Analysis Seminar

The Nonlinear Heat Equation with High Order Mixed Derivatives of the Dirac Delta as Initial Values and Applications

Speaker: Slim Tayachi, University of Tunis El Manar

Location: Warren Weaver Hall 1302

Date: Thursday, January 16, 2014, 11 a.m.

Synopsis:

In this talk we prove local existence of solutions for the nonlinear heat equation $$u_t = \Delta u + |u|^\alpha u, \; t\in(0,T),\; x\in \mathbb{R}^N,\; \alpha>0,$$ with initial values in high order negative Sobolev spaces. In particular, we consider high order mixed derivatives of the Dirac Delta as initial values.

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 and $\alpha<2/(N+m).$ Here, $f$ satisfies in particular $f\in C_0(\mathbb{R}^N)\cap L1(\mathbb{R}^N)$ and is anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\; x_m,\; 1\leq m \leq N,$ where $x:=(x_1,x_2,\cdots,x_N)\in \mathbb{R}^N.$ Moreover we require, $\int_{\mathbb{R}^N} x_1\cdots x_mf(x) dx\not=0$. This extends known “small lambda” blowup results.

If $f$ is also in $H1(\mathbb{R}^N)$, then by standard energy arguments, the solution with initial value $u_0 = \lambda f$ blows up in finite time if $\lambda > 0$ is sufficiently large. We prove the existence of a function $f_0$ for which, the solution with initial value $u_0 = \lambda f_0,\; \alpha<2/(N+m)$ would blow up in finite time for large and small $\lambda > 0$, but would be global for $\lambda = 1.$

We show also that the condition $\alpha<2/(N+m)$ is sharp. If $\alpha>2/(N+m)$, then all initial data $u_0 = \lambda f$ anti-symmetric in $x_1,x_2,\cdots, x_m,$ $\lambda>0$ is sufficiently small and $f$ satisfying the same conditions as above produce solutions which are global in time.

This is a joint work with Fred B. Weissler.