Event
01_1
| 제출번호(No.) | 0046 |
|---|---|
| 분류(Section) | Contributed Talk |
| 분과(Session) | Analysis (AN) |
| 영문제목 (Title(Eng.)) |
A new condition for the concavity method of blow-up solutions to semilinear heat equations |
| 저자(Author(s)) |
Chung Soon Yeong1, Choi Min Jun1 Sogang University1 |
| 초록본문(Abstract) | In this talk, we consider the semilinear heat equations under Dirichlet boundary condition \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right),\\ u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right),\\ u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \end{cases} \end{equation*} where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of our work is to introduce a new condition \[ (C)\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0 \] for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is the first eigenvalue of Laplacian $\Delta$ in order to use the concavity method to obtain the blow-up solutions to the semilinear heat equations. In fact, it will be seen that the condition $(C)$ improves the conditions ever known so far. |
| 분류기호 (MSC number(s)) |
39A14, 35K57 |
| 키워드(Keyword(s)) | semilinear heat equations, concavity method, blow-up |
| 강연 형태 (Language of Session (Talk)) |
Korean |