Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative is also a function of at most linear growth. In this note, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution

We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.

We consider the Cauchy problem for Laplacian. Using the single layer representation, we obtain an equivalent system of boundary integral equations. We show the singular values of the ill-posed Cauchy operator decay exponentially, which means that a small error is exponentially amplified in the solution of the Cauchy problem. We show the decaying rate is dependent on the geometry of the domain, ...