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Generalized solutions of a periodic Goursat problem

Valmorin V.

Journal of Applied Analysis

2001

Vol. 7, nr 1

43-59

Under reasonable assumptions on the data u, v and the function f, we show that the nonlinear periodic Goursat problem (wzór) ; u(x, 0) = v(x); u(0, y) = w(y) which cannot be posed in the general theory of distributions, may be studied and solved in a differential algebra of periodic new generalized functions on R2. This algebra contains, in a canonical way, the space of periodic distributions on R2 as a linear subspace. When the data are Dirac measures, a particular study is done, showing the nature of the singularities involved in this case.