Remark on the Cauchy problem for the evolution p-Laplacian equation

Abstract In this paper, we prove that the semigroup S ( t ) $S(t)$ generated by the Cauchy problem of the evolution p-Laplacian equation ∂ u ∂ t − div ( | ∇ u | p − 2 ∇ u ) = 0 $\frac{\partial u}{\partial t}-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0$ ( p > 2 $p>2$ ) is continuous form a weighted L ∞ $L^{\infty}$ space to the continuous space C 0 ( R N ) $C_{0}(\mathbb{R}^{N})$ . Then we use this property to reveal the fact that the evolution p-Laplacian equation generates a chaotic dynamical system on some compact subsets of C 0 ( R N ) $C_{0}(\mathbb{R}^{N})$ . For this purpose, we need to establish the propagation estimates and the space-time decay estimates for the solutions first

Complicated asymptotic behavior exponents for solutions of the evolution p-Laplacian equation with absorption

Abstract In this paper, we investigate how the initial value belonging to spaces W σ ( R N ) $W_{\sigma}(\mathbb{R}^{N})$ ( 0 < σ < N $0<\sigma<N$ ) affects the complicated asymptotic behavior of solutions for the Cauchy problem of the evolution p-Laplacian equation with absorption. In fact, we reveal the fact that σ = p q − p + 1 $\sigma=\frac{p}{q-p+1}$ is the critical exponent for the complicated asymptotic behavior of the solutions