Global Existence and Gradient Estimates for the Quasilinear Parabolic Equations of m-Laplacian Type with a Nonlinear Convection Term

Abstract

In this paper, we derive precise estimates for {u(t) including smoothing effects near t=0 and decay as t Ä as well as global existence of the solutions u(t) to the initial-boundary value problem in a bounded domain in R n for the quasilinear parabolic equation of the m Laplacian type with a nonlinear convection term b(u) {u. For the initial data u 0 we only assume u 0 # L q (0), 1 q< . 2000Academic Press

L ∞ Estimates of solution for m-Laplacian parabolic equation with a nonlocal term

“…As the proof of Proposition 2, we can show that there exist a bounded sequence {y n } and a convergent sequence {z n } such that (4.6) ∇u(t) pn y n t −zn , 0 < t T, for which z n → µ = (2(1 + 2λ(α + β)) + N 2 )/(2m + (m − 2)N 2 ), see [11]. Then the estimate (2.6) is obtained from (4.6) as n → ∞.…”

“…Noticing that the estimate constants C 1 , C p in (3.2)-(3.5) are independent of i, we can obtain the desired solution u(t) as the limit of {u i }(or a subsequence) by the standard compactness argument in [10,11]. The solution u(t) of (1.1) also satisfies (3.2)-(3.5) and (2.3)-(2.5).…”

“…For L ∞ estimates, we use Moser's technique as in [2]- [4], [11]- [13]. To obtain an estimate of ∇u(t) ∞ , we also make the assumption that the mean curvature H(x) of ∂Ω at x is non-positive with respect to the outward normal; such assumption is made also in [2], [7].…”

L∞ estimates of solutions for the quasilinear parabolic equation with nonlinear gradient term and L1 data

“…We remark that the methods used in our article are different from that of [1]. In L ∞ estimates, we use an improved Morser's technique as in [8][9][10]. Since the equation in (1.1) contains the nonlinear gradient term u|u| a-2 |∇u| p and u|u| b-2 |∇u| q , it is difficult to derive L ∞ estimates for u(t) and ∇u(t).…”

On Global Attractors for a Nonlinear Parabolic Equation of m-Laplacian Type in RN

“…For the proof we use again Moser's technique (cf. [1,11,12,13]). Indeed similar estimates are derived in [11] for the problem in bounded domains and here we modify the argument there.…”