On The Stability Of The Positive Radical Steady States For A Nonhomogeneous Semilinear Cauchy Problem

In this paper, we consider the stability of the positive radical steady states, which are positive solutions ofof the following Cauchy problem:where initial function 0 is a bounded non-negative continuous function in Rn, and K(r) > Crl for some l > -2 and r large. The norm of is defined as:where A is a real number.For the physical reasons, we consider the regular positive radical solutions of (0.1) that have limits at r = 0, when K(x) = K(r), f(x) = f(r), where r =| x |, this leads us to consider the initial problemFor the homogeneous equation, when K(x) = 1, Cui, Li and Wang obtained that the positive radical steady states of (0.2) are stable with respect to the norm || . ||m+1 and weakly asymptotically stable with respect to the norm || . ||m+2. Deng, Li and Liu extended the result to a more general class of K(x), With the topology introduced in (0.3), we prove the stability and asympototic stability of the steady states of (0.2). Since K(x) is not homogeneous and f(x) 0, hence steady states can not be obtained by scalling. To overcome this, we construct super and subsolutions in a different way and give some very delicate estimates on them.In this paper, the monotone property with respect to a of the positive solutions for nonhomogeneous equation (0.4) is first obtained by O.D.E theory and stated as:for some K(x) and f(x), when p > pC, any two solutions can not intersect each otherand when, any two solutions will intersect infinity many times.And then the main result is obtained by barrier method, comparative principle and maximum principle, which is: when p > pc, the positive radical steady states of (0.2) are stable with respect to the norm and weakly asymptotically stable with respect to the norm .