Characteristic Cauchy problems for some non-Fuchsian partial differential operators

A Construction of asymptotic solutions and the existence of smooth null-solutions for a class of non-Fuchsian partial differential operators

“…The author considered the characteristic Cauchy problems for a class of operators wider than the Fuchsian operators in [3]. In that result, he showed the unique solvability of the characteristic Cauchy problems in the category of functions which are of class C°° with respect to t and real-analytic with respect to x.…”

“…Further, using these asymptotic solutions, we prove the conjecture in [3] mentioned above under an additional assumption. The C°° null-solution constructed here is one of the most fastly decaying nontrivial solutions of Pu -0.…”

“…(A-l) For all μ e S(P), there exist no (j, a) e I u (P) such that a Φ 0. for μ > 0 is weaker than the condition (A-2; 0), and (A-2;0) is equivalent to (A-2) in [3]. Now, the following is one of the three main theorems in this article.…”

“…Especially, from their results it easily follows that if P is a Fuchsian operator with real-analytic coefficients, then there exist no sufficiently smooth null-solutions. Here, a Schwartz distribution u in a neighborhood of (0,0) is called a null-solution for P at (0,0), if Pu = 0 in a neighborhood of (0,0) and (0,0) e supp u c {t > 0}, where supp u denotes the support of u.The author considered the characteristic Cauchy problems for a class of operators wider than the Fuchsian operators in [3]. In that result, he showed the unique solvability of the characteristic Cauchy problems in the category of functions which are of class C°° with respect to t and real-analytic with respect to x.…”

“…In this article, we construct an asymptotic solution of Pu = 0 in the form for a class of operators wider than that considered in [3].…”