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Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $

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    * Corresponding author
The research is supported by National Natural Science Foundation of China (Grant No. 11671367) and the Doctor Foundation of Henan University of Technology, China (No. 2019BS041). The first author is supported by the Doctor Foundation of Henan University of Technology, China (Grant No. 2019BS041). The second author is supported by NSFC (Grant No. 11671367)
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  • The paper investigates the well-posedness and the existence of global attractor for a strongly damped wave equation on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u_{t}-\Delta u+u_{t}+u+g(u) = f(x) $. It shows that when the nonlinearity $ g(u) $ is of supercritical growth $ p $, with $ \frac{N+2}{N-2}\equiv p^*< p< p^{**} \equiv\frac{N+4}{(N-4)^+} $, (i) the initial value problem of the equation is well-posed and its weak solution possesses additionally partial regularity as $ t>0 $; (ii) the related solution semigroup has a global attractor in natural energy space. By using a new double truncation method on frequency space $ \mathbb{R}^N $ rather than approximating physical space $ \mathbb{R}^N $ by a sequence of balls $ \Omega_R $ as usual, we break through the longstanding existed restriction on this topic for $ p: 1\leqslant p\leqslant p^* $.

    Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35B33, 35B65.


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