Abstract
We consider the Dirichlet initial boundary value problem ∂tum(x)-div∇upx,t-2∇u=ax,tuq(x,t), where the exponents p(x,t)>1, q(x,t)>0, and m(x)>0 are given functions. We assume that a(x,t) is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if esssupp(x,t)-1<essinfm(x), then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when esssupm(x)<essinfp(x,t)-1, we get the positivity of solutions for large t. In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that esssupm(x)<essinfp(x,t)-1 and either q(x,t)=m(x) or a(x,t)≤0 a.e. Finally, we give a uniform localization of the support of solutions for all t>0, in the case where a(x,t)<a1<0 a.e. and esssupqx,t<essinfp(x,t)-1.
Highlights
This paper is devoted to studying qualitative properties of nonnegative weak solutions for the following doubly nonlinear parabolic problem with variable exponents{{{{ ∂b (x, ∂t u) − Δ p(x,t)u = a (x, t)|u|q(x,t)−1 u in QT = Ω × (0, T),P : {{{{u = 0 on ∂Ω × (0, T), (1){b (x, u (x, 0)) = b (x, u0 (x))in Ω, where Ω is a bounded domain of RN, N ≥ 1, with smooth boundary ∂Ω, b(x, u) = |u|m(x)−1u and Δ p(x,t)u is defined as
We prove that if ess sup p(x, t) − 1 < ess inf m(x), any weak solution will be extinct in finite time when the initial data is small enough
For any p(x) ∈ C+(Ω), we introduce the variable exponent Lebesgue space as follows: Lp(x) (Ω) = {u : Ω
Summary
The phenomenon of finite speed of propagation was investigated by Kalashnikov in [15] He considered, for N = 1, the equation ∂b(u)/dt − Δu = 0 in R × (0, ∞) and, under specific conditions, proved that if the initial condition u0 has a compact support, the condition ∫0+ (1/b(s))ds < +∞ is necessary and sufficient for solutions to have compact support. Under certain regularity hypotheses on m(x), p(x, t), and under the sign condition a(x, t) ≤ 0 a.e, they studied properties of finite speed of propagation and extinction in finite time in [9, 10] Their results were established by using the local energy method.
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