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**Extra resources for Foundations of the Classical Theory of Partial Differential Equations **

**Example text**

J. 2. An Example of Nonexistence of au Analytic Solution. 1) even in the case when the right-hand sides may contain derivatives DtD,“Uj with k + JoI > nj, k < nj (we previously required k + Ial 5 nj, k < nj). However, in this case an analytic solution may fail to exist for some data of the problem. 12. Consider Ut = %x, 2 E (-1, the Cauchy problem l), t > 0; z&O for the heat equation = $-& 2 E (-l,l). tk, k=O which diverges for any t # 0. 10) under consideration has no analytic solution in any neighborhood of the origin.

He proved that under this assumption there exists one and only one solution u(t) holomorphic in a neighborhood of the point to. The idea of this proof is very simple. 1), then ae = ue, al = f(tc, UO), and all the subsequent coefficients as, as, . . can be found by differentiating both sides of the differential equation and setting t = to. , the solution is unique. To prove the existence it suffices to show that the power series with coefficients found in this way converges in some neighborhood of the point to.

The question naturally arises: by which smooth functions a can any tempered distribution be multiplied without going outside the class of tempered distributions? , that multiplication by a be a continuous linear operator in S(lP). 9), ‘p E S(F). In particular one can multiply polynomial in 2. 10) and iV, are constants depending the class of distributions S’(F) in which it is necessary to take a distribution of s’(P) by any Chapter 58 Example 2. 4. Let us compute x6’(z), so that &Y(z) Classical Theory where x E R.