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

  • Chunhua Wang
  • Juncheng Wei
  • Suting Wei
  • Yifu ZhouEmail author
Article
  • 153 Downloads

Abstract

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 

Notes

Acknowledgements

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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