English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we consider the following semi-linear complex heat equation\begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C}\end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. Inparticular, $p$ can be non integer and even irrational. We construct for thisequation 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 theasymptotics 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 anew phenomenon unkown for the standard heat equation in the real case: a nonconstant sign near the singularity, with the existence of a vanishing surfacefor the imaginary part, shrinking to the origin. In our work, we have succeededto extend for any power $p$ where the non linear term $u^p$ is not continuousif $ p$ is not integer. In particular, the solution which we have constructedhas a positive real part. We study our equation as a system of the real partand 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 topologicalargument based on the index theory to get the conclusion.

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