We introduce relative twisted homology and cohomology groups associated with Euler type integrals of solutions to Lauricella's system (a,b,c) of hypergeometric differential equations. We define an intersection form between relative twisted homology groups and that between relative twisted cohomology groups, and show their compatibility. We prove that the relative twisted homology group is canonically isomorphic to the space of local solutions to (a,b,c) for any parameters a, b, c. Through this isomorphism, we study (a,b,c) by the relative twisted homology and cohomology groups and the intersection forms without any conditions on a, b, c.

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Hardy parabolic equation. Using these conditions, we attempt to identify the optimal strength of the singularity of the initial data for the existence of nonnegative solutions to this equation. Furthermore, we also obtain analogous results for a fractional case.

We consider the Cauchy problems in Rⁿ for the plate equation with a weighted L¹-initial data. We derive new optimal estimates of the L²-norm of solutions for n = 1,2,3,4. In particular, such obtained results express infinite time blowup property in the case when the 0-th moment of the initial velocity does not vanish. From a series of estimates obtained in this paper one can observe much complicated growth properties of the solution itself of the plate equations.

In this paper, we consider the Cauchy problem for the linear dispersive equations. We establish the probabilistic radial Strichartz estimates for a randomization preserving the radial symmetry. As an application, we prove almost sure local well-posedness along with small data global existence and scattering for the radial mass-critical fractional Schrödinger equation.

In this note, we prove the nondegeneracy of the explicit finite-mass solution to the N-Laplace Liouville equation on the whole space, which is recently shown to be unique up to scaling and translation.