We consider the Cauchy problem for Klein–Gordon equation in $${\mathbb{R}}^{d},d\ge 1,$$ with random initial data. introduce a family of measures $${\mu }_{0}^{\varepsilon },\varepsilon >0$$ depending on small parameter ε. The }$$ are assumed to be locally homogeneous or slowly varying under spatial shifts order o(ε−1) and inhomogeneous ε−1. Moreover, correlation functions decrease uniformly ε ...

We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.

We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on R in a class of Banach spaces whose norms only depend on the size of the space-time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which ca...

in this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional cauchy problem with an integral initial condition in banach spaces.

A generalized version of Chernoff’s theorem has been obtained. Namely, the version of Chernoff’s theorem for semigroups obtained in a paper by Smolyanov, Weizsäcker, and Wittich [1] is generalized for a time-inhomogeneous case. The main theorem obtained in the current paper, Chernoff’s theorem for evolution families, deals with a family of time-dependent generators of semigroups At on a Banach ...

We investigate time dependent parabolic problems of diiusion type on open subsets of R N+1 and on networks, where the domains are possibly unbounded or non-cylindrical. The coeecients are assumed to be continuous and may be singular or degenerate at the boundary. We are looking for solutions which belong locally to suitable Sobolev spaces and vanish at the boundary. The well-posedness of the ho...