Speaker : Jin Takahashi (Tohoku University)
Title : Time-dependent singularities in a semilinear heat equation
Time : 2018-01-30 (Tue) 11:00 - 12:00
Place : Seminar Room 722, Institute of Mathematics (NTU Campus)
Abstract: This talk is based on the results of $\left [1,2 \right ]$. We consider the following semilinear heat equation $u_t-\Delta u=u^p$, $x\in \mathbf{R}^N\setminus\{\xi(t)\}$, $t\in I$, where $N\geq3$, $p>1$, $I\subset\mathbf{R}$ is an open interval and $\xi:\overline{I}\rightarrow\mathbf{R}^N$ is a prescribed curve which is smooth enough. The aim of this talk is to study time-dependent singularities of nonnegative solutions of the above equation under the assumption that $1< p< N/(N-2)$. In the first part of this talk, we prove that every solution $u$ can be extended as a distributional solution of the following equation $u_t -\Delta u =u^p +(\delta_0\otimes\mu)\circ\mathcal{T}$ in $\mathcal{D}'(\mathbf{R}^N \times I)$. Here $\mathcal{T} (\varphi) (x,t):=\varphi (x+\xi(t),t)$, $\delta_0$ is the Dirac measure on $\mathbf{R}^N$ concentrated at the origin and $\mu$ is a Radon measure on $I$ determined by the solution $u$. In addition, we show relations between the exponent $p$ and the local growth rate of $\mu$ and specify the behavior of solutions at the time-dependent singularity. In the second part, we give sharp conditions on $\mu$ for the existence and the nonexistence of solutions of the above extended equation. This is a joint work with Prof. Toru Kan (Tokyo Institute of Technology).

References:

[1] T. Kan and J. Takahashi, Time-dependent singularities in semilinear parabolic equations: behavior at the singularities. J. Differential Equations 260 (2016), 7278--7319.

[2] T. Kan and J. Takahashi, Time-dependent singularities in semilinear parabolic equations: Existence of solutions. J. Differential Equations 263 (2017), 6384--6426.