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

Abstract

In this paper, we consider the following semi-linear complex heat equation∂tu=Δu+up,u∈C in Rn, with an arbitrary power p, p>1. We construct for this equation a complex solution u=u1+iu2, 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:u(x,T)∼[(p−1)2|x−a|28p|ln⁡|x−a||]−1p−1,u2(x,T)∼2p(p−1)2[(p−1)2|x−a|28p|ln⁡|x−a||]−1p−11|ln⁡|x−a||>0, as x→a. In addition to that, since we also have u1(0,t)→+∞ and u2(0,t)→−∞ as t→T, the blowup in the imaginary part shows a new phenomenon unknown 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 up 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 u1 and u2. 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.