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 solutio...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.展开更多
基金Supported by the National Natural Science Foundation of China(No.2015J01592,No.2019J01858)
文摘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.
