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Register a new userOn the well-posedness of the time-fractional diffusion equation with Robin boundary condition
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The diffusion system with time-fractional order derivative is of great importance mathematically due to the nonlocal property of the fractional order derivative, which can be applied to model the physical phenomena with memory effects. We consider an initial-boundary value problem for the time-fractional diffusion equation with inhomogenous Robin boundary condition. Firstly, we show the unique existence of the weak/strong solution based on the eigenfunction expansions, which ensures the well-posedness of the direct problem. Then, we establish the Hopf lemma for time-fractional diffusion operator, generalizing the counterpart for the classical parabolic equation. Based on this new Hopf lemma, the maximum principles for this time-fractional diffusion are finally proven, which play essential roles for further studying the uniqueness of the inverse problems corresponding to this system.
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In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term $t^{-alpha}$ as $ttoinfty$.
The Vlasov-Poisson-Boltzmann equation is a classical equation governing the dynamics of charged particles with the electric force being self-imposed. We consider the system in a convex domain with the Cercignani-Lampis boundary condition. We construct a uniqueness local-in-time solution based on an $L^infty$-estimate and $W^{1,p}$-estimate. In particular, we develop a new iteration scheme along the characteristic with the Cercignani-Lampis boundary for the $L^infty$-estimate, and an intrinsic decomposition of boundary integral for $W^{1,p}$-estimate.
Let $M$ be a smooth manifold with boundary $partial M$ and bounded geometry, $partial_D M subset partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $partial M smallsetminus partial_D M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f mbox{ in } M, u = 0 mbox{ on } partial_D M, partial^P_ u u + bu = 0 mbox{ on } partial M setminus partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s geq 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincar{e} inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a emph{uniform Agmon condition} and show that it is equivalent to the Gaa rding inequality. This extends an important result of Agmon (1958).
A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is established via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schrodinger, Korteweg-de Vries and good Boussinesq equations. Thus, the present work extends the recently introduced unified transform method approach to well-posedness from dispersive equations to diffusive ones.
We prove global well-posedness for the microscopic FENE model under a sharp boundary requirement. The well-posedness of the FENE model that consists of the incompressible Navier-Stokes equation and the Fokker-Planck equation has been studied intensively, mostly with the zero flux boundary condition. Recently it was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl. Math., 68(5):1304--1315] that any preassigned boundary value of a weighted distribution will become redundant once the non-dimensional parameter $b>2$. In this article, we show that for the well-posedness of the microscopic FENE model ($b>2$) the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function. Under this condition, it is shown that there exists a unique weak solution in a weighted Sobolev space. Moreover, such a condition still ensures that the distribution is a probability density. The sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero as fast as the distance function.
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