When an integrable Hamiltonian system, possessing an m-resonant lower dimensional normally parabolic torus is perturbed, a parabolic m-resonance occurs. If, in addition, the iso-energetic non-degeneracy condition for the integrable system fails, the near integrable Hamiltonian exhibits a at parabolic m-resonance. It is established that most kinds of parabolic resonances are persistent in n (n 3...

We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst–Planck and the Debye–Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and o...

We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst–Planck and the Debye–Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and o...

Abstract. Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on Rn. In particular, in the case when n = 2 they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second o...

We prove local in time solvability of the nonlinear initial-boundary problem to nonlinear nondiagonal parabolic systems of equations (multidimensional case). No growth restrictions are assumed on generating the system functions. In the case of two spatial variables we construct the global in time solution to the Cauchy-Neumann problem for a class of nondiagonal parabolic systems. The solution i...

When an integrable Hamiltonian system, possessing an m-resonant lower dimensional normally parabolic torus is perturbed, a parabolic m-resonance occurs. If, in addition, the isoenergetic nondegeneracy condition for the integrable system fails, the near integrable Hamiltonian exhibits a at parabolic m-resonance. It is established that most kinds of parabolic resonances are persistent in n (n 3) ...

A parabolic resonance is formed when an integrable two-degrees-of-freedom (d.o.f.) Hamiltonian system possessing a circle of parabolic fixed points is perturbed. It is proved that its occurrence is generic for one parameter families (co-dimension one phenomenon) of near-integrable, two d.o.f. Hamiltonian systems. Numerical experiments indicate that the motion near a parabolic resonance exhibits...

This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study. AMS MSC 2010: 65N30, 65M60, 65K1...