| Abstract: |
| We consider an initial boundary value problem for the viscous Cahn-Hilliard equation. The Cahn-Hilliard equation is proposed in 1958 to describe the phase separation in an alloy. When one takes into account internal microforces, some viscous term appears in the Cahn-Hilliard equation, which is called the viscous Cahn-Hilliard equation. In Bui, et al. (2014), they proved the existence of a strong solution under the condition that the nonlinear term $\varphi(u)$ appearing in the chemical potential satisfies $\varphi(u)u\geq0$ for all $u\in\mathbb{R}$ and Sobolev subcritical growth condition. In this talk, we exclude these conditions by decomposing the nonlinear term $\varphi(u)$ into the sum of a monotone function and a locally Lipschitz perturbation and show the existence of a strong solution. In physics, $\varphi(u)=u^3-u$ is often used as a typical example. However, the previous result cannot cover this case. Our framework can cover not only this case but also more general cases $\varphi(u)=|u|^{p-2}u-|u|^{q-2}u$ with $p>q\geq2$. |
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