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Exact solutions of Cauchy problem for partial differential equations with double characteristics and singular coefficients. (English) Zbl 0863.35025
Various Cauchy problems for the equation $$L_{a,b}=0$$ in the first quadrant resp. in the upper half-plane are studied where $$L_{a,b}= (\partial_x - ax^k\partial_t) (\partial_x - bx^k\partial_t) +kbx^{k-1}\partial_t - \frac{k}{x}\partial_x$$ ($$a,b$$ reals with $$ab\neq 0, a\neq b$$, $$k>0$$ an odd integer). Necessary and sufficient conditions are given for the existence of classical solutions of the problems and expressions for the solutions are presented.
##### MSC:
 35C15 Integral representations of solutions to PDEs 35L99 Hyperbolic equations and hyperbolic systems
##### Keywords:
exact solutions; Cauchy problem; singular coefficients
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