Maximum principle for the generalized time-fractional diffusion equation

In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.     

Analysis of a Multi-Term Variable-Order Time-Fractional Diffusion Equation and Its Galerkin Finite Element Approximation

In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time $t = 0$. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to $C^2([0,T])$ in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimal-order convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.     

Absolute uniqueness of minimal surfaces bounded by contours with a one-to-one projection onto a plane

In this paper the uniqueness of the always existing Douglas-Radó solution to Plateau’s Problem of finding a minimal surface of the type of the disc to a given rectifiable Jordan Curve Γ in ?³ is considered in combination with the question: When is the solution surface also a graph? So only contours with a one-to-one projection onto a plane are candidates. The uniqueness is generalized to all genera and orientations and the non orientable case and then called “absolute”. It is demonstrated, that if it we have a continuous family of catenoids as support surfaces or barriers at the non convex projected part of the C ² contour, assured by the non negativity of a function of the radii of the necks of the catenoids and the contour, the Douglas-Radó solution is absolute unique and a graph. Especially there are no further restrictions to the projection of the contour than C ²-smoothness. As easy first consequences, the earlier result of Sauvigny, using Scherk’s surface as support surfaces for certain contours with a non convex projection, is generalized to absolute uniqueness and the problem initiating, famous uniqueness and existence result in case of a convex projection of Radó, Kneser and Meeks is also carried over to absolute uniqueness. Finally, by a generic example using our mentioned main fruit of investigation, the non necessity of any curvature bound, any bound on function norms, any height bound, and Williams local tangential Lipschitz condition for the whole contour for the existence of a solution for the minimal surface equation or uniqueness of parametric minimal surfaces is shown. This covers more than all known sufficient conditions, i.e. the well known 4π curvature bound of Nitsche, Sauvigny’s uniform concavity-, Lau’s smallness in the C ^0,1-norm condition and Meeks restriction of the deviation from the plane in the C ²-norm. The article is strongly geometrical in spirit so enabling the use of geometrical imagination and intuition in the realms of results and techniques from many different, including very advanced and abstract, mathematical fields. As the proofs are very complete, transparent and proceed in little steps, covering also new elementary results, the article is predestined for educational employment. Especially for its use of the maximum principle in a very sophistcated, long run, merged with topology, way and the new setting of surfaces with a Riemannian structure and usual derivations for handling the non-orientable case.     

Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana–Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana–Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial–boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.     

Existence and Uniqueness for Some 3rd Order Dissipative Problems with Various Boundary Conditions

We briefly present new results on the existence and uniqueness of solutions u of a large class of initial-boundary-value problems characterized by a quasi-linear third order equation on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in Superconductor Theory and in the Theory of Viscoelastic Materials.     

Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in L ₁(R) on each interval.     

GENERALIZED SOLUTION OF SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

The singularly perturbed initial boundary value problem for reaction diffusion equation is considered. Under suitable conditions the existence, uniqueness and asymptotic behavior of the generalized solution for the problems are studied.     

A generalized quasi-boundary value method for the backward semi-linear time-fractional heat problem in a cylinder

We are concerned with a backward problem associated with a semi-linear time-fractional heat equation in an axis-symmetric cylinder, which arises from the modeling of the blast furnace steelmaking in metallurgy. Under some assumptions, the existence and uniqueness of the solution to the semi-linear problem is first established. The ill-posedness of the backward problem is then established, and we obtain the error estimates by a generalized quasi-boundary value regularization method. Finally, the numerical experiment is presented to demonstrate the effectiveness of the proposed method.     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

Stochastic partial differential equations with unbounded and degenerate coefficients

In this article, using DiPerna-Lions theory (DiPerna and Lions, 1989) [1], we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the L¹-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.     

一非线性双曲型方程解的爆破

本文应用压缩映射原理证明一非线性双曲型方程的初边值问题存在唯一局部广义解，并给出此问题的广义解爆破的充分条件。     

Mixed Problem for an Ultraparabolic Equation in Unbounded Domain

We investigate a mixed problem for a nonlinear ultraparabolic equation in a certain domain Q unbounded in the space variables. This equation degenerates on a part of the lateral surface on which boundary conditions are given. We establish conditions for the existence and uniqueness of a solution of the mixed problem for the ultraparabolic equation; these conditions do not depend on the behavior of the solution at infinity. The problem is investigated in generalized Lebesgue spaces.     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process

Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.     

On 2nth-order Lidstone boundary value problems

This paper is concerned with the solutions of a class of 2nth-order Lidstone boundary value problems. Sufficient conditions for the existence and uniqueness of a solution are given. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution. The approach to the problem is by the method of upper and lower solutions with a new maximum principle.     

Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time dependent coefficients. New uniqueness results are formulated in Theorem 3.1. We also show optimality of the conditions under which uniqueness holds by explicitly constructing counterexamples, that is by constructing more than one solution in the case when the conditions for uniqueness are violated.     

A priori estimates of solutions of linear parabolic problems with coefficients from Sobolev spaces

We consider the general initial-boundary-value problem for a linear parabolic equation of arbitrary even order in anisotropic Sobolev spaces. We prove the existence and uniqueness of a solution and establish ana priori estimate for it. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1534–1548, November, 1999.     

Optimal solvability for a nonlocal problem at critical growth

We provide optimal solvability conditions for a nonlocal minimization problem at critical growth involving an external potential function a. Furthermore, we get an existence and uniqueness result for a related nonlocal equation.     

Generalized Reflected BSDE and an Obstacle Problem for PDEs with a Nonlinear Neumann Boundary Condition

Abstract

In this article, we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equation involving the integral with respect to a continuous process, which is the local time of the diffusion on the boundary, in using the penalization method. We also give a characterization of the solution as the value function of an optimal stopping time problem. Then we give a probabilistic formula for the viscosity solution of an obstacle problem for PDEs with a nonlinear Neumann boundary condition.     

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.