On the representation by special series of solutions to nonlinear Cauchy--Kovalevskaya type equations with nonanalytic initial data (Q2773664)

The author describes some methods for representing a solution to a partial differential equation in the form of an absolutely convergent series. The emphasis is on the nonlinear hyperbolic equation NEWLINE\[NEWLINE u_{tt} = u_{xx} + F(x,u,u_x,u_{xx})\tag{1}NEWLINE\]NEWLINE which is furnished with the Cauchy data NEWLINE\[NEWLINE u(x,0)=u_0(x),\qquad u_t(x,0)=u_1(x).\tag{2}NEWLINE\]NEWLINE The Cauchy data are not assumed to be analytic, i.e., the Cauchy-Kovalevskaya theorem is not applicable. It is assumed that the function \(F\) is representable as NEWLINE\[NEWLINE F=\sum\limits_{m+n+k\geq 2}h_{mnk}(x)u^mu_x^nu_{xx}^k, NEWLINE\]NEWLINE where the series is absolutely convergent. It is demonstrated that, for special classes of initial data, a classical solution to this problem can be written as NEWLINE\[NEWLINE u(x,t)= \sum_{n=1}^\infty u_n(x)P_1^n(x,t), NEWLINE\]NEWLINE where \(P_1(x,t)=(t+f(x))^{-1}\) (\(f(x)\) is some sufficiently smooth function) and the coefficients \(u_n\) are determined recursively. Some other representations of a solution are also examined. For instance, it is shown that in some cases a solution may be written as NEWLINE\[NEWLINE u=\sum_{n=1}^\infty u_n(t)Q^n(x), NEWLINE\]NEWLINE with the function \(Q\) suitably chosen.