Marko Nedeljkov

Singular shock waves in interactions

In a number of papers it has been shown that there exist one-dimensional systems such that they contain solutions with so-called overcompressive singular shock waves besides the usual elementary waves (shock and rarefaction waves as well as contact discontinuities). One can see their definition for a general 2 × 2 system with fluxes linear in one of the dependent variables in Nedeljkov, Delta and singular delta locus for one dimensional systems of conservation laws, Math. Method Appl. Sci. 27 (2004), 931-955. This paper is devoted to examining their interactions with themselves and elementary waves. After a discussion of systems given in a general form, a complete analysis will be given for the ion-acoustic system given in Keyfitz and Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differ. Equations 118 (1995), no. 2, 420-451.

Heat equation with singular potential and singular data

Nets of Schrodinger C 0 -semigroups ( S e ) e with the polynomial growth with respect to e are used for solving the Cauchy problem (∂ t − Δ) U + VU = f ( t, U ), U (0, x ) = U 0 ( x ) in a suitable generalized function algebra (or space), where V and U 0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

HYPOELLIPTIC DIFFERENTIAL OPERATORS WITH GENERALIZED CONSTANT COEFFICIENTS

The space Q of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter e. There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on e and behave like powers of e as e -* 0. The solutions of such family of equations should also satisfy the power order estimate with respect to e.

Generalized solution to a semilinear hyperbolic system with a non-Lipshitz nonlinearity

AbstractLet(1)

Dynamics of a Fractional Derivative Type of a Viscoelastic Rod With Random Excitation

(\partial _t + \Lambda (x,t)\partial _x )y(x,t) = F(x,t,y(x,t)),y(x,0) = A(x)

The linear theory of Colombeau generalized functions

be a semilinear hyperbolic system, whereA is a real diagonal matrix and a mappingy→F(x, t, y) is in

Interactions of delta shock waves in a strictly hyperbolic system of conservation laws

\mathcal{O}_M (\mathbb{C}^n )