Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity
Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity
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DOI：
10.1007/s00033-006-0062-9
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发表时间：
2006-05
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影响因子：
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通讯作者：
K. Nishihara
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作者：
K. Nishihara
我们考虑一维空间上具有椭圆性的非线性耗散演化系统的柯西问题 $$ \left\{{\begin{array}{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} \end{array}} \right。 $$ 与S。 Q. Tang 和 H. Zhu [4] 已经考虑了这个问题，并针对适当的小数据获得了最佳衰减特性。在本文中，我们使用高斯核 G(t, x) 导出渐近轮廓，它显示了随着时间趋于无穷大解的精确行为。事实上，我们将证明渐近公式 $$ \left\| {\left( {\begin{array}{*{20}c} \psi \\ \theta \\ \end{array} } \right)\left( {t,x} \right) - D_0 e^{ - \left( {1 - \alpha - \frac{{\nu ^2 }} {{4\alpha }}} \right)t} G\left( {t,x} \right)\left( {\begin{array}{*{20}c} {\cos \left( {\frac{\nu } {{2\alpha }}x + \frac{\pi } {4} + \beta _0 } \right)} \\ { - \nu \sin \left( {\frac{\nu } {{2\alpha }}x + \frac{\pi } {4} + \beta _0 } \right)} \\ \end{array} } \right)} \right\|_{L^p \left( {R_x } \right)} = e^{^{ - \left( {1 - \alpha - \frac{{\nu ^2 }} {{4\alpha }}} \right)t} } o\left( {t^{ - \frac{1} {2}\left( {1 - \frac{1} {p}} \right)} } \right),$$成立，其中D0、β0由数据决定。通过适当的变化变量将系统重构为非线性抛物线系统是关键。
We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space $$ \left\{{\begin{array}{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} \end{array}} \right. $$ withS. Q. Tang and H. Zhao [4] have considered the problem and obtained the optimal decay property for suitably small data. In this paper we derive the asymptotic profile using the Gauss kernelG(t, x), which shows the precise behavior of solution as time tends to infinity. In fact, we will show that the asymptotic formula $$ \left\| {\left( {\begin{array}{*{20}c} \psi \\ \theta \\ \end{array} } \right)\left( {t,x} \right) - D_0 e^{ - \left( {1 - \alpha - \frac{{\nu ^2 }} {{4\alpha }}} \right)t} G\left( {t,x} \right)\left( {\begin{array}{*{20}c} {\cos \left( {\frac{\nu } {{2\alpha }}x + \frac{\pi } {4} + \beta _0 } \right)} \\ { - \nu \sin \left( {\frac{\nu } {{2\alpha }}x + \frac{\pi } {4} + \beta _0 } \right)} \\ \end{array} } \right)} \right\|_{L^p \left( {R_x } \right)} = e^{^{ - \left( {1 - \alpha - \frac{{\nu ^2 }} {{4\alpha }}} \right)t} } o\left( {t^{ - \frac{1} {2}\left( {1 - \frac{1} {p}} \right)} } \right), $$ holds, whereD0, β0are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing variables.