摘要
The paper studies an evolutionary p(x)-Laplacian equation with a convection term ut=div(ρα|■u|p(x)-2■u)+∑N i=1■bi(u)/■xi,whereρ(x)=dist(x,■Ω),ess inf p(x)=p^->2.To assure the well-posedness of the solutions,the paper shows only a part of the boundary,Σp■■Ω,on which we can impose the boundary value.Σp is determined by the convection term,in particular,when 1<α<(p^--2)/2,Σp={x∈■Ω:bi′(0)ni(x)<0}.So,there is an essential difference between the equation and the usual evolutionary p-Laplacian equation.At last,the existence and the stability of weak solutions are proved under the additional conditionsα<(p^--2)/2 andΣp=■Ω.
The paper studies an evolutionary p(x)-Laplacian equation with a convection term ■,where ρ(x) = dist(x, ??), ess inf p(x) = p^-> 2. To assure the well-posedness of the solutions, the paper shows only a part of the boundary, Σ_p ? ??, on which we can impose the boundary value. Σ_p is determined by the convection term, in particular, when 1 < α <(p^--2)/2, Σ_p = {x ∈ ?? : b_i′(0)n_i(x) < 0}. So, there is an essential difference between the equation and the usual evolutionary p-Laplacian equation. At last, the existence and the stability of weak solutions are proved under the additional conditions α <(p^--2)/2 and Σ_p = ??.
基金
Supported by the National Natural Science Foundation of China(No.2015J01592,No.2019J01858)
