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

Abstract

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^* $.