Cauchy–Kovalevskaya theorem

Remarks

The equation and the initial conditions determine all derivatives of $u$ at $(0, x_0)$, so the Taylor series of a real-analytic solution is fixed; this gives uniqueness. For existence, the proof shows that this Taylor series converges by the method of majorants: one compares with an explicit system whose analytic solution dominates the coefficients. The form of the equation matters: the equation must be solved for $\partial_t^k u$, and the right side may contain neither $\partial_t^k u$ nor any derivative of total order greater than $k$. Without this, the conclusion fails: for the heat equation $\partial_t u = \partial_x^2 u$ with $u(0, x) = 1/(1 - x)$, the formal power series solution is $\sum_{j \ge 0} \frac{(2j)!}{j!} \frac{t^j}{(1 - x)^{2j + 1}}$, which diverges for every $t \ne 0$. Analyticity of the data is also needed; for smooth data, existence can fail even for a linear equation, as Lewy's example of an operator without local solvability shows. Uniqueness among all smooth solutions, not only analytic ones, holds for linear equations by Holmgren's uniqueness theorem.

Going deeper

For a student

The theorem gives local existence in the analytic class for any noncharacteristic Cauchy problem, after a change of variables that makes the initial surface $t = 0$. It does not say that the problem is well posed: Hadamard's example shows that for the Laplace equation the analytic solution exists but depends discontinuously on the data. So for evolution equations one looks for theories in Sobolev or $C^\infty$ spaces, such as the energy method for hyperbolic equations. For ordinary differential equations, with $n = 0$, the theorem reduces to the existence of analytic solutions to analytic initial value problems.