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Global solutions and self-similar solutions of nonlinear wave equations. (Solutions globales et solutions auto-similaires de l’équation des ondes non linéaire.) (French) Zbl 0933.35140
Summary: We prove some results about existence uniqueness and regularity properties of global solutions of the nonlinear wave equations of the type \[ \begin{cases} \partial^2_t u-\Delta u=-\lambda|u|^{\alpha -1}u,\\ u(0,x)= f(x),\quad \partial_tu(0,x) =g(x),\end{cases} (\alpha>1). \] In particular, we show existence of regular self-similar solutions. We build also a family of finite energy solutions which converge asymptotically to self-similar solutions.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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