Existence and asymptotics of the solution of an implicit Cauchy problem (Q1902720)

This article deals with the Cauchy problem \(\sum^m_{k = l} a_kt(x')^k + bt^\lambda + f(t,x,x') = 0\), \(x(0) = 0\) in a domain \(D = \{(t,x,y) : 0 < t \leq \tau\), \(|x |< rt^{1 + \nu}\), \(|y |< (1 + \nu) rt^\nu\}\) \((\nu = (\lambda - 1)/l)\). The main result is a theorem on unique existence of a solution \(x(t)\) to the above problem satisfying the inequalities \(|x(t) |< rt^{1 + \lambda}\), \(|x'(t) |< (1 + \nu) rt^\nu\), and \(|x(t) - ct^{1 + \nu} |\leq Mt^{1 + 2 \nu}\), \(|x'(t) - (1 + \nu) ct^\nu |\leq Mt^{2 \nu}\) \((0 < t \leq \rho)\) for suitable \(\rho \in (0, \tau)\), \(M > 0\) under natural conditions for \(a_k\), \(b\), \(f(t,x,x')\), \(r, \tau\).