Nonlinear Hyperbolic Smoothing at a Focal Point

Joly, Jean-Luc; Métivier, Guy; Rauch, Jeffrey

Joly, Jean-Luc ¹ ; Métivier, Guy ² ; Rauch, Jeffrey ³

¹ MAB, Université de Bordeaux I, 33405 Talence, FRANCE
² IRMAR, Université de Rennes I,35042 Rennes, FRANCE
³ Department of Mathematics, University of Michigan, Ann Arbor 48109 MI, USA

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1998-1999), Exposé no. 5, 11 p.

Résumé

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

MR Zbl

Affiliations des auteurs :

Joly, Jean-Luc ¹ ; Métivier, Guy ² ; Rauch, Jeffrey ³

¹ MAB, Université de Bordeaux I, 33405 Talence, FRANCE
² IRMAR, Université de Rennes I,35042 Rennes, FRANCE
³ Department of Mathematics, University of Michigan, Ann Arbor 48109 MI, USA

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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"},
     note = {talk:5},
     pages = {1--11},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1998-1999},
     zbl = {1059.35516},
     mrnumber = {1721323},
     language = {en},
     url = {http://archive.numdam.org/item/SEDP_1998-1999____A5_0/}
}

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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://archive.numdam.org/item/SEDP_1998-1999____A5_0/

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