The balance between nonlinear inwards and outwards boundary flux for a nonlinear heat equation

[摘要] We study nonnegative solutions of {u(t) = (u(m))(xx), (x,t) is an element of (0,L) X (0,T), {-(u(m))(x)(0,t) = u(p)(0,t), t is an element of (0, T), {(u(m))(x)(L,t) = -lambdau(q)(L,t), t is an element of (0,T), {u(x,0) = u(0)(x), x is an element of (0,L), where m, p, q, lambda and L are positive parameters. For different values of the parameters three situations may occur: (1) all solutions of this problem exist for all t > 0; (2) for certain initial data functions the solution exists for all t > 0 while for others the solution blows up as t NE arrow T for some finite T; (3) excepting the trivial solution when u(0) equivalent to 0, all solutions blow up as t NE arrow T for some finite T. We identify in terms of the parameters which of them actually happens. For solutions which blow up we find the blow-up rate and the blow-up set. (C) 2002 Elsevier Science (USA).

[发布日期] 2002-09-01