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arXiv 1804.00560math.AP
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具有无理幂的复半线性heat equation方程的爆破解

A blowup solution of a complex semi-linear heat equation with an irrational power

Giao Ky Duong

备注：
arXiv admin note: text overlap with arXiv:1712.07183
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摘要：

本文考虑半线性复热方程\begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C} \end{eqnarray*}在具有任意幂$p,$ $ p > 1$的$\mathbb{R}^n,$中的解.特别地，$p$可以是非整数，甚至是无理数。我们为这个方程构造了一个复解$u = u_1 + i u_2$，它在有限时间$T$中爆破，并且只在一个爆破点$a.$上爆破。此外，我们还通过以下最终轮廓描述了解的渐近性：\begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}},\\ u_2(x,T) &\sim & \frac{2 p}{(p-1)^2} \left[ \frac{ (p-1)^2|x-a|^2}{ 8p |\ln|x-a||}\right]^{-\frac{1}{p-1}}\frac{1}{ |\ln|x-a||} > 0 , \text{ as } x \to a. \end{eqnarray*}除此之外，由于我们也有$u_1 (0,t) \to + \infty$和$u_2(0,t) \to - \infty$作为$t \to T,$，虚部的爆破显示了一个新的现象，在真实情况下标准热方程是未知的：奇点附近的一个非常数符号，虚部存在一个消失面，收缩到原点。在我们的工作中，我们已经成功地扩展到任何幂$p$，其中非线性项$u^p$是不连续的，如果$ p$不是整数。特别地，我们构造的解具有正实部。我们研究我们的方程作为一个系统的实部和虚部$u_1$和$u_2$。我们的工作依赖于两个主要的参数：减少到有限维的问题和拓扑参数的基础上的指数理论得到的结论。

英文摘要：

In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = Δu + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be non integer and even irrational. We construct for this equation a complex solution $u = u_1 + i u_2$, which blows up in finite time $T$ and only at one blowup point $a.$ Moreover, we also describe the asymptotics of the solution by the following final profiles: \begin{eqnarray*} u(x,T) &\sim & \left[ \frac{(p-1)^2 |x-a|^2}{ 8 p |\ln|x-a||}\right]^{-\frac{1}{p-1}},\\ u_2(x,T) &\sim & \frac{2 p}{(p-1)^2} \left[ \frac{ (p-1)^2|x-a|^2}{ 8p |\ln|x-a||}\right]^{-\frac{1}{p-1}}\frac{1}{ |\ln|x-a||} > 0 , \text{ as } x \to a. \end{eqnarray*} In addition to that, since we also have $u_1 (0,t) \to + \infty$ and $u_2(0,t) \to - \infty$ as $t \to T,$ the blowup in the imaginary part shows a new phenomenon unkown for the standard heat equation in the real case: a non constant sign near the singularity, with the existence of a vanishing surface for the imaginary part, shrinking to the origin. In our work, we have succeeded to extend for any power $p$ where the non linear term $u^p$ is not continuous if $ p$ is not integer. In particular, the solution which we have constructed has a positive real part. We study our equation as a system of the real part and the imaginary part $u_1$ and $u_2$. Our work relies on two main arguments: the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion.

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