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 R N (N. It shows that when the nonlinearity g(u) is of supercritical growth p, with N +2 N −2 ≡ p * < p < p * * ≡ 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 R N rather than approximating physical space R N by a sequence of balls Ω R as usual, we break through the longstanding existed restriction on this topic for p : 1 p p * .

A sufficient condition for global existence of the solution to nonlinear damped wave equations at arbitrary positive initial energy

“…However, the global existence of the solution to problem (1.1)–(1.3) with $$E(0)&gt;d$$ has not been solved for a long time. As we know, the global existence of the solution is the foundation for further study of the long time dynamics of the solution, and global solutions obtained from arbitrary large initial data can extensively service for the theoretical study of the dynamical behavior such as the asymptotic and stability of solutions to the damped wave equation, see [5, 15]. Unexpectedly, up to now, only a result on the global existence of solutions to the Cauchy problem for nonlinear weakly damped wave equations at arbitrary initial energy has been mentioned in [9], but the conclusion stated in [9] restricts the nonlinear damping exponent m and the nonlinear source exponent p to satisfy $$p\le \min \lbrace m,2(n-1)(n-2)^{-1}\rbrace$$.…”

Forward dynamics of strongly wave equations with asymptotically vanishing damping on $ {\mathbb{R}}^3 $

“…Here and below, B r (x 0 ) denotes the ball centered at x 0 with radius r in R n . Clearly, λ 1 depends only on the dimension n. It follows from (10) that…”

“…to guarantee the dissipative bound of (1), see e.g. [10,12,26,30]. The condition (2) can be weaken to…”