in…

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators. Our main results is a stochastic representation of…

We prove the existence and weak uniqueness of weak solutions of It\^o's stochastic time dependent equations with irregular diffusion and drift terms of Morrey class with mixed norms.

We prove the existence of weak solutions of It\^o's stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are…

We consider a family of Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t -> +infinity. It is well-known that, when the decay of…

We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency…

We study a fully discrete finite element method for variable-order time-fractional diffusion equations with a time-dependent variable order. Optimal convergence estimates are proved with the first-order accuracy in time (and second order…

We consider time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction…

In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is…

Fractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the…

By exploiting a suitable Trudinger-Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential…

We investigate existence and regularity of weak solutions of a 1-dimensional parabolic differential equation with a non-constant H\"older diffusion coefficient and a rough forcing term. Such an equation appears in studying the 1-dimensional…

Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are H\"{o}lder continuous, which also generalizes the classic…

In this paper we give a survey of results on various analytical aspects of time fractional diffusion equations. We describe the approach via abstract Volterra equations and collect results on strong solutions in the $L_p$ sense. We further…

We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space $R^n,n\geq 1$. Here, the fractional order $\alpha$ is related to the diffusion-type source term behaving as the usual diffusion term on the high…

In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate…

We revisit a well-established model for highly re-entrant semi-conductor manufacturing systems, and analyze it in the setting of states, in- and outfluxes being Borel measures. This is motivated by the lack of optimal solutions in the…

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of…