Global solution and smoothing effect for a non-local regularization of a hyperbolic equation

J. Droniou (), T. Gallouët () and J. Vovelle ()
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J. Droniou: Université Montpellier II Place Eugène Bataillon, Département de Mathématiques CC 051
T. Gallouët: CMI, Université de Provence
J. Vovelle: CMI, Université de Provence

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 499-521 from Springer

Abstract: Abstract We study the problem 1.1 $$ \left\{ \begin{gathered} {{\partial }_{t}}u\left( {t,x} \right) + {{\partial }_{x}}\left( {f\left( u \right)} \right)\left( {t,x} \right) + g\left[ {u\left( {t,\cdot } \right)} \right]\left( x \right) = 0 t \in ]0,\infty [,x \in \mathbb{R} \hfill \\ u\left( {0,x} \right) = {{u}_{0}}\left( x \right) x \in \mathbb{R}, \hfill \\ \end{gathered} \right. $$ where $$ f \in {C^\infty }\left( \mathbb{R} \right)$$ is such that f (0) = 0(there is not loss of generality in assuming this), $$ {u_0} \in {L^\infty }\left( \mathbb{R} \right)$$ and gis the non-local (in general) operator defined through the Fourier transform by 1.1 $$ \mathcal{F}\left( {g\left[ {u\left( {t, \cdot } \right)} \right]} \right)\left( \xi \right) = {\left| \xi \right|^\lambda }\mathcal{F}\left( {u\left( {t, \cdot } \right)} \right)\left( \xi \right), with \lambda \in \left] {1,2} \right]. $$

Keywords: Maximum Principle; Global Solution; Detonation Wave; Hyperbolic Equation; Spatial Derivative (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_27

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DOI: 10.1007/978-3-0348-7924-8_27

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