具有退化粘性的非齐次双曲守恒律方程的Cauchy问题

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Cauchy Problem for the Nonhomogeneous Hyperbolic Conservation

Laws with the Degenerate Viscous Term

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Abstract: In this paper, the authors consider Cauchy problem for the nonhomogeneous hyperbolic conservation
laws with the degenerate viscous term

{\begin{cases} u_{t} + f (u)_{x} = a^{2} t^{α} u_{x x} + g (u), x \in R, t > 0, \\ u (x, 0) = u_{0} (x) \in L^{\infty} (R) . \end{cases} \eqno (I)

$\left\{\begin{array}{l} u_t+f(u)_x=a^2t^\alpha u_{xx}+g(u),\ \ \ x\in{\bf R},\ \ \ t>0,\\ u(x,0)=u_0(x) \in L^\infty({\bf R}). \end{array}\right. \eqno{({\rm I})}$
where here

f (u), g (u)

$f(u),g(u)$ is a one order continuous and differentiable function defined on

R, a > 0, 0 < α < 1

${\bf R}, a>0, 0<\alpha <1$ 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}

$L^\infty$ estimate of solution
by 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

CLC Number:

35L80

Wang Bing;Xu Xuewen.

Cauchy Problem for the Nonhomogeneous Hyperbolic Conservation

Laws with the Degenerate Viscous Term

[J].Acta mathematica scientia,Series A, 2008, 28(1): 109-115.

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