Abstract
A global attractor in $L^2$ is shown for weakly dissipative $p$-Laplace equations on the entire Euclid space, where the weak dissipativeness means that the order of the source is lesser than $p-1$. Half-time decomposition and induction techniques are utilized to present the tail estimate outside a ball. It is also proved that the equations in both strongly and weakly dissipative cases possess an $(L^2,L^r)$-attractor for $r$ belonging to a special interval, which contains the critical exponent $p$. The obtained attractor is proved to be approximated by the corresponding attractor inside a ball in the sense of upper strictly and lower semicontinuity.
Highlights
This paper is concerned with the dynamical behavior of the following p-Laplace equation ut
The absorption can reach the level of p-norm. We use this absorption together with an half-time decomposition to derive that the p-Laplace system is asymptotically small outside a large ball and obtain a global attractor in the weakly dissipative case
We prove the upper semi-continuity (93)
Summary
This paper is concerned with the dynamical behavior of the following p-Laplace equation ut. If the equation is strongly dissipative (q ≥ p), this absorption reaches automatically the level of p-norm by an interpolation This is the reason why the literatures mentioned above assumed the strong dissipativeness. If the equation is weakly dissipative (q < p), the interpolation lose its effectiveness To overcome this difficulty, we define an increasing sequence qm = m(q − 2) + 2 and prove the absorption under qN -norm by an induction argument, where N is given in (3) and it is just the minimal integer such that qN ≥ p. The absorption can reach the level of p-norm We use this absorption together with an half-time decomposition to derive that the p-Laplace system is asymptotically small outside a large ball and obtain a global attractor in the weakly dissipative case. |uxi |p−2uxi vxi for u, v ∈ W 1,p
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