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Volume 12, Issue 2
Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values

Hongjun Yuan

J. Part. Diff. Eq., 12 (1999), pp. 149-178.

Published online: 1999-12

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  • Abstract
The aim of this paper is to discuss the Cauchy problem for degenerate quasilinear hyperbolic equations of the form \frac{∂u}{∂t} + \frac{∂u^m}{∂x} = -u^p, m > 1, p > 0  with measures as initial conditions. The existence and uniqueness of solutions are obtained. In particular, we prove the following results: (1) 0 < p < 1 is a necessary and sufficient condition for the above equations to have extinction property; (2) 0 < p < m is a necessary and sufficient condition for the above equations to have localization property of the propagation of perturbations.
  • Keywords

Degenerate quasilinear hyperbolic equations existence and uniqueness extinction and positivity localization

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@Article{JPDE-12-149, author = {}, title = {Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values}, journal = {Journal of Partial Differential Equations}, year = {1999}, volume = {12}, number = {2}, pages = {149--178}, abstract = { The aim of this paper is to discuss the Cauchy problem for degenerate quasilinear hyperbolic equations of the form \frac{∂u}{∂t} + \frac{∂u^m}{∂x} = -u^p, m > 1, p > 0  with measures as initial conditions. The existence and uniqueness of solutions are obtained. In particular, we prove the following results: (1) 0 < p < 1 is a necessary and sufficient condition for the above equations to have extinction property; (2) 0 < p < m is a necessary and sufficient condition for the above equations to have localization property of the propagation of perturbations.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5532.html} }
TY - JOUR T1 - Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values JO - Journal of Partial Differential Equations VL - 2 SP - 149 EP - 178 PY - 1999 DA - 1999/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5532.html KW - Degenerate quasilinear hyperbolic equations KW - existence and uniqueness KW - extinction and positivity KW - localization AB - The aim of this paper is to discuss the Cauchy problem for degenerate quasilinear hyperbolic equations of the form \frac{∂u}{∂t} + \frac{∂u^m}{∂x} = -u^p, m > 1, p > 0  with measures as initial conditions. The existence and uniqueness of solutions are obtained. In particular, we prove the following results: (1) 0 < p < 1 is a necessary and sufficient condition for the above equations to have extinction property; (2) 0 < p < m is a necessary and sufficient condition for the above equations to have localization property of the propagation of perturbations.
Hongjun Yuan . (2019). Cauchy's Problem for Degenerate Quasilinear Hyperbolic Equations with Measures as Initial Values. Journal of Partial Differential Equations. 12 (2). 149-178. doi:
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