Interest is directed to a moving boundary problem with a gradient flow structure which generalizes surface-tension driven Hele-Shaw flow to the case of nonconstant surface tension coefficient taken along with the liquid particles at the boundary. In the case with kinetic undercooling regularization well-posedness of the resulting evolution problem in Sobolev scales is proved, including cases in...

In this paper we consider implicit Runge–Kutta methods for the time-integration of linear Maxwell’s equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauß and Radau collocation methods. The results for the abstr...

1.3. De nition. ABSTRACT. We apply enriched category theory to study Cauchy completeness in continuity spaces. Our main result is the equivalence in continuity spaces of the category theoretic and the uniform notions of Cauchy completeness. This theorem, which generalizes a result of Lawvere for quasi-metric spaces, makes a natural connection between the category-theoretic and topological aspec...

Abstract. In this paper we discuss the existence of PC-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative. By utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a PC-mild solution for our problem is introduced. Our main techniques based on ...

Abstract. We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup ar...

Abstract. In the present paper we are concerned with the Novikov–Veselov equation at negative energy, i.e. with the (2+1)–dimensional analog of the KdV equation integrable by the method of inverse scattering for the two–dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non–singular scattering data behaves asymptotically a...

Abstract. Given a system of linear partial differential operators with constant coefficients whose affine algebraic varieties V (℘̌) have dimension 1, we establish in which classes of (small) Gevrey functions the associated Cauchy problem admits at least one solution, looking at the Puiseux series expansions on the branches at infinity of the algebraic curves V (℘̌). We focus, in particular, on t...