| Abstract: |
| In this talk we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem
$$
\partial_t u + (-\Delta)^{\frac{\theta}{2}} u = u^p, x\in{\bf R}^N , t>0,
\quad
u(0) = \mu \ge 0\quad\mbox{in}\quad{\bf R}^N,
$$
where $N\ge1$, $01$ and $\mu$ is a Radon measure or a measurable function in ${\bf R}^N$. Here $(-\Delta)^{\theta/2}$ denotes the fractional power of the Laplace operator $-\Delta$ in ${\bf R}^N$. This is defined by
$$
{\cal F} [(-\Delta)^\frac{\theta}{2} u ](\xi) = |\xi|^\theta {\cal F}[u](\xi),\quad \xi\in{\bf R}^N,
$$
where ${\cal F}[v]$ is the Fourier transform of $v$. Our conditions identify the strongest singularity of initial data for the solvability of the problem. And as an application we obtain optimal estimates of the life span of the solution with small initial data. |
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