Infinite time blow-up for critical heat equation with drift terms

  • Chunhua Wang
  • Juncheng Wei
  • Suting Wei
  • Yifu ZhouEmail author


We construct infinite time blow-up solution to the following heat equation with Sobolev critical exponent and drift terms
$$\begin{aligned} {\left\{ \begin{array}{ll} u_t \,=\, \Delta u\,+\,\nabla b (x) \cdot \nabla u\,+\, u^{\frac{n+2}{n-2}} ~ \text{ in } ~ \mathbb {R}^n\times (0,+\infty ),\\ u(\cdot ,0)=u_0 ~ \text{ in } ~ \mathbb {R}^n, \end{array}\right. } \end{aligned}$$
where b(x) is a smooth bounded function in \(\mathbb {R}^{n}\) with \(n\ge 5\) and the initial datum \(u_0\) is positive and smooth. Let \(q_j \in \mathbb {R}^n,j=1,\ldots ,k\), 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(xt) which blows up at infinite time near those points with the form
$$\begin{aligned} u(x,t) \approx \sum _{j=1}^k \alpha _n \left( \frac{ \mu _j(t)}{ \mu _j(t)^2 \,+\, |x-\xi _j(t)|^2 } \right) ^{\frac{n-2}{2}}, \quad \text{ as } t\rightarrow +\infty . \end{aligned}$$
Here \(\xi _j(t) \rightarrow q_j\) and \(0<\mu _j(t)\rightarrow 0\) exponentially as \(t\rightarrow +\infty \).

Mathematics Subject Classification

Primary 35K58 Secondary 35B40 



C. Wang is partially supported by NSFC with Nos. 11671162 and CCNU18CXTD04. J. Wei is partially supported by NSERC of Canada.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Chunhua Wang
    • 1
  • Juncheng Wei
    • 2
  • Suting Wei
    • 3
  • Yifu Zhou
    • 2
    Email author
  1. 1.School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Department of MathematicsSouth China Agricultural UniversityGuangzhouPeople’s Republic of China

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