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Anti-symmetric solutions of the nonlinear heat equation on $\mathbb{R} ^N$ : local existence and finite time blowup

PDE and Applied Math Seminar

Speaker: Fred Weissler, Paris 13
Location: 2112 MSB
Start time: Thu, Oct 3 2019, 4:30PM

We consider the nonlinear heat equation $u_t - \Delta u =|u|^\alpha u$ on $\mathbb{R}^N$, where $\alpha >0$, and we study solutions which are anti-symmetric with respect to the first $m$ variables, $x_1, \dots,x_m$, where $1 \le m \le N$. In particular, we show that very singular initial values, i.e. derivatives of the Dirac delta of the form $\partial_1\partial_2\cdots\partial_m\delta$, give rise to local (regular) solutions.These solutions exist when $0 < \alpha < 2/(N + m)$, which is consistent with the scaling properties of the equation. Similarly, singular initial values of the form $\partial_1\partial_2\cdots\partial_m|\cdot|^{-\gamma}$, $0 < \gamma < N$, when $0 < \alpha < 2/(\gamma + m)$, give rise to local (regular) solutions. These results enable us to obtain new finite time blowup results for certain classes of regular anti-symmetric initial values of the form $u_0 = \lambda f$. For example, in the case $0 < \alpha < 2/(N + m)$, suppose $f \in C_0(\mathbb{R} ^N)$ is anti-symmetric with respect to $x_1, \dots,x_m$, decays exponentially, and has non-zero weighted mean value in the sense that $\int_{\mathbb{R} ^N}x_1\cdots x_mf(x) dx \neq 0$. It follows that if $\lambda > 0$ is sufficiently small, then the solution with initial value $u_0 = \lambda f$ blows up in finite time. An analogous result holds in the case where $f$ has a specific power decay. For these results, there is no sign condition on $f$ other than anti-symmetry. This is joint work with S. Tayachi