Nonlinear Hyperbolic Smoothing at a Focal Point
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 5, 11 p.

The nonlinear dissipative wave equation ${u}_{tt}-\Delta u+{|{u}_{t}|}^{h-1}{u}_{t}=0$ in dimension $d>1$ has strong solutions with the following structure. In $0\le t<1$ the solutions have a focusing wave of singularity on the incoming light cone $|x|=1-t$. In $\left\{t\ge 1\right\}$ that is after the focusing time, they are smoother than they were in $\left\{0\le t<1\right\}$. The examples are radial and piecewise smooth in $\left\{0\le t<1\right\}$

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author = {Joly, Jean-Luc and M\'etivier, Guy and Rauch, Jeffrey},
title = {Nonlinear {Hyperbolic} {Smoothing} at a {Focal} {Point}},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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year = {1998-1999},
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Joly, Jean-Luc; Métivier, Guy; Rauch, Jeffrey. Nonlinear Hyperbolic Smoothing at a Focal Point. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 5, 11 p. http://www.numdam.org/item/SEDP_1998-1999____A5_0/

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