The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values

Abstract:

In this paper we prove local existence of solutions of the nonlinear heat equation $u_t = \Delta u + |u|^\alpha u, \; t\in (0,T),\; x\in \mathbb {R}^N,$ with initial value $u(0)=K\partial _{1}\partial _{2}\cdot \cdot \cdot \partial _{m}\delta ,\; K\not =0,\; m\in \{1,\; 2,\; \cdots ,\; N\},\; 0<\alpha <2/(N+m)$ and $\delta$ is the Dirac distribution. In particular, this gives a local existence result with an initial value in a high order negative Sobolev space $H^{s,q}(\mathbb {R}^N)$ with $s\leq -2.$

As an application, we prove the existence of initial values $u_0 = \lambda f$ for which the resulting solution blows up in finite time if $\lambda >0$ is sufficiently small. Here, $f$ satisfies in particular $f\in C_0(\mathbb {R}^N)\cap L^1(\mathbb {R}^N)$ and is anti-symmetric with respect to $x_1,\; x_2,\; \cdots ,\; x_m.$ Moreover, we require $\int _{\mathbb {R}^N} x_1\cdots x_mf(x) dx\not =0$. This extends the known “small lambda” blow up results which require either that $\int _{\mathbb {R}^N}f(x) dx\not =0$ (Dickstein (2006)) or $\int _{\mathbb {R}^N} x_1f(x) dx\not =0$ (Ghoul (2011), (2012)).