Complex valued semi-linear heat equations in super-critical spaces $E_{\sigma }^s$

Abstract

We consider the Cauchy problem for the complex valued semi-linear heat equation $${\partial }_{t}u-\Delta u-u^m=0,u(0,x)=u_0(x),$$ where $m\ge 2$ is an integer and the initial data belong to super-critical spaces $E_{\sigma }^s$ for which the norms are defined by $$\Vert f{\Vert }_{E_{\sigma }^s}=\Vert ⟨\xi {⟩}^{\sigma }{2}^{s|\xi |}\hat{f}(\xi ){\Vert }_{L^2},\sigma \in \mathbb{R},s<0.$$ If $s<0$, then any Sobolev space $H^r$ is a subspace of $E_{\sigma }^s$, i.e., ${\cup }_{r\in \mathbb{R}}{H}^r\subset E_{\sigma }^s$. We obtain the global existence and uniqueness of the solutions if the initial data belong to $E_{\sigma }^s$ ($s<0,\sigma \ge d/2-2/(m-1)$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E_{\sigma }^s$ are not required for the global solutions. Moreover, we show that the error between the solution $u$ and the iteration solution $u^{(j)}$ is $C^j/(j!{)}^2$. Similar results also hold if the nonlinearity $u^m$ is replaced by an exponential function $e^u-1$.

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