论文标题
超临界空间中的复杂有价值的半线性加热方程$ e^s_σ$
Complex valued semi-linear heat equations in super-critical spaces $E^s_σ$
论文作者
论文摘要
我们考虑了复杂有价值的半线性热方程$$的库奇问题 \ partial_t u-ΔU -Δu -u^m = 0，\ \ u（0，x）= u_0（x），$$，其中$ m \ geq 2 $是整数，初始数据属于超级临界空间$ e^s_σ$，其规范由$ f \ | f \ | __________ ξ\ rangle^σ2^{s |ξ|} \ hat {f}（ξ）\ | _ {l^2}，\ \σ\ in \ mathbb {r}，\ s <0。 $$如果$ s <0 $，则任何Sobolev space $ h^{r} $是$ e^s_σ$的子空间，即$ \ cup_ {r \ in \ sathbb {r}} h^r \ subset e^s_σ$。如果最初的数据属于$ e^s_σ$（$ s <0，\σ\ geq d/2-2/（m-1）$），我们将获得解决方案的全局存在和独特性，并且在第一个八分代中支持了它们的傅立叶变换，即全球解决方案中$ e^s_σ$中的初始数据的较小条件。此外，我们表明解决方案$ u $和迭代解决方案$ u^{（j）} $之间的错误是$ c^j/（j \，！）^2 $。如果非线性$ u^m $被指数函数$ e^u-1 $取代，也可以保持类似的结果。
We consider the Cauchy problem for the complex valued semi-linear heat equation $$ \partial_t u - Δu - u^m =0, \ \ u (0,x) = u_0(x), $$ where $m\geq 2$ is an integer and the initial data belong to super-critical spaces $E^s_σ$ for which the norms are defined by $$ \|f\|_{E^s_σ} = \|\langle ξ\rangle^σ2^{s|ξ|}\hat{f}(ξ)\|_{L^2}, \ \ σ\in \mathbb{R}, \ s<0. $$ If $s<0$, then any Sobolev space $H^{r}$ is a subspace of $E^s_σ$, i.e., $\cup_{r \in \mathbb{R}} H^r \subset E^s_σ$. We obtain the global existence and uniqueness of the solutions if the initial data belong to $E^s_σ$ ($s<0, \ σ\geq d/2-2/(m-1)$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E^s_σ$ 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 $e^u-1$.
