Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

Abstract.

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

$$ \left\{{\begin{array}{ll} {\psi_t=-\left({1-\alpha}\right)\psi-\theta_x+\alpha\psi_{xx},}&{\left( {t,x} \right) \in \left( {0,\infty } \right) \times {\bf R}}\\ {\theta _t = - \left( {1 - \alpha } \right)\theta + \nu ^2 \psi _x + \alpha \theta _{xx} + 2\psi \theta _x ,} \end{array}} \right. $$

with $0 < \nu ^2 < 4\alpha \left( {1 - \alpha } \right),0 < \alpha < 1.$ S. 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 kernel G(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, where D₀, β₀ are determined by the data. It is the key point to reformulate the system to the nonlinear parabolic one by suitable changing variables.