Cauchy Problem for the Nonhomogeneous Hyperbolic Conservation
Laws with the Degenerate Viscous Term
Acta mathematica scientia,Series A
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Wang Bing;Xu Xuewen
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Abstract: In this paper, the authors consider Cauchy problem for the nonhomogeneous hyperbolic conservationlaws with the degenerate viscous term$$\left\{\begin{array}{l} u_t+f(u)_x=a^2t^\alpha u_{xx}+g(u),\ \ \ x\in{\bf R},\ \ \ t>0,\\u(x,0)=u_0(x) \in L^\infty({\bf R}).\end{array}\right.\eqno{({\rm I})}$$where here $f(u),g(u)$ is a one order continuous and differentiable function defined on ${\bf R}, a>0, 0<\alpha <1$ are both constants. Under these conditions, the authors obtain the local existence of solutions of the Cauchy problem (I). Then, the authors get $L^\infty$ estimate of solutionby the maximum principle and make use of the extension theorem to obtain the global existence.
Key words: Hyperbolic conservation laws, Degenerate viscosity, Maximum principle, L∞ estimate, Global existence
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Wang Bing;Xu Xuewen.
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URL: http://121.43.60.238/sxwlxbA/EN/
http://121.43.60.238/sxwlxbA/EN/Y2008/V28/I1/109
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