The weak solutions of a doubly nonlinear parabolic equation related to the $p(x)$-Laplacian

Abstract

A nonlinear degenerate parabolic equation related to the p(x)-Laplacian u t = div(b(x) ∇a(u) p(x)-2 ∇a(u)) + N i=1 ∂b i (u) ∂x i + c(x, t)-b 0 a(u) is considered in this paper, where b(x)| x∈Ω > 0, b(x)| x∈∂Ω = 0, a(s) ≥ 0 is a strictly increasing function with a(0) = 0, c(x, t) ≥ 0 and b 0 > 0. If Ω b(x)-1 p-1 dx ≤ c and | N i=1 b i (s)| ≤ ca (s), then the solutions of the initial-boundary value problem is well-posedness. When Ω b(x)-(p(x)-1) dx < ∞, without the boundary value condition, the stability of weak solutions can be proved.

Positive solutions of a nonlinear parabolic equation with double variable exponents

“…In addition, we are able to prove (2.2) as in [19,21]. Thus, we have proved u(x, t) is the weak solution of equation (1.1) with the initial value (1.2) in the sense of Definition 2.1.…”

“…were explored in [18,19]. In comparison with equations (2.7)(2.8), the essential different characteristic of equation (1.1) is that the variable exponents α(x, t) and p(x, t) both depend on the time variable t. The interactions between α(x, t) and p(x, t) in equation (1.1) bring some essential difficulties.…”

“…In this paper, we assume that for any t ∈ [0, T ], a(x, t) satisfies (1.9), and try to establish the stability of the weak solutions of equation (1.1) with the initial value condition (1.2), but without the boundary value condition (1.3). Since the existence of weak solutions can be proved in a similar way as [19], the main aim of this paper is to study the stability of weak solutions.…”