Coupling local and nonlocal evolution equations

Nonlocal higher order evolution equations

In this paper we study the asymptotic behavior of solutions to the nonlocal operator ut(x, t) = (−1) n−1 (J ∗ Id − 1)n (u(x, t)), x ∈ R which is the nonlocal analogous to the higher order local evolution equation vt = (−1)(∆)v. We prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. Moreover, we prove that the solutions of the nonlocal problem conver...

Nonlocal Impulsive Cauchy Problems for Evolution Equations

Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolution equations. Combining the techniques of operator semigroups, approximate solutions, noncompact measures and the fixed point theory, new existence theorems are obtained, which generalize and improve some previous results since neither the Lipschitz continuity nor compactness assumption on the impulsive fu...

Nonlocal Symmetries of Systems of Evolution Equations

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of evolution equations having non-trivial Lie symmetry. Next, we modify the above method to generate more general nonlocal symmetries, which yields a purely al...

LOCAL AND NONLOCAL WEIGHTED p-LAPLACIAN EVOLUTION EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

Transformations of Ordinary Differential Equations: Local and Nonlocal Symmetries

The brief review of new methods of factorization, autonomization and exact linearization of the ordinary differential equations is represented. These methods along with the method of the group analysis based on using both point and nonpoint, local and nonlocal transformations are effective tools for study of nonlinear autonomous and nonautonomous dynamical systems. Thus a scope of exactly solva...