We construct infinite time blow-up solution to the following heat equation with Sobolev critical exponent and drift terms ut=Δu+∇b(x)·∇u+un+2n-2inRn×(0,+∞),u(·,0)=u0inRn,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} u_t \,=\, \Delta u\,+\,\nabla b (x) \cdot \nabla u\,+\, u^{\frac{n+2}{n-2}} ~ \text{ in } ~ \mathbb {R}^n\times (0,+\infty ),\\ u(\cdot ,0)=u_0 ~ \text{ in } ~ \mathbb {R}^n, \end{array}\right. } \end{aligned}$$\end{document}where b(x) is a smooth bounded function in Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^{n}$$\end{document} with n≥5\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 5$$\end{document} and the initial datum u0\documentclass[12pt]{minimal}
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\begin{document}$$u_0$$\end{document} is positive and smooth. Let qj∈Rn,j=1,…,k\documentclass[12pt]{minimal}
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\begin{document}$$q_j \in \mathbb {R}^n,j=1,\ldots ,k$$\end{document}, be distinct nondegenerate local minimum points of b(x). Assume that an eigenvalue condition (1.6) is satisfied. We prove the existence of a positive smooth solution u(x, t) which blows up at infinite time near those points with the form u(x,t)≈∑j=1kαnμj(t)μj(t)2+|x-ξj(t)|2n-22,ast→+∞.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u(x,t) \approx \sum _{j=1}^k \alpha _n \left( \frac{ \mu _j(t)}{ \mu _j(t)^2 \,+\, |x-\xi _j(t)|^2 } \right) ^{\frac{n-2}{2}}, \quad \text{ as } t\rightarrow +\infty . \end{aligned}$$\end{document}Here ξj(t)→qj\documentclass[12pt]{minimal}
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\begin{document}$$\xi _j(t) \rightarrow q_j$$\end{document} and 0<μj(t)→0\documentclass[12pt]{minimal}
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\begin{document}$$0<\mu _j(t)\rightarrow 0$$\end{document} exponentially as t→+∞\documentclass[12pt]{minimal}
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\begin{document}$$t\rightarrow +\infty $$\end{document}.
