Abstract:
We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image of the unit square into the torus. We study the structure of the maximal invariant set, and in a generic case we prove the sensitive dependence on the initial conditions. We study the decay of correlations and the diffusion in the corresponding system on the plane. We also demonstrate how the rationality of the real numbers defining the map influences the dynamical properties of the system.

Abstract:
In this paper we consider $C^{1+\epsilon}$ area-preserving diffeomorphisms of the torus $f,$ either homotopic to the identity or to Dehn twists. We suppose that $f$ has a lift $\widetilde{f}$ to the plane such that its rotation set has interior and prove, among other things that if zero is an interior point of the rotation set, then there exists a hyperbolic $\widetilde{f}$-periodic point $\widetilde{Q}$$\in {\rm I}\negthinspace {\rm R^2}$ such that $W^u(\widetilde{Q})$ intersects $W^s(\widetilde{Q}+(a,b))$ for all integers $(a,b)$, which implies that $\bar{W^u(\widetilde{Q})}$ is invariant under integer translations. Moreover, $\bar{W^u(\widetilde{Q})}=\bar{W^s(\widetilde{Q})}$ and $\widetilde{f}$ restricted to $\bar{W^u(\widetilde{Q})}$ is invariant and topologically mixing. Each connected component of the complement of $\bar{W^u(\widetilde{Q})}$ is a disk with uniformly bounded diameter. If $f$ is transitive, then $\bar{W^u(\widetilde{Q})}=$${\rm I}\negthinspace {\rm R^2}$ and $\widetilde{f}$ is topologically mixing in the whole plane.

Abstract:
We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Green's function, integral representation of solutions, the method of upper and lower solutions, and fixed point theorems for multivalued operators.

Abstract:
We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Green's function, integral representation of solutions, the method of upper and lower solutions, and fixed point theorems for multivalued operators.

Abstract:
We will show that the same type of estimates known for the fundamental solutions for scalar parabolic equations with smooth enough coefficients hold for the first order derivatives of fundamental solution with respect to space variables of scalar parabolic equations of divergence form with discontinuous coefficients. The estimate is very important for many applications. For example, it is important for the inverse problem identifying inclusions inside a heat conductive medium from boundary measurements.

Abstract:
This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for a linear non-stationary convection-diffusion problem that is discretised with a backward Euler dG method. An adaptive algorithm is then proposed to utilise the error estimator. The effectiveness of both the error estimator and the proposed algorithm is shown through a series of numerical experiments. Moving on to nonlinear problems, we investigate the numerical approximation of blow-up. To begin this study, we first look at the numerical approximation of blow-up in nonlinear ODEs through standard time stepping schemes. We then derive an a posteriori error estimator for an implicit-explicit (IMEX) dG discretisation of a semilinear parabolic PDE with quadratic nonlinearity. An adaptive algorithm is proposed that uses the error estimator to approach the blow-up time. The adaptive algorithm is then applied in a series of test cases to gauge the effectiveness of the error estimator. Finally, we consider the adaptive numerical approximation of a nonlinear interface problem that is used to model the mass transfer of solutes through semi-permiable membranes. An a posteriori error estimator is proposed for the IMEX dG discretisation of the model and its effectiveness tested through a series of numerical experiments.

Abstract:
The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. In this paper, existence and regularity results are obtained under some Cordes type restrictions on the coefficients. The results are applied to diffusion processes.

Abstract:
We compare two combinatorial definitions of parabolic sets of roots. We show that these definitions are equivalent for simple finite dimensional Lie algebras, affine Lie algebras, and toroidal Lie algebras. In contrast, these definitions are not always equivalent for simple finite dimensional Lie superalgebras.

Abstract:
This paper is concerned with a weakly coupled system of quasilinear parabolic equations where the coefficients are allowed to be discontinuous and the reaction functions may depend on continuous delays. By the method of upper and lower solutions and the associated monotone iterations and by difference ratios method and various estimates, we obtained the existence and uniqueness of the global piecewise classical solutions under certain conditions including mixed quasimonotone property of reaction functions. Applications are given to three 2-species Volterra-Lotka models with discontinuous coefficients and continuous delays.