Local existence for quasilinear symmetric hyperbolic systems

Reinhard Racke
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Reinhard Racke: University of Konstanz, Department of Mathematics and Statistics

Chapter 5 in Lectures on Nonlinear Evolution Equations, 2015, pp 57-77 from Springer

Abstract: Abstract Theorem 1.1 and Theorem 1.2 will be proved in detail for the initial value problem 5.1 $$ y_{tt} - \Delta y = f\left( {Dy,\,\nabla Dy} \right), $$ 5.2 $$ y\left( {t = 0} \right) = y_0 ,\,\,\,\,\,y_t \left( {t = 0} \right) = y_1 , $$ with 5.3 $$ f\left( {Dy,\,\nabla Dy} \right) = \sum\limits_{i,\,j = 1}^n {a_{ij} \left( {Dy} \right)\partial _i \partial _j y} , $$ where 5.4 $$ a_{ij} = \bar a_{ji} \in C^\infty \left({\mathrm{I}\!\mathrm{R}^{n + 1} } \right),\,\,\,\,\,i,\,j = 1, \ldots ,\,n, $$ 5.5 $$ a_{ij} \left( 0 \right) = 0\,\,\,\,\,,\,\,\,\,\,i,\,j = 1, \ldots ,\,n. $$

Keywords: Fractional Derivative; Cauchy Sequence; Local Existence; Nonlinear Wave Equation; High Norm (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-21873-1_6

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