Nonlinear Equations with Infinitely many Derivatives

@article{Grka2011NonlinearEW,
  title={Nonlinear Equations with Infinitely many Derivatives},
  author={Przemysław G{\'o}rka and Humberto Prado and Enrique G. Reyes},
  journal={Complex Analysis and Operator Theory},
  year={2011},
  volume={5},
  pages={313-323}
}
We study the generalized bosonic string equation$$ \Delta e^{-c\,\Delta}\phi = U(x,\phi), \quad c > 0$$on Euclidean space $${\mathbb {R}^n}$$ . First, we interpret the nonlocal operator $${\Delta e^{-c\,\Delta}}$$ using entire vectors of Δ in $${L^2(\mathbb{R}^n)}$$ , and we show that if $${U(x, \phi) = \phi(x) + f(x)}$$ , in which $${f \in L^2(\mathbb {R}^n)}$$ , then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space $${{\mathcal H}^{c,\infty… CONTINUE READING