Reviews in Mathematical Physics

We consider time-dependent perturbations by unbounded potentials of Schrödinger operators with scalar and magnetic potentials which are almost critical for the selfadjointness. We show that the corresponding time-dependent Schrödinger equations generate a unique unitary propagator if perturbations of scalar and magnetic potentials are differentiable with respect to the time variable and they increase at the spatial infinity at most quadratically and at most linearly, respectively, where both have mild local singularities. We use time-dependent gauge transforms and apply Kato's abstract theorem on evolution equations.

For monochromatic solutions of D'Alembert's wave equation and Maxwell's equations, we obtain sharp bounds on the sup norm as a function of the far field energy. The extremizer in the scalar case is radial. In the case of Maxwell's equation, the electric field maximizing the value at the origin follows longitude lines on the sphere at infinity. In dimension d = 3, the highest electric field for Maxwell's equation is smaller by a factor 2/3 than the highest corresponding scalar waves.

The highest electric field densities on the balls B_R(0) occur as R → 0. The density dips to half max at R approximately equal to one third the wavelength. For these small R, the extremizing fields are identical to those that attain the maximum field intensity at the origin.

We look for "static" spherically symmetric solutions of Einstein's Equations for perfect fluid source with equation of state p = wρ, for constant w. We consider all four cases compatible with the standard ansatz for the line element, discussed in previous work. For each case, we derive the equation obeyed by the mass function or its analogs. For these equations, we find all finite-polynomial solutions, including possible negative powers.

For the standard case, we find no significantly new solutions, but show that one solution is a static phantom solution, another a black hole-like solution. For the dynamic and/or tachyonic cases we find, among others, dynamic and static tachyonic solutions, a Kantowski–Sachs (KS) class phantom solution, another KS-class solution for dark energy, and a second black hole-like solution.

The black hole-like solutions feature segregated normal and tachyonic matter, consistent with the assertion of previous work. In the first black hole-like solution, tachyonic matter is inside the horizon, in the second, outside.

The static phantom solution, a limit of an old one, is surprising at first, since phantom energy is usually associated with super-exponential expansion. The KS-phantom solution stands out since its "mass function" is a ninth order polynomial.

In this paper, a nonlinear stochastic viscoelastic wave equation with linear damping is considered. By an appropriate energy inequality and estimations, we show that the local solution of the stochastic equations will blow up with positive probability or explosive in L² sense under some sufficient conditions. Moreover, the upper bound of the blow-up time is given.

The Cauchy problems for nonlinear complex Ginzburg–Landau equations and nonlinear dissipative wave equations are considered in Sobolev spaces. The relation between the order of the nonlinear terms and the regularity of solutions is considered in terms of the scaling arguments, and the existence of local solutions and small global solutions is shown in Sobolev and Besov spaces.