The amount of non-uniqueness for factored equations with Euler-Poisson-Darboux factors
Of concern are factored Euler-Poisson-Darboux equations of the type Πj=1N (d2/dt2 + (ρ/t)d/dt + Aj)u(t) = 0, where, for example, Aj = - cjΔ, Δ being the Dirichlet Laplacian acting on L2(Ω), Ω ⊂ ℝn, and 0 < c1 < ⋯ < cN. More generally -Aj can be the square of the generator of a (C0) group on a Banach space. When the constant ρ is negative, the initial value problem for the factored EPD equation is ill-posed. Nevertheless, we determine how many initial conditions are necessary to guarantee uniqueness of a solution. This number jumps up as ρ crosses a negative integer from right to left. © 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.
Mathematical Methods in the Applied Sciences
Goldstein, J., Lightbourne, J., & Sandefur, J. (1997). The amount of non-uniqueness for factored equations with Euler-Poisson-Darboux factors. Mathematical Methods in the Applied Sciences, 20 (16), 1449-1457. https://doi.org/10.1002/(SICI)1099-1476(19971110)20:16<1449::AID-MMA939>3.0.CO;2-K