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We show that the viscous Burgers equation ut+uux = uxx consid- ered for complex valued functions u develops finite-time singularities from compactly supported smooth data. By means of the Cole-Hopf transformation, the singularities of u are related to zeros of complex- valued solutions v of the heat equation vt = vxx. We prove that such zeros are isolated if they are not present in the initial data.

Zeros of complex caloric functions and singularities of complex viscous Burgers equation