The Cauchy problem in Rn of (i) ut = Δuα+1+uβ and (ii) with non-negative initial value is studied. Here α>0, β>1. It is proved that for (i) with β = α+2/n being the critical case every non-trivial solutio...The Cauchy problem in Rn of (i) ut = Δuα+1+uβ and (ii) with non-negative initial value is studied. Here α>0, β>1. It is proved that for (i) with β = α+2/n being the critical case every non-trivial solution blows up in finite time. A similar result is proved for (ii).展开更多
The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject T...The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject The principles derived may be used to deduce bounds on the gradient q.展开更多
In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth ord...In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth order of B(x, u, ζ) with respect to ζ is supercritical. Many aspects can be solved satisfactorily.展开更多
文摘The Cauchy problem in Rn of (i) ut = Δuα+1+uβ and (ii) with non-negative initial value is studied. Here α>0, β>1. It is proved that for (i) with β = α+2/n being the critical case every non-trivial solution blows up in finite time. A similar result is proved for (ii).
基金Supported by the National Natural Science Foundation of China （No.60174007） and PNSFS.
文摘The Hopf＇s maximum principles are utilized to obtain maximum principles for functions defined on solutions of nonlinear elliptic equations in divergence form （g（u）u,i）,i ＋f（x,u,q）=0（q=｜△↓u｜^2）, subject The principles derived may be used to deduce bounds on the gradient q.
基金supported by the National Natural Science Foundation of China under Grant Nos.11401096 and 11326123the Natural Science Foundation of Guangdong Province under Grant Nos.S2013010014485,2014A030313619 and S2013010012463+2 种基金Special fund of the Guangdong College discipline construction under Grant Nos.2013KJCX0189 and 2013B020314020Guangdong provincial major school projects:2014KZDXM063Guangdong Provincial Innovation Project features 2014KTSCX150
文摘In this paper, we introduce quasilinear elliptic equations in divergent form. It is permitted that the growth orders of A(x, u, ζ) and B(x, u, ζ) with respect to u arrive at the critical exponents and the growth order of B(x, u, ζ) with respect to ζ is supercritical. Many aspects can be solved satisfactorily.
