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Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

  • * Corresponding author: Xingchang Wang

    * Corresponding author: Xingchang Wang 
The first author is supported by the Ph.D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (3072021CF2404), the second author is supported by NSFC grant 11871017
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  • We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.

    Mathematics Subject Classification: Primary: 35L05, 35A01; Secondary: 35B45.


    \begin{equation} \\ \end{equation}
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  • Table 1.  The values for $ m $ and $ m' $

    The value Results
    $ n $ $ \frac{2n}{n-2} $ $ m\in\left(n,\frac{2n}{n-2}\right) $ $ m'\in\left(n,\frac{2n}{n-2}\right) $ $ \frac{1}{m}+\frac{1}{m'}=\frac{1}{2} $
    $ \le 2 $ $ \infty $ $ (n, \infty) $ $ (n,\infty) $ $ (m,m')=(3,6) $ valid
    $ 3 $ $ 6 $ $ (3,6) $ $ (3,6) $ $ (m,m')=\left(5,\frac{10}{3}\right) $ valid
    $ 4 $ $ 4 $ $ - $ $ - $ $ - $ invalid
    $ 5 $ $ \frac{10}{3} $ $ - $ $ - $ $ - $ invalid
    $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ invalid
     | Show Table
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