Studies On Borel Summability Of Formal Solutions To Some Equations With Irregular Singularity

In this dissertation, we mainly study the solvability and uniqueness problems of the following singular partial differential equation:(t(?)t)mu=F(t,x,(t(?)j(?)xαu),(t,x)∈C×Cxn. Here m is a positive integer, j,α satisfy j+|α|<m and j<m.The study of the singular differential equations can date back to the year1856when Briot-Bouquet carried out a study of Briot-Bouquet type differential equations, and the year1913when Gevrey initiated the study of the forward-backward diffusion equations. Further studies on these problems will at last come to the study of the above singular differential equations. Meanwhile, for the classical Cauchy-Kowalewski Theorem, when we consider the singular solutions along a hypersurface, will also contribute to the study of the singular differential equation above. Hence the study of this kind of singular equations is quite interesting.For the case m=1and n=1, the study is somehow very complete. In particular, for the solutions of totally characteristic equations, Chen-Tahara, Chen-Luo-Tahara and Chen-Luo-Zhang studied the summability and Gevrey divergency.In this dissertation, we will generalize their work to the case m=1and n=2. Namely, we will study the solvability of the equation:t(?)tu=F(t,x,y,u,(?)xu,(?)yu),(t,x, y)∈C3. More precisely, we will study the following content:In the first chapter, we first introduced some background of the study of the singular equations. Next, we stated some recent process for the case m=1and n=1.In the second chapter, we recalled some basic notations and properties of the Gevrey asymptotic expansion and Borel summarability of Gevrey-type formal power series. We also proved some basic lemmas that will be used later. In the third chapter, we consider the summability of the following equation: t(?)tu=F(t,x,y,u,(?)xu,(?)yu), u(0,x,y)=0,(4) Under some suitable conditions, we proved that the unique solution u(t,x,y)∈C{t,y}[[x]]k, whose singular points are discretely distributed on some lattice in the plane region.In the fourth chapter, we consider the summability of the following equation: By making use of the Nagumo norm, we proved that the equation above has a unique formal power series solution u(t,x,y), which is Borel summarable with respect to the variable x in a small polydiscs DR1×DR2={(t,y)∈C2:|t|<R1,|y|<R2}.In the fifth chapter,we consider the following equation with one space variable: We proved that in the sector domain G(d,θ) with a small aperture angle(θ<π/k),we have infinity many real analytic solutions.