Colloquium

Speaker: | Prof. Shuichi Kawashima (Kyushu University) |

Title: | Asymptotic behavior of solutions to a hyperbolic Cahn-Hilliard equation |

Time: | 2015-05-14 (Thu.) 15:00 - 16:00 |

Place: | Auditorium, 6 Floor, Institute of Mathematics (NTU Campus) |

Abstract: | We studies a hyperbolic Cahn-Hilliard equation % \begin{equation} (1-\Delta)u_{tt}+\Delta^2u+u_t=\Delta f(u) \end{equation} % with the initial data $u(0)=u_0$, $u_t(0)=u_1$ in the $n$-dimensional whole space, where $f(u)=u^{\nu}+O(u^{\nu+1})$ with $\nu={\rm max}\big\{1+\frac{2}{n}, 2\big\}$. It is known that the dissipative structure of the equation is of the regularity-loss type. For this initial value problem we prove the global existence and asymptotic decay of solution $u$ under enough regularity and smallness assumption on the initial data. Moreover, we show that when $n\geq 3$, our solution $u$ is asymptotic to the linear diffusion wave $v_*$ which is defined in terms of the fundamental solution $G_0$ to the linear parabolic equation $ v_t+\Delta^2v=0. $ Here $v_*$ is given explicitly as $v_*(x,t):=MG_0(x,t+1)$, where $G_0(x,t)=t^{-{n\over4}}G_*(xt^{-{1\over4}})$ with $G_*(x)=\mathcal{F}^{-1}[e^{-|\xi|^4}](x)$, and the mass $M$ is given by $M=\int(u_0+u_1)dx$. On the other hand, when $n=1$ or $2$, we show that the solution $u$ is asymptotic to the nonlinear diffusion wave $v^*$ which is the self-similar solution $V_M$ of the semilinear parabolic equation $ v_t+\Delta^2v=\Delta v^{1+\frac{2}{n}} $ with the same mass $M$. More explicitly, $v^*(x,t):=V_M(x,t+1)$, where $V_M(x,t)=t^{-{n\over4}}\Phi_M(xt^{-{1\over4}})$ and $M=\int\Phi_Mdx$. These results are based on the recent joint work with Hiroshi Takeda and Yasunori Maekawa. |

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