拟线性抛物型偏微分方程组解的振动性

Oscillation for Solutions of Systems of Quasilinear Parabolic Partial Differential Equations

关键词:  微分方程  偏微分方程  拟线性  振动性

Key words:  differential equation  partial differential equation  quasilinear  oscillation

摘要:

利用Green定理和微分不等式,研究一类拟线性抛物型偏微分方程组:(∂u_i(x,t)/∂t)=a_i(t)Δu_i(x,t)+∑_k=1^s a_ik(t)Δu_i(x,d_k(t))-p_i(x,t)u_i(x,t)-∑_j=1^m f_ij[t,x,u_j(x,^e(t))],i=1,2,…,m解的振动性,获得该类方程组在两类不同边值条件:(∂u_i(x,t)/∂N)+g_i(x,t)u_i(x,t)=0,(x,t)∈∂Ω×R₊,i=1,2,…,m和u_i(x,t)=0,(x,t)∈∂K×R₊,i=1,2,…,m所有解振动的若干充分条件:lim_t→∞ inf∫_e(t)^t q(s) exp∫_e(s)^s p(r) drds > 1/e.

Abstract:

The oscillation of solutions of the systems of a class of quasilinear parabolic partial differential equations:(∂u_i(x,t)/∂t)=a_i(t)Δu_i(x,t)+∑_k=1^s a_ik(t)Δu_i(x,d_k(t))-p_i(x,t)u_i(x,t)-∑_j=1^m f_ij[t,x,u_j(x,^e(t))],i=1,2,…,m is studied by Green's theorem and differential inequalities.The sufficient conditions:lim_t→∞ inf∫_e(t)^t q(s) exp∫_e(s)^s p(r) drds > 1/e.for the oscillation of all solutions of the systems are obtained under two kinds of different boundary conditions:(∂u_i(x,t)/∂N)+g_i(x,t)u_i(x,t)=0,(x,t)∈∂Ω×R₊,i=1,2,…,m and u_i(x,t)=0,(x,t)∈∂K×R₊,i=1,2,…,m.