In this paper, we examine the unique solvability (in sense of strong solutions) Cauchy problem for a linear inhomogeneous equation in Banach space solved with respect to Caputo fractional derivative. We assume that operator acting on unknown function right-hand side generates an analytic resolving family corresponding homogeneous equation. obtain representation solution and optimal control prob...

In this paper we investigate the exact peroidic wave shock solutions of the Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr ε and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock type solution for the cauchy problem of the in...

In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field V on an open subset of C and a discrete set Y ⊂ R with n elements, we define a nonlinear operator ∆V,Y on L (R) which coincides with the free Laplacian when restricted to regular functions vanishing at Y , and which reduces to the usual Laplacian with point interactions placed ...

The Huygens' property is exploited to study propagation relations for solutions of certain types of linear higher order Cauchy problems. Motivated by the solution properties of the abstract wave problem, addition formulas are developed for the solution operators of these problems. The application of these alternative forms of the solution operators to data leads to connecting operator relations...

We study the Cauchy problem for a system of two coupled nonlinear focusing Schrödinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we give a bound, depending on the coupling parameter, for the blow-up threshold. 2000 Mathematics Subject Classif...

In this work we consider the abstract Cauchy problem with Caputo fractional time derivative of order $$\alpha \in (0,1]$$ , and discuss continuity respective solutions regarding parameter $$ . We also present a study about Mittag-Leffler families operators (for ), when they are induced by sectorial operators.