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A trial to construct specific self-similar solutions to non-linear wave equations. (English) Zbl 1455.35138
Summary: An attempt to construct self-similar solutions to nonlinear wave equations \(\square u=|u|^p\) is explained. The existence of self-similar solutions has been already established by H. Pecher [Math. Ann. 316, No. 2, 259–281 (2000; Zbl 0960.35067); NoDEA, Nonlinear Differ. Equ. Appl. 7, No. 3, 323–341 (2000; Zbl 0967.35099)], J. Kato and T. Ozawa [ Indiana Univ. Math. J. 52, No. 6, 1615–1630 (2003; Zbl 1053.35029); Math. Z. 247, No. 4, 747–764 (2004; Zbl 1063.35042)], etc. based on the standard fixed point theorem. In this note, we will discuss it by a constructive method using the theory of hypergeometric differential equations.
35L05 Wave equation
35L71 Second-order semilinear hyperbolic equations
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